Small Amplitude Periodic Orbits in Three-Dimensional Quadratic Vector Fields with a Zero-Hopf Singularity

Abstract

We consider some families of three-dimensional quadratic vector fields having a fixed zero-Hopf equilibrium. We are interested in the bifurcation of periodic $\nu$-orbits from the singularity, that is, those small amplitude orbits that make a fixed arbitrary number $\nu$ of revolutions about a rotation axis and then returns to the initial point closing the orbit. When the parameters of the family are restricted to certain explicitly computable open semi-algebraic sets $\Lambda$, we characterize those parameters that give rise to the appearance of local two-dimensional periodic invariant manifolds through the singularity. Also we use a Bautin-type analysis to study the maximum number of small-amplitude $\nu$-limit cycles that can be made to bifurcate from the equilibrium when the parameters of the family are restricted to $\Lambda$. We obtain global upper bounds on the number of bifurcated $\nu$-limit cycles.

1 Introduction and Main Results

In this work, we consider polynomial families of quadratic differential systems

$$\begin{aligned} \begin{array}{ll} {\dot{x}}=-y+ F_1(x,y,z; \lambda ) \\ {\dot{y}}=\ \ x+ F_2(x,y,z; \lambda ) \\ {\dot{z}} = F_3(x,y,z; \lambda ), \end{array} \end{aligned}$$

(1)

parameterized by the admissible coefficients $\lambda \in {\mathbb {R}}^p$ (with $p \le 18$) where $F_j$ are homogeneous polynomials in the variables (x, y, z) of degree 2. We will denote by ${\mathcal {X}}_\lambda = (-y+ F_1(x,y,z; \lambda )) \partial _x + (x+ F_2(x,y,z; \lambda )) \partial _y + F_3(x,y,z; \lambda ) \partial _z$ the associated vector field to (1).

With this structure, the eigenvalues of the linear part of (1) at the origin are $\{ \pm i, 0 \}$ with $i^2 = -1$ and the origin is called a zero-Hopf (or fold-Hopf) singularity of system (1). The reader can consult for example the books [12, 13] where zero-Hopf bifurcations in some two-parameter unfolding is studied.

The periodic orbits of (1) near the origin can be isolated (limit cycles) or not. The origin of system (1) is called a three-dimensional center if a neighborhood in ${\mathbb {R}}^3$ of it is foliated by periodic orbits of (1), including an invariant singular curve passing through the origin. In [9] it is shown that three-dimensional centers are characterized by its completely analytically integrable behaviour, that is, the existence of two functionally independent local analytic first integrals. With less integrability still continua of periodic orbits can appear giving rise to local two-dimensional periodic invariant manifolds ${\mathcal {M}} \subset {\mathbb {R}}^3$ through the origin of (1).

The linear part of system (1) generates a rotation, hence the extension of the classical Poincaré-Lyapunov center problem and the Hopf bifurcation for planar vector fields to the 3-dimensional zero-Hopf singularity is quite natural. To the best of my knowledge, the most general approach to the study of the center problem and the bifurcation of small-amplitude limit cycles at zero-Hopf equilibria is given in the recent work [10], where an extension to (1) of the seminal Bautin ideas [1] for planar vector fields is given, see the textbooks [14, 15]. The main goal of this work is to apply the techniques developed in [10] to some quadratic families (1). It is worth to recall here that other approaches like the classical averaging procedures at first order to count the number of periodic fails in this framework as it is explained in [10].

As usual in three-dimensional vector fields, there may occur infinitely many limit cycles if we do not assume some additional constrains. This is the main reason why we restrict our search to the periodic orbits of (1) characterized by the number $\nu \in {\mathbb {Z}}^+$ of turns about the z-axis. Following [10], a periodic orbit of (1) that returns to the starting point after $\nu$ (or a divisor of $\nu$) turns around the z-axis will be called periodic $\nu$-orbit or $\nu$-limit cycle if it is isolated inside the set of periodic $\nu$-orbits.

In particular, we will only consider families (1) having the z-axis as rotation axis, that is, $F_j(0,0,z; \lambda ) = 0$ for $j=1,2$. Moreover, due to technical reason that we will explain later, we also restrict family (1) by the non-degenerate condition $F_3(0, 0, z; \lambda ) \not \equiv 0$, meaning geometrically that the z-axis is non-singular. Even with all these constrains, the full quadratic family (1) escapes from a full analysis and therefore we only study some particular cases.

With the technique at hand it is not possible to control all the possible bifurcations in the full parameter space ${\mathbb {R}}^p$. Instead we will restrict the parameters of (1) to lie in some open semi-algebraic sets called $\Lambda _{\nu }^{\mathbf{N}(\Delta )} \subset {\mathbb {R}}^p$, see (12) for its explicit definition. We will analyze the maximum number of $\nu$-limit cycles that can bifurcate from the origin when the family of vector fields ${\mathcal {X}}_\lambda$ is restricted to the parameters space $\Lambda _{\nu }^{\mathbf{N}(\Delta )}$. This number is called the $\nu$-cyclicity of the origin and is denoted by $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )})$. Notice that, although $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )})$ is bounded for any $\nu \in {\mathbb {Z}}^+$, the maximum number $\mathrm{Cyc}({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )})$ of limit cycles bifurcating from the origin of (1) with parameters in $\Lambda _{\nu }^{\mathbf{N}(\Delta )}$ can be unbounded since $\nu$ is too.

We summarize the main results of this work. The following three theorems have the same structure:

• (a) First we show the structure of the small periodic orbits near the origin of the considered family characterizing under what parameter restrictions ($V_{\mathcal {C}} \cap \Lambda$ with $V_{\mathcal {C}} \subset \Lambda$ a center variety) there is a local two-dimensional periodic invariant manifold ${\mathcal {M}}$ through the origin.

• (b) Next we give $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \backslash V_{\mathcal {C}})$, the $\nu$-cyclicity of the origin in the family when we take perturbations from parameters not belonging to the center variety, that is restricting the family to $\Lambda \backslash V_{\mathcal {C}}$.

• (c) Finally we deal with $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \cap V_{\mathcal {C}})$, the $\nu$-cyclicity of the origin when the parameters lie in $\Lambda \cap V_{\mathcal {C}}$ and we produce either lower bounds on this $\nu$-cyclicity and sometimes the $\nu$-cyclicity itself.

First we analyze family

$$\begin{aligned} {\dot{x}}=-y + x (a_2 y + a_3 z), \ \ {\dot{y}}= x + b_2 y^2, \ \ {\dot{z}} = z (c_1 x + c_3 z), \end{aligned}$$

(2)

having the invariant plane $\{z=0\}$ through the origin with parameter space

$$\begin{aligned} \Lambda = \{ \lambda = (a_2, a_3, b_2, c_1, c_3) \in {\mathbb {R}}^{5} : c_3 b_2 c_1 \ne 0 \}. \end{aligned}$$

(3)

Theorem 1

We consider the family of vector fields ${\mathcal {X}}_\lambda$ associated to (2) with parameter space (3). Then the following holds:

• (a) There is a local two-dimensional periodic invariant manifold ${\mathcal {M}} \ne \{z=0\}$ of (2) through the origin if and only if the parameters $\lambda$ lie in $V_{\mathcal {C}} \cap \Lambda$, being the variety $V_{\mathcal {C}} = \{ \lambda \in {\mathbb {R}}^5 : a_3 = 0 \}$. Family (2) on $V_{\mathcal {C}} \cap \Lambda$ only has two periodic invariant manifolds near the origin which are ${\mathcal {M}}$ and $\{z=0\}$, and both are foliated by periodic 1-orbits.

• (b) For any $\nu \in {\mathbb {Z}}^+$, $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \backslash V_{\mathcal {C}}) = 0$.

• (c) For any $\nu \in {\mathbb {Z}}^+$, $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \cap V_{\mathcal {C}}) = 0$.

The analogous result but now for family

$$\begin{aligned} {\dot{x}}=-y + x f(x,y,z), \ \ {\dot{y}}= x + y f(x,y,z), \ \ {\dot{z}} = z (c_1 x + c_2 y+ c_3 z), \end{aligned}$$

(4)

with $f(x,y,z) = a_1 x + a_2 y + a_3 z$ and parameter space

$$\begin{aligned} \Lambda = \{ \lambda = (a_1, a_2, a_3, c_1, c_2, c_3) \in {\mathbb {R}}^{6} : c_3(a_2 c_1 - a_1 c_2) \ne 0 \}. \end{aligned}$$

(5)

is the following one.

Theorem 2

Let ${\mathcal {X}}_\lambda$ denote the family of vector fields associated to (4) with parameter space (5). Then the next statements holds:

• (a) There is a local two-dimensional periodic invariant manifold ${\mathcal {M}} \ne \{z=0\}$ of (4) through the origin if and only if the parameters $\lambda$ lie in $V_{\mathcal {C}} \cap \Lambda$, being the variety $V_{\mathcal {C}} = \{ \lambda \in {\mathbb {R}}^5 : a_3 = 0 \}$. Family (4) on $V_{\mathcal {C}} \cap \Lambda$ only has two periodic invariant manifolds near the origin which are ${\mathcal {M}}$ and $\{z=0\}$, and both are foliated by periodic 1-orbits.

• (b) For any $\nu \in {\mathbb {Z}}^+$, $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \backslash V_{\mathcal {C}}) = 0$.

• (c) For any $\nu \in {\mathbb {Z}}^+$, $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \cap V_{\mathcal {C}}) = 0$.

A straightforward consequence of the statements (b) and (c) of Theorems 1 and 2 is the following corollary.

Corollary 3

The family of vector fields ${\mathcal {X}}_\lambda$ associated to either family (2) with parameter space $\Lambda$ given by (3) or to family (4) with parameters $\Lambda$ in (5) has $\mathrm{Cyc}({\mathcal {X}}_\lambda , 0, \Lambda ) = 0$.

The last example that we analyze is the Liénard-like family

$$\begin{aligned} \begin{array}{lll} {\dot{x}} &{}=&{} -y, \\ {\dot{y}} &{}=&{} f(x,z; \lambda ) + y g(x,z; \lambda ) \\ {\dot{z}} &{}=&{} F(x,y,z; \lambda ) \end{array} \end{aligned}$$

(6)

where $f(x, z; \lambda ) = x + a_0 x^2 + a_1 x z$, $g(x, z; \lambda ) = b_0 x + b_1 z$ and $F(x, y, z; \lambda ) = c_0 x^2 + c_1 y^2 + c_2 z^2$ with parameter space

$$\begin{aligned} \Lambda = \{ \lambda = (a_0, a_1, b_0, b_1, c_0, c_1, c_2) \in {\mathbb {R}}^{7} : c_2(c_0+ c_1) \ne 0 \}. \end{aligned}$$

(7)

Theorem 4

Let ${\mathcal {X}}_\lambda$ be the family of vector fields associated to (6) with parameter space (7). Then the following holds:

• (a) If $c_2 (c_0+c_1) > 0$ then there is no small amplitude periodic orbit while when $c_2 (c_0+c_1) < 0$ there is a local two-dimensional periodic invariant manifold ${\mathcal {M}}$ of (6) through the origin if and only if the parameters are on the semi-algebraic set $V_{\mathcal {C}} \cap \Lambda$ being the variety $V_{\mathcal {C}} = \{ \lambda \in {\mathbb {R}}^7 : b_1 = a_1 = a_0 b_0 = 0 \}$. Moreover, these manifolds appear in pairs and are foliated by periodic 1-orbits.

• (b) For any $\nu \in {\mathbb {Z}}^+$, $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \backslash V_{\mathcal {C}}) = 8$.

• (c) For any $\nu \in {\mathbb {Z}}^+$, $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \cap V_{\mathcal {C}}) \ge 2$.

The work is structured as follows. In Sect. 2 we present a summary of the technique developed in [10] to compute or to bound $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )})$ by using a Bautin-type method. This is the main tool to prove the results of this paper. In the next Sect. 3 we prove Theorems 1, 2 and 4 . We also include, in Sect. 3.5 a challenging quadratic family (28) to explain technical difficulties. Finally in Sect. 4 we give short appendix with some computations in the first degenerate cases.

2 Preliminary Results and Background

The analysis performed in this work is based on the technique developed in [10]. In this section we give a brief summary of this method.

The first step is the construction of a Poincaré $\nu$-map $\Pi _\nu$ such that its fixed points pick up the periodic $\nu$-orbits of (1) in a small neighborhood of the origin. Since the z-axis is invariant for the flow of family (1), the cylindrical coordinates $(x,y,z) \mapsto (\theta , \rho , z)$ with $x = \rho \cos \theta$, $y = \rho \sin \theta$, transform (1) into the analytic system

$$\begin{aligned} \frac{d \rho }{d \theta } = R(\theta , \rho , z; \lambda ) \ , \ \ \frac{d z}{d \theta } = {\mathcal {Z}}(\theta , \rho , z; \lambda ) \end{aligned}$$

(8)

defined on the cylinder $\{(\theta , \rho , z) \in {\mathbb {S}}^1 \times {\mathbb {R}}^2 : \rho \ge 0, |z| \ll 1 \}$ with ${\mathbb {S}}^1 = {\mathbb {R}} / 2 \pi {\mathbb {Z}}$. Letting $\Psi (\theta ; \rho _0, z_0; \lambda ) = (\rho (\theta ; \rho _0, z_0; \lambda ), z(\theta ; \rho _0, z_0; \lambda ))$ be the solution of (8) with initial condition $\Psi (0; \rho _0, z_0; \lambda ) = (\rho _0, z_0) \in {\mathbb {R}}^+ \times {\mathbb {R}}$ being both $\rho _0$ and $|z_0|$ sufficiently small. Then, for any $\nu \in {\mathbb {Z}}^+$, we have $\Pi _\nu (\rho _0, z_0; \lambda ) = \Psi (2 \nu \pi ; \rho _0, z_0; \lambda )$ and the analytic displacement $\nu$ -map $d_\nu (\rho _0, z_0; \lambda ) = (d^{[1]}(\rho _0, z_0; \nu , \lambda ), d^{[2]}(\rho _0, z_0; \nu , \lambda )) = \Pi _\nu (\rho _0, z_0; \lambda ) - (\rho _0, z_0)$ has its zeroes in one-to-one correspondence with the small-amplitude periodic $\nu$-orbits of (1) near the origin. Notice that the origin of system (1) is a three-dimensional center when $d_1(\rho _0, z_0; \lambda ) \equiv 0$.

The displacement $\nu$-map has the following convergent Taylor expansion at $(\rho _0, z_0)=(0,0)$:

$$\begin{aligned} d_\nu (\rho _0, z_0; \lambda ) = \left( \rho _0 \sum _{i+j \ge 1} v_{ij}^{[1]}(\nu , \lambda ) \, \rho _0^i z_0^j, \sum _{i+j \ge 2} v_{ij}^{[2]}(\nu , \lambda ) \, \rho _0^i z_0^j \right) \end{aligned}$$

with $v_{ij}^{[1]}, v_{ij}^{[2]} \in {\mathbb {R}}[\nu , \lambda ]$ and $v_{10}^{[1]} = v_{11}^{[2]} = 0$. The coefficients $v_{ij}^{[1]}(\nu , \lambda )$ and $v_{ij}^{[2]}(\nu , \lambda )$ are called Poincaré-Lyapunov quantities. As a taste, the first Poincaré-Lyapunov quantities are $v_{01}^{[1]}(\nu , \lambda ) = \nu \pi \left( \partial ^2_{xz} F_1(0,0,0; \lambda ) \right. \left. + \partial ^2_{yz} F_2(0,0,0; \lambda ) \right)$, $v_{20}^{[2]}(\nu , \lambda ) = \frac{\nu \pi }{2} \left( \partial ^2_x F_3(0,0,0; \lambda ) + \partial ^2_y F_3(0,0,0; \lambda ) \right)$, and $v_{02}^{[2]}(\nu , \lambda ) = \nu \pi \partial ^2_z F_3(0,0,0; \lambda )$. In particular $v_{ij}^{[1]}(1, \lambda ^c) = v_{ij}^{[2]}(1, \lambda ^c) = 0$ for all admissible $(i,j) \in {\mathbb {N}}^2$ if and only if family (1) with $\lambda = \lambda ^c$ possesses a three-dimensional center at the origin.

The second step of the method is based on a reduction of the dimension of the Poincaré $\nu$-map using branching theory based on Newton’s diagram, see [4, 17]. A branch of solutions of the equation $d^{[2]}(\rho _0, z_0; \nu , \lambda ) = 0$ arises from the trivial solution point $(\rho _0, z_0) = (0,0)$ if there is a continuous function $z^*(\rho _0; \nu , \lambda )$ such that $z^*(0; \nu , \lambda ) = 0$ and $d^{[2]}(\rho _0, z^*(\rho _0; \nu , \lambda ); \nu , \lambda ) \equiv 0$ for any $\rho _0 > 0$ sufficiently small.

Performing the factorization $d^{[2]}(\rho _0, z_0; \nu , \lambda ) = z_0^{m_z} \Delta (\rho _0, z_0; \nu , \lambda )$ with some maximum multiplicity $m_z$ for which $\Delta (\rho _0, z_0; \nu , \lambda ) = \sum _{i, j} v_{ij}^{[2]}(\nu , \lambda ) \rho _0^i z_0^{j-m_z} = \Delta _0(\rho _0; \nu , \lambda ) + \sum _{i \ge 1} \Delta _{i}(\rho _0; \nu , \lambda ) z_0^i$ with $\Delta _0(\rho _0; \nu , \lambda ) \not \equiv 0$ but $\Delta _0(0; \nu , \lambda ) = 0$. If (1) has the invariant plane $\{z=0\}$ then $m_z > 0$. Let $\mathrm{supp}(\Delta ) = \{ (j-m_z, i) \in {\mathbb {N}}^2 : v_{ij}^{[2]}(\lambda ) \ne 0 \}$ be the support of $\Delta$. The Newton diagram of $\Delta$, denoted by ${\mathbf {N}}(\Delta )$ is the boundary of the convex hull of the set $\bigcup _{(j,i) \in \mathrm{supp}(\Delta )} \{ (j,i) + {\mathbb {R}}^2_+\}$ where ${\mathbb {R}}^2_+$ is the positive quadrant. When we look at the local zero-set of $\Delta$ near $(\rho _0, z_0)=(0,0)$ we will be only interested in the different edges or segments of that polygon with negative slopes $-r/s \in {\mathbb {Q}}^-$. The generic case for a fixed $\nu$ is described by parameters $\lambda$ that lie in the set $\Lambda _{\nu }^{\mathbf{N}(\Delta )} = \{ \lambda \in {\mathbb {R}}^p : v_{20}^{[2]}(\nu , \lambda ) \ne 0, v_{02}^{[2]}(\nu , \lambda ) \ne 0 \}$ so that ${\mathbf {N}}(\Delta )$ has only one edge with vertex (0, 2) and (2, 0).

From Newton-Puiseux Theorem, see [2] for instance, and taking into account that the z-axis is a non-singular invariant axis of (1) so that $\Delta (0, z_0; \nu , \lambda ) \ne 0$, it follows that given an analytic $\Delta (\rho _0, z_0; \nu , \lambda )$ with $\Delta (0, 0; \nu , \lambda )=0$ there exists a finite factorization $\Delta (\rho _0, z_0; \nu , \lambda ) = u(\rho _0, z_0; \nu , \lambda ) \prod _{i} (z_0 - z^*_i(\rho _0; \nu , \lambda ))^{\kappa _i}$ where u is an analytic unit $u(0,0; \nu , \lambda ) \ne 0$, the $z^*_i(.; \nu , \lambda )$ are analytic functions of $\rho _0^{1/ n}$ with $z^*_i(0; \nu , \lambda )=0$ called branches, and the exponents n and $\kappa _i$ are some positive integers. Then ${\mathbf {N}}(\Delta )$ has the initial point $(0, {\bar{k}})$ and the terminal point is (k, 0) where both the multiplicity ${\bar{k}} \ge 2$ and the order $k \ge 2$.

The branches emerging from $(\rho _0, z_0) = (0,0)$ are locally expressed as convergent Puiseux series determined by the descending sections of ${\mathbf {N}}(\Delta )$ of the form

$$\begin{aligned} z^*(\rho _0; \nu , \lambda ) = \alpha _0(\nu , \lambda ) \rho _0^{r/s} + o(\rho _0^{r/s}) \end{aligned}$$

(9)

with $\alpha _0 \in {\mathbb {R}} \backslash \{ 0 \}$ and $0 < r/s \in {\mathbb {Q}}$. Associated to each descending segment, the leading coefficients $\alpha _0$ are the nonzero real roots of the determining polynomial $D(.; \nu , \lambda )$ for the particular descending segment which is defined by the first term of the expansion $\Delta (\sigma ^s, \sigma ^r \eta ; \nu , \lambda ) = \sigma ^{m_1 r+n_1 s} \, \eta ^{m_1} \big [ D(\eta ; \nu , \lambda ) + {\mathcal {O}}(\sigma ) \big ]$ where the segment contains the points $(m_i, n_i) \in {\mathbb {N}}^2$ with $i=1, \ldots , p$ ordered such that $m_1< m_2< \cdots < m_p$. The number $n \in {\mathbb {Z}}^+$ depends on the branch, hence on $\lambda$ and $\nu$, and is called the index of the branch. Therefore any branch associated to a segment of slope $-r/s$ and index n can be expanded analytically near $\sigma =0$ as

$$\begin{aligned} z^*(\sigma ^n; \nu , \lambda ) = \sigma ^\frac{n r}{s} \sum _{i \ge 0} \alpha _i(\nu , \lambda ) \sigma ^{i} \end{aligned}$$

(10)

with $0 < \frac{n r}{s} \in {\mathbb {N}}$ and it satisfies

$$\begin{aligned} d^{[2]}(\sigma ^n, z^*(\sigma ^n; \nu , \lambda ); \nu , \lambda ) \equiv 0 \end{aligned}$$

(11)

for all $\sigma > 0$ close to zero.

We define the open subsets $\Lambda _{\nu }^{\mathbf{N}(\Delta )} \subset {\mathbb {R}}^p$ in the parameter space that keeps the structure of ${\mathbf {N}}(\Delta )$ for a fixed $\nu \in {\mathbb {Z}}^+$. In particular there are two subsets $S_i \subset {\mathbb {N}}^2$, $i=1,2$, such that $\Lambda _{\nu }^{\mathbf{N}(\Delta )}$ is a semi-algebraic set of the form

$$\begin{aligned} \Lambda _{\nu }^{\mathbf{N}(\Delta )} = {\mathbf {V}}\left( v_{ij}^{[2]}(\nu , \lambda ) : (i,j) \in S_1\right) \backslash {\mathbf {V}}\left( \prod _{ij} v_{ij}^{[2]}(\nu , \lambda ) : (i,j) \in S_2 \right) , \end{aligned}$$

(12)

where ${\mathbf {V}}({\mathcal {J}})$ denotes the variety of a polynomial ideal ${\mathcal {J}}$, see for example [6].

We also define the analytic reduced displacement map $\delta : \Sigma \times {\mathbb {Z}}^+ \times \Lambda _{\nu }^{\mathbf{N}(\Delta )} \rightarrow {\mathbb {R}}$ associated to a real branch $z^*$ with index n as

$$\begin{aligned} \delta (\sigma ; \nu , \lambda ) = d^{[1]}(\sigma ^n, z^*(\sigma ^n; \nu , \lambda ); \nu , \lambda ) = \sum _{i \ge 1} \delta _i(\nu , \lambda ) \sigma ^{i}. \end{aligned}$$

(13)

Clearly any small isolated positive zero of $\delta (\cdot ; \nu , \lambda )$ corresponds with a small-amplitude $\nu$-limit cycle of (1) near the origin. If there is one $\sigma _o \in \Sigma$ with $\delta (\sigma _0; \nu , \lambda ) = 0$ then $\delta (\sigma _0; k \nu , \lambda ) = 0$ too for any $k \in {\mathbb {Z}}^+$.

We say that a branch (9) is simple if one of the following two cases holds: (A) its leading coefficient $\alpha _0$ is a simple root of D or (B) the leading coefficient $\alpha _0$ is a multiple root of D but after applying the Newton-Puiseux algorithm there appears a coefficient in its Puiseux series expansion that is simple root of its determining polynomial. A simple branch $z^*$ is always real and for simple branches of type (A) one has index $n = s$.

Remark 5

It is useful to work with a real parameter $\alpha _0 \ne 0$ as a free additional parameter besides $\lambda$ by adding the restriction $D(\alpha _0; \nu , \lambda ) = 0$. Then we can make computations with a fixed $\nu$ in a convenient rational ring ${\mathcal {R}}$ in the parameters $(\alpha _0, \lambda )$ that is related to the set $\Lambda _{\nu }^{\mathbf{N}(\Delta )}$. In particular the coefficients $\alpha _i, \delta _i \in {\mathcal {R}}$. In this context we will denote by $\delta (\sigma ; \alpha _0, \nu , \lambda )$ and $\delta _{i}(\alpha _0, \nu , \lambda )$ the expressions of the reduced displacement map and its coefficients although, since $\alpha _0$ depends on $(\nu , \lambda )$ as a root of $D(.; \nu , \lambda )$, when we make this substitution we will explicitly write $\alpha _0(\nu , \lambda )$ and we use by simplicity the short notation $\delta (\sigma ; \nu , \lambda )$, $\delta _{i}(\nu , \lambda )$.

The ring ${\mathcal {R}}$ is Noetherian and therefore any ideal in ${\mathcal {R}}$ will be generated by a finite number of elements by the Hilbert basis theorem (an unknown basis a priori). For a fixed $\nu$ we define the extended parameter space

$$\begin{aligned} E_\nu = \{ (\alpha _0, \lambda ) \in {\mathbb {R}}^{p+1} : \alpha _0 \ne 0, \lambda \in \Lambda _{\nu }^{\mathbf{N}(\Delta )} \}, \end{aligned}$$

(14)

we have that $\delta _i(\alpha _0, \nu , \lambda ) = P_i(\alpha _0, \nu , \lambda )/Q_i(\alpha _0,\nu , \lambda )$, with $P_i, Q_i \in {\mathbb {R}}[\alpha _0, \lambda ]$ and $Q_i|_{E_\nu } \ne 0$. In this context we also define, for each $\nu$, the Bautin ideal

$$\begin{aligned} {\mathcal {B}}^{\{\nu \}} = \langle D(\alpha _0; \nu , \lambda ), \delta _{i}(\alpha _0, \nu , \lambda ) : i \ge 1 \rangle \end{aligned}$$

in the ring ${\mathcal {R}}$ and the center variety

$$\begin{aligned} {\mathbf {V}}({\mathcal {I}}^{\{\nu \}}) \subset {\mathbb {R}}^{p+1} \end{aligned}$$

(15)

associated to the polynomial ideal ${\mathcal {I}}^{\{\nu \}} = \langle D(\alpha _0; \nu , \lambda ), P_{i}(\alpha _0, \nu , \lambda ) : i \ge 1 \rangle$ in the ring ${\mathbb {R}}[\alpha _0, \lambda ]$. In doing computations we also need to define ${{\hat{\delta }}}_{i} \equiv \delta _{i} \mod \mathcal {B}_{i-1}^{\{\nu \}}$ where ${\mathcal {B}}_{i}^{\{\nu \}} = \langle D, \delta _{1}, \ldots , \delta _{i} \rangle$ is an ideal in ${\mathcal {R}}$. That computation means that ${{\hat{\delta }}}_{i}$ denotes the remainder of $\delta _{i}$ upon division by a Gröbner basis of the ideal ${\mathcal {B}}_{i-1}^{\{\nu \}}$ and can be performed. for example, with the functions GroebnerBasis and PolynomialReduce of the algebraic manipulator Mathematica.

Let $B_{jn}^\nu$ be the set of real branches of $\Delta$ with index n associated to the jth descending segment of ${\mathbf {N}}(\Delta )$ for some fixed $\nu$. Clearly the cardinality of the set $B_{jn}^\nu$ depends on $\lambda$ and the reduced displacement map $\delta (\sigma ; \alpha _0, \nu , \lambda )$ on $E_\nu$ has exactly the same expression for all the branches in $B_{jn}^\nu$.

With these notions it follows that system (1) corresponding to $\lambda = \lambda ^* \in \Lambda _{\nu }^{\mathbf{N}(\Delta )}$ with $d^{[1]}(\rho _0, z_0; \nu , \lambda ^*) \not \equiv 0$ has a local two-dimensional invariant periodic manifold ${\mathcal {M}} \subset {\mathbb {R}}^3$ through the origin foliated by $\nu$-orbits if and only if there is a branch in $B_{jn}^\nu$ such that $(\alpha _0, \lambda ^*) \in {\mathbf {V}}({\mathcal {I}}^{\{\nu \}}) \cap E_\nu$ where ${\mathbf {V}}({\mathcal {I}}^{\{\nu \}})$ is the center variety of the set $B_{jn}^\nu$.

The following theorems aims to analyze (either compute or only obtain an upper bound) the $\nu$-cyclicity $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )} )$ for members of the family (1) when its parameters lie in the open subsets $\Lambda _{\nu }^{\mathbf{N}(\Delta )} \subset {\mathbb {R}}^p$.

Theorem 6

([10]) We consider a polynomial family (1) with the z-axis invariant and non-singular and with the parameters restricted to some open set $\Lambda _{\nu }^{\mathbf{N}(\Delta )} \subset {\mathbb {R}}^p$ where the Newton diagram ${\mathbf {N}}(\Delta )$ remains invariant. Make an specific parameter choice $\lambda = \lambda ^\dag \in \Lambda _{\nu }^{\mathbf{N}(\Delta )}$ and assume that $d^{[1]}(\rho _0, z_0; \nu , \lambda ^\dag ) \not \equiv 0$ (equivalently $v_{ab}^{[1]}(\nu , \lambda ^\dag ) \ne 0$ for some (a, b)). Let ${\mathfrak {s}} \ge 0$ be the number of descending segments of ${\mathbf {N}}(\Delta )$ and $B_{jn}^\nu$ the set of real branches of index n associated to the jth segment of ${\mathbf {N}}(\Delta )$. For $\lambda = \lambda ^\dag$, we define the finite cardinal $0 \le {\mathfrak {b}}_{\nu jn}(\lambda ^\dag ) = \# B_{jn}^\nu$. Let $N_\nu (\lambda ^\dag )$ be the maximum number of small-amplitude $\nu$-limit cycles that can be made to bifurcate from the origin in family (1) by perturbing the parameters from $\lambda ^\dag$ and $N_{\nu jn}(\lambda ^\dag )$ the maximum number of them arising from the branches in $B_{jn}^\nu$. Then

$$\begin{aligned} N_\nu (\lambda ^\dag ) \le \sum _{j=1}^{\mathfrak {s}} \sum _{n \in {\mathbb {N}}} N_{\nu jn}(\lambda ^\dag ) < \infty . \end{aligned}$$

Moreover, let $\delta (\sigma ; \alpha _0, \nu , \lambda )$ be the reduced displacement map associated to the set $B_{jn}^\nu$, ${\mathbf {V}}({\mathcal {I}}^{\{\nu \}})$ its center variety (15) and $E_\nu$ the extended parameter space (14). Then the following holds:

• (i) Let $D(\alpha _0; \nu , \lambda ^\dag )=0$ and $(\alpha _0, \lambda ^\dag ) \not \in {\mathbf {V}}({\mathcal {I}}^{\{\nu \}}) \cap E_\nu$, hence $\delta (\sigma ; \alpha _0, \nu , \lambda ^\dag ) \not \equiv 0$ and there is a positive $\ell \in {\mathbb {N}}$ such that $\delta _i(\alpha _0, \nu , \lambda ^\dag ) = 0$ for $i \leqslant \ell -1$ but $\delta _{\ell }(\alpha _0, \nu , \lambda ^\dag ) \ne 0$. Then $N_{\nu jn}(\lambda ^\dag ) \le {\mathfrak {b}}_{\nu jn}(\lambda ^\dag ) (\ell -1)$.

• (ii) Assume $(\alpha _0, \lambda ^\dag ) \in {\mathbf {V}}({\mathcal {I}}^{\{\nu \}}) \cap E_\nu$, that is, $\delta (\sigma ; \alpha _0, \nu , \lambda ^\dag ) \equiv 0$ or equivalently $\delta _i(\alpha _0, \nu , \lambda ^\dag ) = 0$ for all $i \ge 1$. If $m+1$ is the cardinality of the minimal basis of the Bautin ideal ${\mathcal {B}}^{\{\nu \}} = \langle D, \delta _i : i \in {\mathbb {N}} \rangle$ in the associated ring ${\mathcal {R}}$, then $N_{\nu jn}(\lambda ^\dag ) \le {\mathfrak {b}}_{\nu jn}(\lambda ^\dag )(m-1)$. Moreover, system (1) with parameters $\lambda ^\dag$ has ${\mathfrak {b}}_{\nu jn}(\lambda ^\dag )$ local two-dimensional periodic invariant manifolds foliated by $\nu$-orbits through the origin (one associated to each branch in $B_{jn}^\nu$).

From Theorem 6 we see that how to get a global upper bound on $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )} )$ as follows:

$$\begin{aligned} \mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )} ) = \sup _{\lambda \in \Lambda _{\nu }^{\mathbf{N}(\Delta )} } \{ N_\nu (\lambda ) \} \le \sup _{\lambda \in \Lambda _{\nu }^{\mathbf{N}(\Delta )} } \left\{ \sum _{j=1}^{\mathfrak {s}} \sum _{n \in {\mathbb {N}}} N_{\nu jn}(\lambda ) \right\} . \end{aligned}$$

The bound on $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )} )$ is derived after we bound all the involved $N_{\nu jn}(\lambda )$ by using Theorem 6.

Remark 7

We emphasize a bound on $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda _{\nu }^{\mathbf{N}(\Delta )} )$ for any $\nu \in {\mathbb {Z}}^+$ does not imply that the cyclicity of the origin of (1) with parameters in $\Lambda _{\nu }^{\mathbf{N}(\Delta )}$ has an upper bound since $\nu$ is unbounded.

We recall that the computation of the minimal basis of the Bautin ideal ${\mathcal {B}}^{\{ \nu \}}$ is a non-algorithmic step in Theorem 6(ii). From the practical point of view we present an approach to obtain lower bounds on $N_{\nu jn}$ when perturbations are taken starting at parameters lying on irreducible components of the center variety ${\mathbf {V}}({\mathcal {I}}^{\{ \nu \}})$ associated to $B_{jn}^\nu$. The method is based on the initial ideas developed in [5] for planar non-degenerate centers which were extended to Hopf singularities of three-dimensional systems in [11].

For any $\kappa \in {\mathbb {N}}$, we let $\{ D(\alpha _0; \nu , \lambda ), \delta _{j_1}(\alpha _0, \nu , \lambda ), \ldots , \delta _{j_\kappa }(\alpha _0, \nu , \lambda ) \}$ be the minimal basis of the ideal ${\mathcal {B}}_{j_\kappa }^{\{ \nu \}}$ in the ring ${\mathcal {R}}$. Given a fixed $\nu$, we denote by $d_PF_\kappa$ the $(\kappa +1) \times (p+1)$ Jacobian matrix of the real analytic mapping

$$\begin{aligned} (\alpha _0, \lambda ) \mapsto F_\kappa (\alpha _0, \lambda ) = (D(\alpha _0; \nu , \lambda ), \delta _{j_1}(\alpha _0, \nu , \lambda ), \ldots , \delta _{j_\kappa }(\alpha _0, \nu , \lambda )) \end{aligned}$$

(16)

evaluated at $P \in E_\nu = \{ (\alpha _0, \lambda ) \in {\mathbb {R}}^{p+1} : \alpha _0 \ne 0, \lambda \in \Lambda \}$.

Theorem 8

([10]) Let $C \subset {\mathbf {V}}({\mathcal {I}}^{\{ \nu \}})$ be an irreducible component of the center variety associated to $B_{jn}^\nu$. Let $P = (\alpha _0^*, \lambda ^*) \in C \cap E_\nu$ be a point such that ${\mathfrak {b}}_{\nu jn}(\lambda ^*) = \# B_{jn}^\nu$ and $\mathrm{rank}(d_PF_\kappa ) = \kappa +1$ because the gradient $\nabla _\lambda F_\kappa (P)$ has rank $\kappa +1$. Let $N_{\nu jn}(\lambda ^*)$ be the maximum number of small amplitude $\nu$-limit cycles of (1) coming from all the branches in $B_{jn}^\nu$ for parameter values $\lambda$ sufficiently close to $\lambda ^*$. Assuming that $\kappa \leqslant p$ the following holds:

• (i) $N_{\nu jn}(\lambda ^*) \ge {\mathfrak {b}}_{\nu jn}(\lambda ^*)(\kappa -1)$.

• (ii) If moreover $\mathrm{codim}(C) = \kappa +1$ then $N_{\nu jn}(\lambda ^*) = {\mathfrak {b}}_{\nu jn}(\lambda ^*)(\kappa -1)$.

Remark 9

We recall here a well known method used in bifurcation theory for analytic functions, see for example [16] and also chapter 5.1 of [3]. Let $\{ D, \delta _{j_1}, \dots , \delta _{j_\kappa } \}$ be a minimal basis of the ideal ${\mathcal {B}}_{j_\kappa }^{\{ \nu \}}$ in ${\mathcal {R}}$ and perturb system (1) along a smooth curve $\varepsilon \mapsto \lambda (\varepsilon ) \subset {\mathbb {R}}^{p}$ with $\lambda (0) = \lambda ^\dag$, $D(\alpha _0(\lambda (\varepsilon )); \nu , \lambda (\varepsilon )) \equiv 0$ for any $|\varepsilon | \ll 1$ in such a way that

$$\begin{aligned} |\delta _{j_1}(\nu , \lambda (\varepsilon ))| \ll |\delta _{j_2}(\nu , \lambda (\varepsilon ))| \ll \cdots \ll |\delta _{j_\kappa }(\nu , \lambda (\varepsilon ))| \end{aligned}$$

with $\delta _{j_i}(\nu , \lambda (\varepsilon )) \delta _{j_{i+1}}(\nu , \lambda (\varepsilon )) < 0$ for $i = 1, \ldots , \kappa - 1$, then $\kappa -1$ small amplitude $\nu$-limit cycles (associated to each branch in the corresponding segment) can be made to bifurcate from the origin in family (1) by perturbing the parameters from $\lambda ^\dag$. In this way sometimes we are able to compute the exact value of the numbers $N_{\nu jn}$ given in Theorem 6.

Remark 10

Given a polynomial ideal ${\mathcal {I}}$ one can use the routine minAssChar in the primdec.lib library [8] of Singular [7] to obtain the prime decomposition of $\sqrt{{\mathcal {I}}}$, the radical ideal of ${\mathcal {I}}$. Then, if $\sqrt{{\mathcal {I}}} = \cap _{j=1}^{q} {\mathcal {P}}_j$ is that prime decomposition, the irreducible components $C_j$ of ${\mathbf {V}}({\mathcal {I}})$ are $C_j = {\mathbf {V}}({\mathcal {P}}_j)$.

2.1 The Generic Case

We consider a family (1) satisfying the generic conditions $\partial ^2_x F_3(0,0,0; \lambda ) + \partial ^2_y F_3(0,0,0; \lambda ) \ne 0$ and $\partial ^2_z F_3(0,0,0; \lambda ) \ne 0$. This is equivalent to $v_{20}^{[2]}(\nu , \lambda ) \ne 0$ and $v_{02}^{[2]}(\nu , \lambda ) \ne 0$. Therefore, for any fixed $\nu \in {\mathbb {Z}}^+$ and $\lambda \in {\mathbb {R}}^p \backslash {\mathbf {V}}(v_{20}^{[2]} v_{02}^{[2]})$, ${\mathbf {N}}(\Delta )$ has just one edge with endpoints (0, 2) and (2, 0) and therefore $(r,s) = (2, 2)$. The leading coefficients $\alpha _0 \ne 0$ are the real nonzero roots of $D(\alpha _0; \nu , \lambda ) = \alpha _0^2 v_{02}^{[2]}(\nu , \lambda ) + v_{20}^{[2]}(\nu , \lambda )$, hence if $v_{02}^{[2]} v_{20}^{[2]} < 0$ then there is no real branch while two real branches arise when $v_{02}^{[2]} v_{20}^{[2]} > 0$ and both are simple with index $n=2$.

In this generic case $\alpha _0$ is independent of $\nu$ and we simply write $\Lambda$ and E instead of $\Lambda _{\nu }^{\mathbf{N}(\Delta )}$ and $E_\nu$. We define the parameter space $\Lambda = \{ \lambda \in {\mathbb {R}}^p : v_{02}^{[2]} v_{20}^{[2]} \ne 0 \} = {\mathbb {R}}^p \backslash {\mathbf {V}}(v_{02}^{[2]} v_{20}^{[2]})$ so that ${\mathbf {N}}(\Delta )$ remains invariant when the parameter are restricted to lie in $\Lambda$ although the number of branches may vary.

The coefficients $\alpha _i(\nu , \lambda )$ are obtained from (11), that is, equating to zero the coefficients of the powers of $\sigma$ in

$$\begin{aligned} v_{20}^{[2]} \sigma ^4 + v_{02}^{[2]} \sigma ^4 \left( \sum _{k \ge 0} \alpha _k \sigma ^k \right) ^2 + \sum _{i+j \ge 3} v_{ij}^{[2]} \sigma ^{2(i+j)} \left( \sum _{k \ge 0} \alpha _k \sigma ^k \right) ^j \equiv 0. \end{aligned}$$

It follows that $\alpha _{2j+1} = 0$ for all $j \ge 0$, hence $\delta _{2j+1} = 0$ too. The coefficients $\alpha _{2i}$ restricted to E are analytic functions. The first coefficients $\alpha _{2i}$ are:

$$\begin{aligned} \alpha _2= & {} -(\alpha _0^3 v_{03}^{[2]} + \alpha _0^2 v_{11}^{[2]} + \alpha _0 v_{21}^{[2]} + v_{30}^{[2]})/(2 \alpha _0 v_{02}^{[2]}), \nonumber \\ \alpha _4= & {} - (-5 \alpha _0^6 (v_{03}^{[2]})^2 + 4 \alpha _0^6 v_{02}^{[2]} v_{04}^{[2]} - 8 \alpha _0^5 v_{03}^{[2]} v_{12}^{[2]} -3 \alpha _0^4 (v_{12}^{[2]})^2 + 4 \alpha _0^5 v_{02}^{[2]} v_{13}^{[2]} \nonumber \\&- 6 \alpha _0^4 v_{03}^{[2]} v_{21}^{[2]} - 4 \alpha _0^3 v_{12}^{[2]} v_{21}^{[2]} - \alpha _0^2 (v_{21}^{[2]})^2 + 4 \alpha _0^4 v_{02}^{[2]} v_{22}^{[2]} - 4 \alpha _0^3 v_{03}^{[2]} v_{30}^{[2]} - 2 \alpha _0^2 v_{12}^{[2]} v_{30}^{[2]} \nonumber \\&+ (v_{30}^{[2]})^2 + 4 \alpha _0^3 v_{02}^{[2]} v_{31}^{[2]} + 4 \alpha _0^2 v_{02}^{[2]} v_{40}^{[2]} ) / (8 \alpha _0^3 (v_{02}^{[2]})^2). \end{aligned}$$

(17)

Similarly, the first non-zero coefficients $\delta _{2i}(\alpha _0, \nu , \lambda )$ are

$$\begin{aligned} \delta _4= & {} \alpha _0 v_{01}^{[1]}, \\ \delta _6= & {} \alpha _2 v_{01}^{[1]} + \alpha _0^2 v_{02}^{[1]} + \alpha _0 v_{11}^{[1]} + v_{20}^{[1]}, \\ \delta _8= & {} \alpha _4 v_{01}^{[1]} + 2 \alpha _0 \alpha _2 v_{02}^{[1]} + \alpha _0^3 v_{03}^{[1]} + \alpha _2 v_{11}^{[1]} + \alpha _0^2 v_{12}^{[1]} + \alpha _0 v_{21}^{[1]} + v_{30}^{[1]} \end{aligned}$$

where the $\alpha _j$ with $j > 0$ are replaced by its corresponding expressions, see (17). The coefficients $\alpha _{2i}$ and $\delta _{2i}$ lie in ${\mathcal {R}} = {\mathbb {R}}(\alpha _0, \lambda _1)[\alpha _0, \lambda ]$, the Noetherian ring of polynomials in $(\alpha _0, \lambda )$ with coefficients in the field of rational expressions in $\alpha _0$ and $\lambda _1$ with $\lambda _1 = v_{02}^{[2]}/(\nu \pi ) = \partial ^2_z F_3(0,0,0; \lambda )$.

The formulas for the degenerate case $v_{20}^{[2]}(\lambda ) v_{02}^{[2]}(\lambda ) = 0$ are in the Appendix of Sect. 4.

3 Some Quadratic Families and the Proofs of the Main Theorems

We consider some particular cases of the quadratic family.

3.1 A Family with an Invariant Plane through the Zero-Hopf Singularity

Consider family

$$\begin{aligned} \begin{array}{ll} {\dot{x}}=-y + x (a_1 x + a_2 y + a_3 z), \\ {\dot{y}}=\ \ x + y (b_1 x + b_2 y + b_3 z), \\ {\dot{z}} = z (c_1 x + c_2 y + c_3 z). \end{array} \end{aligned}$$

(18)

The parameters of the family are $\lambda = (a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3) \in {\mathbb {R}}^9$, the z-axis is invariant and if $c_3 \ne 0$ it is non-singular for all the family. Since the plane $\{ z=0\}$ is invariant for the flow of (18) it follows that $d^{[2]}(\rho _0, 0; \nu , \lambda ) = 0$ or, in other words, $v_{i0}^{[2]}(\nu , \lambda ) = 0$ for any $i \ge 2$. The first Poincaré-Lyapunov quantities are:

$$\begin{aligned} v_{01}^{[1]}(\nu , \lambda )= & {} \nu \pi (a_3 + b_3), \\ v_{20}^{[1]}(\nu , \lambda )= & {} \nu \pi (a_1 a_2 - b_1 b_2)/4, \\ v_{11}^{[1]}(\nu , \lambda )= & {} - \nu \pi (a_3 + b_3) (a_2 + 2 b_2 - 3 c_2)/3, \\ v_{02}^{[1]}(\nu , \lambda )= & {} \nu ^2 \pi ^2 (a_3 + b_3) (a_3 + b_3 + 2 c_3)/2, \\ v_{21}^{[1]}(\nu , \lambda )= & {} \nu \pi (-4 a_1^2 a_3 - 5 a_2^2 a_3 + 5 a_1 a_3 b_1 - a_3 b_1^2 + a_2 a_3 b_2 + 4 a_3 b_2^2 + 20 a_1^2 b_3 \\&+\, 7 a_2^2 b_3 - 7 a_1 b_1 b_3 - 13 b_1^2 b_3 + 13 a_2 b_2 b_3 - 20 b_2^2 b_3 - 12 a_1 a_3 c_1 \\&+\, 6 a_3 b_1 c_1 + 30 b_1 b_3 c_1 - 6 a_3 c_1^2 - 18 b_3 c_1^2 + 22 a_2 a_3 c_2 + 32 a_3 b_2 c_2 \\&-\, 2 a_2 b_3 c_2 + 20 b_2 b_3 c_2 - 42 a_3 c_2^2 - 30 b_3 c_2^2 - 24 a_1 a_2 a_3 \nu \pi + 24 a_3 b_1 b_2 \nu \pi \\&-\, 24 a_1 a_2 b_3 \nu \pi + 24 b_1 b_2 b_3 \nu \pi + 24 a_3 b_2 c_1 \nu \pi + 24 b_2 b_3 c_1 \nu \pi \\&- \, 24 a_1 a_3 c_2 \nu \pi - 24 a_1 b_3 c_2 \nu \pi )/48, \\ v_{02}^{[2]}(\nu , \lambda )= & {} 2 \nu \pi c_3, \\ v_{03}^{[2]}(\nu , \lambda )= & {} 4 \nu ^2 \pi ^2 c_3^2, \\ v_{21}^{[2]}(\nu , \lambda )= & {} - \nu \pi (b_2 c_1 - a_1 c_2) . \end{aligned}$$

We have $v_{02}^{[2]} \ne 0$ since $c_3 \ne 0$ and we see that family (18) is generic if $b_2 c_1 - a_1 c_2 \ne 0$ in which case $v_{21}^{[2]} \ne 0$. Consequently $d^{[2]}(\rho _0, z_0; \nu , \lambda ) = z_0 \Delta (\rho _0, z_0; \nu , \lambda )$ with

$$\begin{aligned} \Delta (\rho _0, z_0; \nu , \lambda ) = \left[ v_{21}^{[2]}(\nu , \lambda ) \rho _0^2 + O(\rho _0^3) \right] + \left[ v_{02}^{[2]}(\nu , \lambda ) + O(\rho _0) \right] z_0 + O(z_0^2), \end{aligned}$$

which implies that the Newton’s diagram ${\mathbf {N}}(\Delta )$ has just one descending segment with endpoints (0, 2) and (1, 0) so that $(r,s) = (2, 1)$. Therefore we consider the parameter space $\Lambda = {\mathbb {R}}^9 \backslash {\mathbf {V}}(v_{02}^{[2]} v_{21}^{[2]})$ that is independent of $\nu$. More precisely,

$$\begin{aligned} \Lambda = \{ \lambda = (a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3) \in {\mathbb {R}}^9 : c_3(b_2 c_1 - a_1 c_2) \ne 0 \}. \end{aligned}$$

(19)

Some computations give the determining polynomial $D(\alpha _0; \nu , \lambda ) = \alpha _0 v_{02}^{[2]}(\nu , \lambda ) + v_{21}^{[2]}(\nu , \lambda )$, that is, $D(\alpha _0; \nu , \lambda ) = \nu \pi [2 c_3 \alpha _0 - (b_2 c_1 - a_1 c_2)]$ and in consequence there is a unique branch associated to the segment which is simple. Moreover, the leading coefficient $\alpha _0$ is independent of $\nu$ so we will write $\alpha _0(\lambda ) = (b_2 c_1 - a_1 c_2)/(2 c_3) \ne 0$. The branch has index $n=1$ and its expansion (10) is $z^*(\sigma ; \nu , \lambda ) = \sigma ^2 \left( \alpha _0(\lambda ) + \sum _{i \ge 1} \alpha _i(\nu , \lambda ) \sigma ^{i} \right)$ with first coefficients

$$\begin{aligned} \alpha _1= & {} -(\alpha _0 v_{12}^{[2]} + v_{31}^{[2]})/ v_{02}^{[2]}, \\ \alpha _2= & {} -(\alpha _0^2 v_{02}^{[2]} v_{03}^{[2]} - \alpha _0 (v_{12}^{[2]})^2 + \alpha _0 v_{02}^{[2]} v_{22}^{[2]} - v_{12}^{[2]} v_{31}^{[2]} + v_{02}^{[2]} v_{41}^{[2]})/(v_{02}^{[2]})^2. \end{aligned}$$

Furthermore, the reduced displacement map has an expansion (13) given by $\delta (\sigma ; \nu , \lambda ) = \sum _{i \ge 3} \delta _i(\nu , \lambda ) \sigma ^{i}$ with first coefficients

$$\begin{aligned} \delta _3= & {} \alpha _0 v_{01}^{[1]} + v_{20}^{[1]}, \\ \delta _4= & {} \alpha _1 v_{01}^{[1]} + \alpha _0 v_{11}^{[1]} + v_{30}^{[1]}, \\ \delta _5= & {} \alpha _2 v_{01}^{[1]} + \alpha _0^2 v_{02}^{[1]} + \alpha _1 v_{11}^{[1]} + \alpha _0 v_{21}^{[1]} + v_{40}^{[1]}. \end{aligned}$$

Working in the ring ${\mathcal {R}} = {\mathbb {R}}(\alpha _0, c_3)[\alpha _0, \lambda ]$ and taking into account the expression of D, the first non-identically zero reduced coefficients ${{\hat{\delta }}}_{i}$ are, up to positive constants, independent of $\nu$ and given by:

$$\begin{aligned} {{\hat{\delta }}}_3(\alpha _0, \lambda )= & {} a_1 a_2 - b_1 b_2 + 4 a_3 \alpha _0 + 4 b_3 \alpha _0, \\ {{\hat{\delta }}}_5(\alpha _0, \lambda )= & {} (27 a_1^2 a_3 b_2 c_1 - 36 a_1 a_3 b_1 b_2 c_1 + 9 a_3 b_1^2 b_2 c_1 - 27 a_3 b_2^3 c_1 + 27 a_1^2 b_2 b_3 c_1 \\&-\, 36 a_1 b_1 b_2 b_3 c_1 + 9 b_1^2 b_2 b_3 c_1 - 27 b_2^3 b_3 c_1 -\, 9 a_2 a_3 b_1 b_2 c_2 + 36 a_3 b_1 b_2^2 c_2 \\&-\, 9 a_2 b_1 b_2 b_3 c_2 + 36 b_1 b_2^2 b_3 c_2 - 18 a_1^3 b_2 c_3 +\, 15 a_1^2 b_1 b_2 c_3 + 3 a_2^2 b_1 b_2 c_3 \\&+\, 6 a_1 b_1^2 b_2 c_3 - 3 b_1^3 b_2 c_3 - 6 a_2 b_1 b_2^2 c_3 + 18 a_1 b_2^3 c_3 - 15 b_1 b_2^3 c_3 \\&+\, 36 a_1 a_3 b_3 c_1 \alpha _0 - 36 a_3 b_1 b_3 c_1 \alpha _0 + 36 a_1 b_3^2 c_1 \alpha _0 - 36 b_1 b_3^2 c_1 \alpha _0 \\&+\, 36 a_3 b_3 c_1^2 \alpha _0 + 36 b_3^2 c_1^2 \alpha _0 - 108 a_3^2 b_2 c_2 \alpha _0 + 36 a_2 a_3 b_3 c_2 \alpha _0 \\&-\, 252 a_3 b_2 b_3 c_2 \alpha _0 + 36 a_2 b_3^2 c_2 \alpha _0 - 144 b_2 b_3^2 c_2 \alpha _0 + 36 a_3^2 c_2^2 \alpha _0 \\&+\, 36 a_3 b_3 c_2^2 \alpha _0 + 20 a_1^2 a_3 c_3 \alpha _0 + 5 a_2^2 a_3 c_3 \alpha _0 - a_1 a_3 b_1 c_3 \alpha _0 - a_3 b_1^2 c_3 \alpha _0 \\&+\, 41 a_2 a_3 b_2 c_3 \alpha _0 + 98 a_3 b_2^2 c_3 \alpha _0 - 16 a_1^2 b_3 c_3 \alpha _0 - 13 a_2^2 b_3 c_3 \alpha _0 \\&+ \,17 a_1 b_1 b_3 c_3 \alpha _0 + 17 b_1^2 b_3 c_3 \alpha _0 + 23 a_2 b_2 b_3 c_3 \alpha _0 + 134 b_2^2 b_3 c_3 \alpha _0 \\&-\, 18 a_1 a_3 c_1 c_3 \alpha _0 + 9 a_3 b_1 c_1 c_3 \alpha _0 - 36 a_1 b_3 c_1 c_3 \alpha _0 - 27 b_1 b_3 c_1 c_3 \alpha _0 \\&-\, 9 a_3 c_1^2 c_3 \alpha _0 + 9 b_3 c_1^2 c_3 \alpha _0 - 27 a_2 a_3 c_2 c_3 \alpha _0 - 36 a_3 b_2 c_2 c_3 \alpha _0 \\&+\, 9 a_2 b_3 c_2 c_3 \alpha _0 - 18 b_2 b_3 c_2 c_3 \alpha _0 + 9 a_3 c_2^2 c_3 \alpha _0 - 9 b_3 c_2^2 c_3 \alpha _0)/c_3, \\ {{\hat{\delta }}}_7(\alpha _0, \lambda )= & {} (-9762768 a_2^2 a_3^3 b_1 b_3 c_1^5 + \cdots - 235892736 b_3 c_3^7 \alpha _0^4)/(\alpha _0 c_3^5). \end{aligned}$$

The expression of ${{\hat{\delta }}}_7$ is huge and, for simplicity, we only display the beginning and the end of it.

The surprising behaviour that the computed ${{\hat{\delta }}}_{i}$ do not depend on $\nu$ because it only appears in the positive multiplicative constants that we remove implies that the center variety ${\mathbf {V}}({\mathcal {I}})$ does not depend on $\nu$.

The computations to obtain a characterization of the center variety are heavy and therefore in the next two subsections we only study the subfamilies (2) and (4) of (18) in $\Lambda$ given in the introductory section.

Before, we recall a classical result that we will use intensively in several proofs. It shows the relationship between Riccati equations and second order linear differential equations.

Remark 11

The general Riccati equation ${\dot{z}} = P(t) + Q(t) z + R(t) z^2$, under the transformation $z \mapsto u$ defined by $z = - {\dot{u}}/(R u)$, becomes the second order linear equation $-R \ddot{u} + ({\dot{R}} + Q R) {\dot{u}} - P R^2 u = 0$.

3.2 Proof of Theorem 1

In this section we study the subfamily of (18) with parameters (19) restricted to $a_1 = b_1 = b_3 = c_2 = 0$, that is, we study family (2) in $\Lambda$ given by (3). Statements (a), (b) and (c) of Theorem 1 are just consequences of the forthcoming Propositions 12, 14 and 15 , respectively.

Proposition 12

We consider family (2) with parameter space $\Lambda$ given by (3). Then there is a unique descending segment in the Newton’s diagram of $\Delta$ whose associated center variety is ${\mathbf {V}}({\mathcal {I}}) = {\mathbf {V}}({\mathcal {I}}_5)$ where

$$\begin{aligned} {\mathcal {I}}_5 = \langle 2 c_3 \alpha _0 - b_2 c_1, a_3 \alpha _0, a_3 b_2^3 c_1 \rangle . \end{aligned}$$

Moreover there is a local two-dimensional periodic invariant manifold ${\mathcal {M}} \ne \{z=0\}$ of (2) through the origin if and only if the parameters $\lambda$ lie in $V_{\mathcal {C}} \cap \Lambda$, being the variety $V_{\mathcal {C}} = \{ \lambda \in {\mathbb {R}}^5 : a_3 = 0 \}$. Family (2) on $V_{\mathcal {C}} \cap \Lambda$ only has two periodic invariant manifold near the origin which are ${\mathcal {M}}$ and $\{z=0\}$, and both are foliated by periodic 1-orbits.

Proof

Using the notation ${{\hat{\delta }}}_j(\alpha _0, \lambda ) = P_j(\alpha _0, \lambda )/Q_j(\alpha _0, \lambda ) \in {\mathcal {R}}$ with $P_j$ and $Q_j$ polynomials with denominator $Q_j|_E \ne 0$ and restricting to $a_1 = b_1 = b_3 = c_2 = 0$ the computed expressions of ${{\hat{\delta }}}_j$ we get, up to positive constants, that $P_3 = a_3 \alpha _0$, ${\hat{P}}_5 = a_3 b_2^3 c_1$ and ${\hat{P}}_7 =0$, where ${\hat{P}}_j \equiv P_j \mod \mathcal {I}_{j-1}$ in the ring ${\mathbb {R}}[\alpha _0, \lambda ]$. We are going to prove that ${\mathbf {V}}({\mathcal {I}}) = {\mathbf {V}}({\mathcal {I}}_5)$ with ${\mathcal {I}}_5 = \langle D, P_3, P_5\rangle$ by showing that if

$$\begin{aligned} (\alpha _0^*, \lambda ^*) \in {\mathbf {V}}({\mathcal {I}}_5) \cap E = \{ (\alpha _0, \lambda ) \in E : a_3 = 0 \} = \{ (\alpha _0, \lambda ) \in E : \lambda \in V_{\mathcal {C}} \}, \end{aligned}$$

and $d^{[1]}(\rho _0, z_0; \nu , \lambda ^*) \not \equiv 0$ (that is $v_{ij}^{[1]}(\nu , \lambda ^*) \ne 0$ for some (i, j)) then system (18) with $\lambda = \lambda ^*$ has a local periodic invariant manifold ${\mathcal {M}}$ through the origin associated with the branch (10) given by $z^*(\rho _0; \nu , \lambda ) = \rho _0^2 \sum _{i \ge 0} \alpha _i(\nu , \lambda ) \rho _0^{i}$ with $\alpha _0 = b_2 c_1/(2 c_3) \ne 0$. The family (18) with $\lambda = \lambda ^*$ becomes

$$\begin{aligned} {\dot{x}}=-y + a_2 x y, \ \ {\dot{y}}= x + b_2 y^2, \ \ {\dot{z}} = z (c_1 x + c_3 z) \end{aligned}$$

(20)

with $c_3 b_2 c_1 \ne 0$. Notice that the invariant plane $\{z=0 \}$ is a periodic invariant manifold foliated by periodic 1-orbits since the two first components of the vector field form a planar field ${\mathcal {X}}_c = (-y + a_2 x y, x + b_2 y^2)$ with a center at the origin because it is reversible with respect to the y-axis. But the plane $\{ z = 0 \}$ is not the manifold ${\mathcal {M}}$, recall that the curve $\{ (x, 0, z^*(x; \nu , \lambda )) : 0 \le x \ll 1 \} \subset {\mathcal {M}} \cap \{ y=0 \}$.

To prove the existence of ${\mathcal {M}}$ we will consider the initial conditions $(x_0, 0, z_0)$ for the solutions of (20). The third equation ${\dot{z}} = z (c_1 x(t; x_0, \lambda ) + c_3 z)$ is a Riccati equation for the function $z(t; x_0, z_0, \lambda )$ and therefore it can be transformed into a linear second order differential equation, see Remark 11. Indeed, the change $z \mapsto u$ with

$$\begin{aligned} z = - \frac{\dot{u}}{c_3 u}, \end{aligned}$$

(21)

well defined off $\{z=0\}$, transforms the Riccati equation into $\ddot{u} - c_1 x(t; x_0, \lambda ) {\dot{u}} = 0$ whose solution is

$$\begin{aligned} u(t; x_0, z_0, \lambda ) = u_0 + u_0' \int _0^t \exp \left( c_1 \int _0^s x(\sigma ; x_0, \lambda ) d \sigma \right) d s, \end{aligned}$$

(22)

being $u_0 = u(0; x_0, z_0, \lambda )$ and $u_0' = {\dot{u}}(0; x_0, z_0, \lambda )$ satisfying $z_0 = - u'_0/(c_3 u_0) \ne 0$. Let $T(x_0, \lambda )$ be the period function of the planar center, that is, $x(.; x_0, \lambda )$ is $T(x_0, \lambda )$-periodic. Then the orbit of (20) starting at $(x_0, 0, z_0)$ is a periodic $\nu$-orbit with $\nu \in {\mathbb {Z}}^+$ if and only if the function $z(.; x_0, z_0, \lambda )$ es $\nu T(x_0, \lambda )$-periodic. From the expression (22) together with (21), the $\nu T(x_0, \lambda )$-periodicity condition $z(\nu T(x_0, \lambda ); x_0, z_0, \lambda ) - z_0 = 0$ leads

$$\begin{aligned} z_0 = \frac{1 - \exp \left( c_1 \int _0^{\nu T(x_0, \lambda )} x(\sigma ; x_0, \lambda ) d \sigma \right) }{c_3 \int _0^{\nu T(x_0, \lambda )} \exp \left( c_1 \int _0^t x(\sigma ; x_0, \lambda ) d \sigma \right) dt} := \xi (x_0; \nu , \lambda ), \end{aligned}$$

(23)

being $\xi$ an analytic function at $x_0=0$. Notice that from the second equation in (20) we have that

$$\begin{aligned} \int _0^{\nu T(x_0, \lambda )} x(\sigma ; x_0, \lambda ) d \sigma = - b_2 \int _0^{\nu T(x_0, \lambda )} y^2(\sigma ; x_0, \lambda ) d \sigma \not \equiv 0 \end{aligned}$$

so that $\xi (x_0; \nu , \lambda ) \not \equiv 0$. Using that $T(0, \lambda ) = 2 \pi$ and that $x(t; 0, \lambda ) \equiv 0$ for all $t \in {\mathbb {R}}$ we get $\xi (0; \nu , \lambda ) = 0$. The above computations show that there is a unique curve $\{ (x_0, 0, \xi (x_0; \nu , \lambda )) \} \subset {\mathbb {R}}^3$ through the origin not contained in the plane $\{ z = 0 \}$ of initial conditions such that the solutions of (20) are periodic. With a little more effort we can see that $\xi (x_0, \nu , \lambda ) = \alpha _0(\lambda ) x_0^2 + \cdots$ as it must be because $\xi (x_0, \nu , \lambda ) = z^*(x_0; \nu , \lambda )$. This is accomplished by expanding in power series of $x_0$ the functions $T(x_0, \lambda ) = 2 \pi + O(x_0^2)$ and $x(t; x_0, \lambda ) = \cos (t) x_0 + 2 (a_2 - b_2 + (2 a_2 + b_2) \cos (t)) \sin ^2(t/2) x_0^2/3 + O(x_0^3)$ and computing the derivatives of $\xi$ at $x_0=0$ from its defining expression (23). The result is

$$\begin{aligned} \frac{\partial \xi }{\partial x_0}(0, \nu , \lambda ) = - \frac{c_1}{2 c_3 \nu \pi } \int _0^{2 \nu \pi } \frac{\partial x}{\partial x_0}(t; 0, \lambda ) \, dt = - \frac{c_1}{2 c_3 \nu \pi } \int _0^{2 \nu \pi } \cos (t) \, dt = 0, \end{aligned}$$

and a similar computation gives $\partial ^2 \xi /\partial x_0^2(0, \nu , \lambda ) = 2 \alpha _0(\lambda )$.

We only need to prove that ${\mathcal {M}}$ is foliated by periodic 1-orbits. This is a consequence of the relevant fact that, even though the branch $z^*(\sigma ; \nu , \lambda )$ explicitly depends on $\nu$, when $\lambda ^* \in V_{\mathcal {C}} \cap \Lambda$ then $z^*(\sigma ; \nu , \lambda ^*)$ does not depend on $\nu$. The first computational evidence of the former claim comes because the first coefficients $\alpha _{i}(\nu , \alpha _0(\lambda ^*), \lambda ^*)$ are independent of $\nu$. But we need to check that indeed all the coefficients $\alpha _{i}(\nu , \alpha _0(\lambda ^*), \lambda ^*)$ with $i \in {\mathbb {N}}$ do not depend on $\nu$. This task is very hard so we change our point of view to prove the claim. The key point is that, by the classical Poincaré-Lyapunov center theorem, the planar field ${\mathcal {X}}_c$ must have an analytic first integral H(x, y) near $(x,y)=(0,0)$. Clearly H is also a first integral of the full family (2) with the parameter space $V_{\mathcal {C}} \cap \Lambda$ which is independet of z. The level surfaces of H near the origin of ${\mathbb {R}}^3$ are therefore topological cylinders surrounding the z-axis. Since the periodic orbit of (2) with any initial condition $(x_0, 0, z^*(x_0; \nu , \lambda ^*))$ for $x_0 > 0$ sufficiently small is contained in some cylinder-like surface $\{H(x,y) = h(x_0) \} \subset {\mathbb {R}}^3$ for some real constant $h(x_0)$, by the topology of this surface (and the fact that the orbits do not cross by the uniqueness of the solutions of (2)) it becomes clear that this periodic orbit must be a periodic 1-orbit. This proves the claim. $\square$

Remark 13

Besides the periodic invariant manifold ${\mathcal {M}}$, family (2) on $\Lambda \cap V_{\mathcal {C}}$ also possesses the analytic first integral

$$\begin{aligned} H(x,y,z) = (-1 + a_2 x)^2 [-1 + 2 b_2 x + b_2(2 b_2-a_2) y^2]^{-a_2/b_2}. \end{aligned}$$

Proposition 14

We consider family (2) with parameter space $\Lambda \backslash V_{\mathcal {C}}$ being $\Lambda$ defined in (3) and $V_{\mathcal {C}}$ the variety defined in Proposition 12. Then, for any $\nu \in {\mathbb {Z}}^+$, there is no $\nu$-limit cycle bifurcating from the origin, hence $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \backslash V_{\mathcal {C}}) = 0$.

Proof

We consider the points $(\alpha _0^\dag , \lambda ^\dag ) \not \in {\mathbf {V}}({\mathcal {I}}) \cap E$, hence with $\lambda ^\dag \not \in V_{\mathcal {C}}$ or equivalently $a_3 \ne 0$. At these points one has ${{\hat{\delta }}}_3(\lambda ^\dag ) \ne 0$ and therefore it is clear that for such points there is no $\nu$-limit cycles bifurcating from the origin associated to each branch. $\square$

Proposition 15

We consider family the family of vector fields ${\mathcal {X}}_\lambda$ associated to (2) with parameter space $\Lambda$ defined in (3) and $V_{\mathcal {C}}$ the variety defined in Proposition 12. Then the center variety ${\mathbf {V}}({\mathcal {I}})$ intersects E as ${\mathbf {V}}({\mathcal {I}}) \cap E = C \cap E$ with the irreducible component $C = \{ (\alpha _0, \lambda ) \in {\mathbb {R}}^{6} : D := 2 c_3 \alpha _0 - b_2 c_1 = a_3 = 0 \}$. Moreover, for any $\nu \in {\mathbb {Z}}^+$ and $(\alpha _0, \lambda ^*) \in C \cap E$, there are no bifurcations of (2) producing $\nu$-limit cycles from the origin for parameter values $\lambda$ sufficiently close to $\lambda ^*$. In other words, $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \cap V_{\mathcal {C}}) = 0$ where $V_{\mathcal {C}}$ the variety defined in Proposition 12.

Proof

Recall that the center variety ${\mathbf {V}}({\mathcal {I}}) = {\mathbf {V}}({\mathcal {I}}_5)$ is independent of $\nu$ and ${\mathcal {I}}_5 = \langle D, P_3, P_5\rangle$ with $P_3 = a_3 \alpha _0$, and $P_5 = a_3 b_2^3 c_1$. The next step is to compute the irreducible components of ${\mathbf {V}}({\mathcal {I}})$. To do it, first we perform the prime decomposition of the ideal $\sqrt{{\mathcal {I}}_5}$ as it is explained in Remark 10. The outcome is $\sqrt{{\mathcal {I}}_5} = {\mathcal {P}}_1 \cap {\mathcal {P}}_2 \cap {\mathcal {P}}_3$ where

$$\begin{aligned} {\mathcal {P}}_1 = \langle D, a_3 \rangle , \ \ {\mathcal {P}}_2 = \langle \alpha _0, c_1 \rangle , \ \ {\mathcal {P}}_3 = \langle \alpha _0, b_2 \rangle . \end{aligned}$$

Let $C_j = {\mathbf {V}}({\mathcal {P}}_j)$, for $j=1,2,3$, denote the three irreducible components of the center variety. Clearly $C=C_1$ and $C_2 \cap E = C_3 \cap E = \emptyset$ because $\alpha _0 \ne 0$ in E, hence $C_2$ and $C_3$ need not be considered.

We will take the map (16) with $\kappa \in \{1,2\}$ which satisfies $\kappa \leqslant p = 5$. Therefore we consider the map $(\alpha _0, \lambda ) \mapsto F_2(\alpha _0, \lambda ) = (D(\alpha _0; \lambda ), {{\hat{\delta }}}_{3}(\alpha _0, \lambda ), {{\hat{\delta }}}_{5}(\alpha _0, \lambda ))$. Now we take an arbitrary point $P=(\alpha _0, \lambda ^*) \in C \cap E$, that is $\lambda ^* = (a_2, 0, b_2, c_1, c_3) \in {\mathbb {R}}^{5}$ with $\alpha _0 c_3 b_2 c_1 \ne 0$. It follows by straight computations that the Jacobian matrix of $F_2$ with respect to $\lambda$ evaluated at P has $\mathrm{rank}(\nabla _\lambda F_2(P)) = 2 \ne 3$ so we cannot apply Theorem 8 to $F_2$. But we may apply it to $F_1$ because $\mathrm{rank}(\nabla _\lambda F_1(P)) = 2$ and we know that $\mathrm{codim}(C) \geqslant 2$. We claim that in fact $\mathrm{codim}(C) = 2$. We observe that $C = G^{-1}(0)$ where $G: {\mathbb {R}}^6 \rightarrow {\mathbb {R}}^2$ is the map $(\alpha _0, \lambda ) \mapsto G(\alpha _0, \lambda )$ whose components are the generators of the ideal ${\mathcal {P}}_1$. It is straightforward to check that, for any point $q \in C \cap E$, $\mathrm{rank}(d_q G) = 2$. That means that G is a submersion at q, so that for some neighborhood U in ${\mathbb {R}}^6$ of q one has ${\mathbf {V}}({\mathcal {I}}) \cap U = G^{-1}(0) \cap U = C \cap U$ is a codimension-2 submanifold of ${\mathbb {R}}^6$ proving the claim. Thus by Theorem 8(ii) our result follows. $\square$

3.3 Proof of Theorem 2

Here we analyze the subfamily of (18) defined by $a_i = b_i$ for $i=1,2,3$, that is, family (4) with parameters (5). Statements (a), (b) and (c) of Theorem 2 are just consequences of the forthcoming Propositions 16, 17 and 18 , respectively.

Proposition 16

We consider family (4) with parameter space (5). Then there is a unique descending segment in the Newton’s diagram of $\Delta$ with center variety ${\mathbf {V}}({\mathcal {I}}) = {\mathbf {V}}({\mathcal {I}}_3)$ where

$$\begin{aligned} {\mathcal {I}}_3 = \langle a_2 c_1 - a_1 c_2 - 2 c_3 \alpha _0, a_3 \alpha _0 \rangle . \end{aligned}$$

Moreover there is a local two-dimensional periodic invariant manifold ${\mathcal {M}}$ of (4) through the origin if and only if the parameters $\lambda$ lie in $V_{\mathcal {C}} \cap \Lambda$, being the variety $V_{\mathcal {C}} = \{ \lambda \in {\mathbb {R}}^6 : a_3 = 0 \}$. Family (4) on $V_{\mathcal {C}} \cap \Lambda$ only has two periodic invariant manifold near the origin which are ${\mathcal {M}}$ and $\{z=0\}$, and both are foliated by periodic 1-orbits.

Proof

Taking the polynomial numerators $P_j(\alpha _0, \lambda )$ of ${{\hat{\delta }}}_j(\alpha _0, \lambda )$ and restricting to $a_i = b_i$ for $i=1,2,3$, the computed expressions of ${{\hat{\delta }}}_j$ we obtain, up to positive constants, the expressions $P_3 = a_3 \alpha _0$ and ${\hat{P}}_5 = {\hat{P}}_7 = 0$ where ${\hat{P}}_j \equiv P_j \mod \mathcal {I}_{j-1}$ in the ring ${\mathbb {R}}[\alpha _0, \lambda ]$.

We claim that the center variety is ${\mathbf {V}}({\mathcal {I}}) = {\mathbf {V}}({\mathcal {I}}_3)$ with ${\mathcal {I}}_3 = \langle D, P_3\rangle$. In other words we are going to prove that if

$$\begin{aligned} (\alpha _0^*, \lambda ^*) \in {\mathbf {V}}({\mathcal {I}}_3) \cap E = \{ (\alpha _0, \lambda ) \in E : a_3 = 0 \} = \{ (\alpha _0, \lambda ) \in E : \lambda \in V_{\mathcal {C}} \}, \end{aligned}$$

and $d^{[1]}(\rho _0, z_0; \nu , \lambda ^*) \not \equiv 0$, then system (4) with $\lambda = \lambda ^*$ has a local periodic invariant manifold ${\mathcal {M}}$ through the origin associated with the unique branch $z^*(\rho _0; \nu , \lambda ) = \alpha _0(\lambda ) \rho _0^2 + \cdots$ with $\alpha _0 = (a_2 c_1 - a_1 c_2)/(2 c_3) \ne 0$.

Family (4) with parameters lying on $V_{\mathcal {C}} \cap \Lambda$ adopts the form

$$\begin{aligned} {\dot{x}}=-y + x (a_1 x + a_2 y), \ \ {\dot{y}}= x + y (a_1 x + a_2 y), \ \ {\dot{z}} = z (c_1 x + c_2 y + c_3 z) \end{aligned}$$

(24)

with $c_3 (a_2 c_1 - a_1 c_2) \ne 0$. First we notice that the two first components of this system form a planar field with a center at the origin. The reason is that it admits the first integral $H(x,y) = (x^2+y^2) / (1-a_2 x+ a_1 y)^2$.

Secondly, considering the initial conditions $(x_0, 0, z_0)$ for the solutions of (24), we see that the third equation ${\dot{z}} = z (c_1 x(t; x_0, \lambda ) + c_2 y(t; x_0, \lambda ) + c_3 z)$ is a Riccati equation for the function $z(t; x_0, z_0, \lambda )$. Therefore we are in a similar position to the one encountered in the proof of Proposition 12 and we left the reader to follow completely analogous arguments to prove the existence of ${\mathcal {M}}$ and its 1-orbit foliation. $\square$

Proposition 17

We consider family (4) with parameter space $\Lambda \backslash V_{\mathcal {C}}$ being $\Lambda$ defined in (5) and $V_{\mathcal {C}}$ the variety defined in Proposition 16. Then, for any $\nu \in {\mathbb {Z}}^+$, there is no $\nu$-limit cycle bifurcating from the origin, hence $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \backslash V_{\mathcal {C}}) = 0$.

Proof

Exactly the same proof than Proposition 14. $\square$

Proposition 18

We consider the family of vector fields ${\mathcal {X}}_\lambda$ associated to (4) with parameter space $\Lambda$ defined in (5) and $V_{\mathcal {C}}$ the variety defined in Proposition 16. Then the center variety ${\mathbf {V}}({\mathcal {I}})$ intersects E as ${\mathbf {V}}({\mathcal {I}}) \cap E = C \cap E$ with the irreducible component $C = \{ (\alpha _0, \lambda ) \in {\mathbb {R}}^{7} : D := \nu \pi (2 c_3 \alpha _0 - a_2 c_1 + a_1 c_2) = a_3 = 0 \}$. Moreover, for any $\nu \in {\mathbb {Z}}^+$ and $(\alpha _0, \lambda ^*) \in C \cap E$, there are no bifurcations of (2) producing $\nu$-limit cycles from the origin for parameter values $\lambda$ sufficiently close to $\lambda ^*$. In other words, $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \cap V_{\mathcal {C}}) = 0$ where $V_{\mathcal {C}}$ the variety defined in Proposition  16.

Proof

Recall that the center variety ${\mathbf {V}}({\mathcal {I}}) = {\mathbf {V}}({\mathcal {I}}_3)$ is independent of $\nu$ and ${\mathcal {I}}_3 = \langle D, P_3\rangle$ with $P_3 = a_3 \alpha _0$. Using Singular we obtain the irreducible components of ${\mathbf {V}}({\mathcal {I}})$ and we easily check that $\sqrt{{\mathcal {I}}_3} = {\mathcal {P}}_1 \cap {\mathcal {P}}_2$ where, defining $C_j = {\mathbf {V}}({\mathcal {P}}_j)$, for $j=1,2$, we get that $C=C_1$ and $C_2 \cap E = \emptyset$.

The map (16) with $\kappa = 1 \leqslant p = 6$ is $(\alpha _0, \lambda ) \mapsto F_1(\alpha _0, \lambda ) = (D(\alpha _0; \lambda ), {{\hat{\delta }}}_{3}(\alpha _0, \lambda ))$. Letting $P=(\alpha _0, \lambda ^*) \in C \cap E$ an arbitrary point with $\lambda ^* = (a_1, a_2, 0, b_1, b_2, b_3) \in {\mathbb {R}}^{6}$ we obtain $\mathrm{rank}(\nabla _\lambda F_1(P)) = 2$. Moreover $\mathrm{codim}(C) = 2$ because $C = G^{-1}(0)$ where $G: {\mathbb {R}}^7 \rightarrow {\mathbb {R}}^2$ is the map $(\alpha _0, \lambda ) \mapsto G(\alpha _0, \lambda )$ whose components are the generators of the ideal ${\mathcal {P}}_1$ and it is a submersion at any point $q \in C \cap E$ since $\mathrm{rank}(d_q G) = 2$. Our result follows after applying Theorem 8(ii). $\square$

3.4 Proof of Theorem 4

In this section we deal with the Liénard-like family (6) in the parameter space (7). Statements (a), (b) and (c) of Theorem 4 are just consequences of the forthcoming Propositions 19, 20 and 21 , respectively.

Proposition 19

We consider family (6) with parameter space $\Lambda$ given in (7). Then there is a unique descending segment in the Newton’s diagram of $d^{[2]}$ and the center variety ${\mathbf {V}}({\mathcal {I}})$ satisfy

$$\begin{aligned} {\mathbf {V}}({\mathcal {I}}) \cap E = {\mathbf {V}}({\mathcal {I}}_{12}) \cap E = \{ (\alpha _0, \lambda ) \in E : \lambda \in V_{\mathcal {C}} \cap \Lambda \} \end{aligned}$$

being the variety $V_{\mathcal {C}} = \{ \lambda \in {\mathbb {R}}^7 : b_1 = a_1 = a_0 b_0 = 0 \}$. If $c_2 (c_0+c_1) > 0$ then there is no small amplitude periodic orbit while when $c_2 (c_0+c_1) < 0$ then there is a local two-dimensional periodic invariant manifold of (4) through the origin if and only if the parameters are on the semi-algebraic set $V_{\mathcal {C}} \cap \Lambda$. Moreover, these manifolds appear in pairs and are foliated by periodic 1-orbits.

Proof

The first Poincaré-Lyapunov quantities are:

$$\begin{aligned} v_{01}^{[1]}(\nu , \lambda )= & {} \nu \pi b_1, \\ v_{20}^{[1]}(\nu , \lambda )= & {} - \nu \pi (2 a_0 b_0 + a_1 (c_0 + 3 c_1) - 4 b_1 (c_0 + c_1) \nu \pi )/8, \\ v_{11}^{[1]}(\nu , \lambda )= & {} - \pi a_0 b_1 \nu /3, \\ v_{02}^{[1]}(\nu , \lambda )= & {} \nu \pi (-a_1 b_1 - a_1 c_2 + b_1^2 \nu \pi + 2 b_1 c_2 \nu \pi )/2, \\ v_{30}^{[1]}(\nu , \lambda )= & {} \nu \pi (-2 a_0^2 b_0 + 7 a_0 a_1 c_0 + 8 b_0 b_1 c_0 + 5 a_0 a_1 c_1 + 16 b_0 b_1 c_1 + 4 a_0 b_1 c_0 \nu \pi \\&+\, 4 a_0 b_1 c_1 \nu \pi )/24, \\ v_{20}^{[2]}(\nu , \lambda )= & {} \nu \pi (c_0 + c_1), \\ v_{02}^{[2]}(\nu , \lambda )= & {} 2 \nu \pi c_2, \\ v_{30}^{[2]}(\nu , \lambda )= & {} 2 \nu \pi a_0 (c_0 + c_1)/3, \\ v_{03}^{[2]}(\nu , \lambda )= & {} \nu \pi c_2 (-a_1 + 4 c_2 \nu \pi ), \\ v_{21}^{[2]}(\nu , \lambda )= & {} \nu \pi (-a_1 c_0 + a_1 c_1 + 2 b_1 c_0 \nu \pi + 2 b_1 c_1 \nu \pi + 4 c_0 c_2 \nu \pi + 4 c_1 c_2 \nu \pi )/2. \end{aligned}$$

If we moreover assume that $c_0 + c_2 \ne 0$ then $v_{20}^{[2]} \ne 0$ and $v_{02}^{[2]} \ne 0$ and this is just the generic case developed with details in Sect. 2.1 where $d^{[2]}(\rho _0, z_0; \nu , \lambda ) = \left[ v_{20}^{[2]}(\nu , \lambda ) \rho _0^2 + O(\rho _0^3) \right] + \left[ v_{21}^{[2]}(\nu , \lambda ) \rho _0^2 + O(\rho _0^3) \right] z_0 + \left[ v_{02}^{[2]}(\nu , \lambda ) + O(\rho _0) \right] z_0^2 + O(z_0^3)$. In consequence the Newton’s diagram ${\mathbf {N}}(\Delta )$ only has one descending segment with endpoints (2, 0) and (0, 2), hence $(r,s) = (2, 2)$. In summary we will work in the parameter space $\Lambda = {\mathbb {R}}^7 \backslash {\mathbf {V}}(v_{02}^{[2]} v_{21}^{[2]})$ that is independent of $\nu$, that is, we will work on (7).

The determining polynomial is $D(\alpha _0; \nu , \lambda ) = \alpha _0^2 v_{02}^{[2]}(\nu , \lambda ) + v_{20}^{[2]}(\nu , \lambda )$, that is, $D(\alpha _0; \nu , \lambda ) = \nu \pi [c_0 + c_1 + 2 c_2 \alpha _0^2]$, hence

$$\begin{aligned} \alpha _0(\lambda ) = \pm \sqrt{- \frac{c_0 + c_1}{2 c_2}}. \end{aligned}$$

(25)

So if $v_{02}^{[2]} v_{20}^{[2]} < 0$ there is no real branch associated to the segment while two real branches appear when $v_{02}^{[2]} v_{20}^{[2]} > 0$ and both are simple with index $n=2$. The branch expansion (10) is $z^*(\sigma ^2; \nu , \lambda ) = \alpha _0(\lambda ) \sigma ^2 + \sum _{i \ge 1} \alpha _{2i}(\nu , \lambda ) \sigma ^{2i}$ with first coefficients

$$\begin{aligned} \alpha _2= & {} -(\alpha _0^3 v_{03}^{[2]} + \alpha _0^2 v_{11}^{[2]} + \alpha _0 v_{21}^{[2]} + v_{30}^{[2]})/(2 \alpha _0 v_{02}^{[2]}), \\ \alpha _4= & {} - (-5 \alpha _0^6 (v_{03}^{[2]})^2 + 4 \alpha _0^6 v_{02}^{[2]} v_{04}^{[2]} - 8 \alpha _0^5 v_{03}^{[2]} v_{12}^{[2]} -3 \alpha _0^4 (v_{12}^{[2]})^2 + 4 \alpha _0^5 v_{02}^{[2]} v_{13}^{[2]} \\&-\, 6 \alpha _0^4 v_{03}^{[2]} v_{21}^{[2]} - 4 \alpha _0^3 v_{12}^{[2]} v_{21}^{[2]} - \alpha _0^2 (v_{21}^{[2]})^2 + 4 \alpha _0^4 v_{02}^{[2]} v_{22}^{[2]} - 4 \alpha _0^3 v_{03}^{[2]} v_{30}^{[2]} - 2 \alpha _0^2 v_{12}^{[2]} v_{30}^{[2]} \nonumber \\&+\, (v_{30}^{[2]})^2 + 4 \alpha _0^3 v_{02}^{[2]} v_{31}^{[2]} + 4 \alpha _0^2 v_{02}^{[2]} v_{40}^{[2]} ) / (8 \alpha _0^3 (v_{02}^{[2]})^2). \end{aligned}$$

The reduced displacement map has an expansion (13) given by $\delta (\sigma ; \nu , \lambda ) = \sum _{i \ge 2} \delta _{2i}(\nu , \lambda ) \sigma ^{i}$ with first coefficients

$$\begin{aligned} \delta _4= & {} \alpha _0 v_{01}^{[1]}, \\ \delta _6= & {} \alpha _2 v_{01}^{[1]} + \alpha _0^2 v_{02}^{[1]} + \alpha _0 v_{11}^{[1]} + v_{20}^{[1]}, \\ \delta _8= & {} \alpha _4 v_{01}^{[1]} + 2 \alpha _0 \alpha _2 v_{02}^{[1]} + \alpha _0^3 v_{03}^{[1]} + \alpha _2 v_{11}^{[1]} + \alpha _0^2 v_{12}^{[1]} + \alpha _0 v_{21}^{[1]} + v_{30}^{[1]}. \end{aligned}$$

Working in the ring ${\mathcal {R}} = {\mathbb {R}}(\alpha _0, c_2)[\alpha _0, \lambda ]$ we obtain, up to positive constants, the first non-identically zero reduced coefficients ${{\hat{\delta }}}_{2i}$ are

$$\begin{aligned} {{\hat{\delta }}}_4(\alpha _0, \lambda )= & {} b_1 \alpha _0, \\ {{\hat{\delta }}}_6(\alpha _0, \lambda )= & {} - a_0 b_0 - a_1 c_1 - a_1 c_2 \alpha _0^2, \\ {{\hat{\delta }}}_8(\alpha _0, \lambda )= & {} - a_1^2 c_1 \alpha _0, \\ {{\hat{\delta }}}_{10}(\alpha _0, \lambda )= & {} - a_1 (60 a_0^2 c_1 - 6 b_0^2 c_1 + 100 a_0^2 c_2 \alpha _0^2 + 15 b_0^2 c_2 \alpha _0^2 - 36 c_1 c_2^2 \alpha _0^2 - 36 c_2^3 \alpha _0^4), \\ {{\hat{\delta }}}_{12}(\alpha _0, \lambda )= & {} - a_1^2 c_2 \alpha _0^3 (45 b_0^2 + 161 c_2^2 \alpha _0^2). \end{aligned}$$

We remark the fact that the computed ${{\hat{\delta }}}_{2i}$ do not depend on $\nu$ (in fact $\nu$ appears as a multiplicative constants that is deleted) and therefore the center variety ${\mathbf {V}}({\mathcal {I}})$ does not depend on $\nu$.

In this example $P_{2j} = {{\hat{\delta }}}_{2j}$ since there are no denominators in ${{\hat{\delta }}}_{2j}$ for $j=2,3,4,5$. We are going to characterize the set ${\mathbf {V}}({\mathcal {I}}_{12}) \cap E$. From $P_4 = P_8 = 0$ we see that either $b_1 = a_1 = 0$ or $b_1 = c_1 = 0$ and $a_1 \ne 0$. Let us study both cases separately:

• (I) Taking $b_1 = a_1 = 0$, $P_6$ vanishes if and only if $a_0 b_0 = 0$ and we fall into ${\mathbf {V}}({\mathcal {I}}_{12}) \cap E$.

• (II) Take $b_1 = c_1 = 0$ and $a_1 \ne 0$. Using the expression of $\alpha _0^2(\lambda )$ from (25), it follows that $P_6 = (2 a_0 b_0 - a_1 c_0)/2$, hence it vanishes if and only if $c_0 = (2 a_0 b_0)/a_1$. Then, up to positive constants, $P_{12} = - a_1^2 c_2 \alpha _0^3 (45 b_0^2 + 161 c_2^2 \alpha _0^2)$ which clearly never vanishes (recall that $c_2 \ne 0$ from (7)). Then there are no points in ${\mathbf {V}}({\mathcal {I}}_{12}) \cap E$.

In summary, introducing the restrictions of case (I) we get ${\mathbf {V}}({\mathcal {I}}_{12}) \cap E = \{ (\alpha _0, \lambda ) \in E : \lambda \in V_{\mathcal {C}} \cap \Lambda \}$ being the variety $V_{\mathcal {C}} = \{ \lambda \in {\mathbb {R}}^7 : b_1 = a_1 = a_0 b_0 = 0 \}$. Now we are going to prove that ${\mathbf {V}}({\mathcal {I}}) \cap E = {\mathbf {V}}({\mathcal {I}}_{12}) \cap E$ showing that if $\lambda ^* \in {\mathbf {V}}({\mathcal {I}}_{12}) \cap E$ and $d^{[1]}(\rho _0, z_0; \nu , \lambda ^*) \not \equiv 0$ then system (6) with $\lambda = \lambda ^*$ possess two local periodic invariant manifolds through the origin associated to both branches $z^*(\sigma ^2; \nu , \lambda ) = \alpha _0(\lambda ) \sigma ^2 + \cdots$ with $\alpha _0$ defined in (25). Depending on the null factor in the constrain $a_0 b_0 = 0$ we have that (6) with $\lambda = \lambda ^*$ is given by either family

$$\begin{aligned} {\dot{x}} = -y, \ \ {\dot{y}} = x + b_0 x y, \ \ {\dot{z}} = c_0 x^2 + c_1 y^2 + c_2 z^2 \end{aligned}$$

(26)

or family

$$\begin{aligned} {\dot{x}} = -y, \ \ {\dot{y}} = x + a_0 x^2, \ \ {\dot{z}} = c_0 x^2 + c_1 y^2 + c_2 z^2. \end{aligned}$$

(27)

In both cases the two first components are planar vector fields with a center at the origin. Actually (26) is reversible with respect to the x-axis while (27) is reversible with respect to the y-axis. Considering the initial conditions $(x_0, 0, z_0)$ for the solutions of (26) or (27), once more we find this familiar situation accompanied by the fact that the third equation ${\dot{z}} = c_0 x^2(t; x_0, \lambda ) + c_1 y^2(t; x_0, \lambda ) + c_2 z^2$ is a Riccati equation for the function $z(t; x_0, z_0, \lambda )$. Therefore we left the reader complete the proof following the same line of reasoning like in the proof of Proposition 12. $\square$

Proposition 20

We consider family (6) with parameter space $\Lambda \backslash V_{\mathcal {C}}$ being $\Lambda$ defined in (7) and $V_{\mathcal {C}}$ the variety defined in Proposition 19. Then, for any $\nu \in {\mathbb {Z}}^+$, the maximum number of $\nu$-limit cycles bifurcating from the origin is 8 and the bound is sharp, hence $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \backslash V_{\mathcal {C}}) = 8$.

Proof

We consider points $(\alpha _0^\dag , \lambda ^\dag ) \not \in {\mathbf {V}}({\mathcal {I}}) \cap E$, hence with $\lambda ^\dag \in \Lambda \backslash V_{\mathcal {C}}$. At some of these points one has ${{\hat{\delta }}}_4(\lambda ^\dag ) = {{\hat{\delta }}}_6(\lambda ^\dag ) = {{\hat{\delta }}}_8(\lambda ^\dag ) = {{\hat{\delta }}}_{10}(\lambda ^\dag ) = 0$ and ${{\hat{\delta }}}_{12}(\lambda ^\dag ) \ne 0$. For example the points $\lambda ^\dag$ with $b_1=c_1=0$, $a_1 = (2 a_0 b_0)/c_0$, $b_0^2 = -2 (50 a_0^2 + 9 c_0 c_2)/15$, $c_0 \ne 0$ and $a_0 (50 a_0^2 + 9 c_0 c_2) (600 a_0^2 + 269 c_0 c_2) \ne 0$ satisfy these restrictions. Using an argument like in Theorem 6(i) we deduce that for such points the maximum number of $\nu$-limit cycles associated to each branch is 4. We will see below that indeed this upper bound can be simultaneously reached in both branches.

Introducing the leading terms of the two branches $\alpha _0^\pm (\lambda ) = \pm \sqrt{- (c_0+c_1)/(2 c_2)}$ in the expressions of ${{\hat{\delta }}}_{i}^\pm (\nu , \lambda ) := {{\hat{\delta }}}_{i}(\alpha _0^\pm (\lambda ), \nu , \lambda )$, for a fixed arbitrary $\nu \in {\mathbb {Z}}^+$, we consider the maps

$$\begin{aligned} \lambda \mapsto {\mathfrak {F}}_\nu ^\pm (\lambda ) = ({{\hat{\delta }}}_4^\pm (\nu , \lambda ), {{\hat{\delta }}}_6^\pm (\nu , \lambda ), {{\hat{\delta }}}_8^\pm (\nu , \lambda ), {{\hat{\delta }}}_{10}^\pm (\nu , \lambda ), {{\hat{\delta }}}_{12}^\pm (\nu , \lambda )). \end{aligned}$$

It is easy to see that the set $\Lambda ^\dag = \{ \lambda \in \Lambda \backslash V_{\mathcal {C}} : \alpha _0^\pm (\lambda ^\dag ) \in {\mathbb {R}}^*, \mathrm{rank}({\mathfrak {F}}_\nu ^\pm )(\lambda ) = 5 \} \ne \emptyset$, check for example that $\lambda ^\dag _0 = (1, 36/115, 1, 0, 115/18, 0, -1) \in \Lambda ^\dag$. Therefore there are perturbations from any point in $\Lambda ^\dag$ producing 4 $\nu$-limit cycles bifurcating from the origin coming from each branch. We will take an explicit perturbation in order to see that the above mentioned bifurcations can be made simultaneously so that the $\nu$-cyclicity of 8 $\nu$-limit cycles is reached. To this end we take a perturbation $\lambda (\varepsilon ; \nu )$ with a real small parameter $\varepsilon$ and $\lambda (0; \nu ) = \lambda ^\dag _0$ such that $D(\alpha _0^\pm (\varepsilon ; \nu ); \nu , \lambda (\varepsilon ; \nu ) = 0$ (and, for simplicity, that $\alpha _0^\pm (\varepsilon ; \nu ) = \alpha _0^\pm (0; \nu )$) for all $\varepsilon$:

$$\begin{aligned} \lambda (\varepsilon ; \nu )= & {} (a_0(\varepsilon ; \nu ), a_1(\varepsilon ; \nu ), b_0(\varepsilon ; \nu ), b_1(\varepsilon ; \nu ), c_0(\varepsilon ; \nu ), c_1(\varepsilon ; \nu ), c_2(\varepsilon ; \nu )) \\= & {} \lambda ^\dag _0 + \left( A_0(\varepsilon ; \nu ), A_1(\varepsilon ; \nu ), 0, \varepsilon ^4, \varepsilon ^2, -\varepsilon ^2, 0 \right) \end{aligned}$$

with

$$\begin{aligned} A_0(\varepsilon ; \nu ) = \frac{36 \varepsilon ^2}{115} + \frac{4 \varepsilon ^3 \alpha _0^\pm }{\nu \pi } + \frac{\varepsilon (115 + 36 \varepsilon ^2) \alpha _0^\pm }{4140 (-4025 + 144 (\alpha _0^\pm )^4)}, \ A_1(\varepsilon ; \nu ) = \frac{\varepsilon \alpha _0^\pm }{115 (-4025 + 144 (\alpha _0^\pm )^4)}, \end{aligned}$$

and $\alpha _0^\pm = \pm \sqrt{115}/6$. The outcome is that ${{\hat{\delta }}}_4^\pm (\lambda (\varepsilon ; \nu )) = \beta _4^\pm (\nu ) \varepsilon ^4 + O(\varepsilon ^5)$, ${{\hat{\delta }}}_6^\pm (\lambda (\varepsilon ; \nu )) = \beta _6^\pm (\nu ) \varepsilon ^3 + O(\varepsilon ^4)$, ${{\hat{\delta }}}_8^\pm (\lambda (\varepsilon ; \nu )) = \beta _8^\pm (\nu ) \varepsilon ^2 + O(\varepsilon ^3)$, ${{\hat{\delta }}}_{10}^\pm (\lambda (\varepsilon ; \nu )) = \beta _{10}^\pm (\nu ) \varepsilon + O(\varepsilon ^2)$, and ${{\hat{\delta }}}_{12}^\pm (\lambda (\varepsilon ; \nu )) = \beta _{12}^\pm (\nu ) + O(\varepsilon )$ where the coefficients $\beta _i^\pm$ are independent of $\varepsilon$ and alternate sign as $\beta _{2i}^\pm \beta _{2i+2}^\pm < 0$ for $i=2,3,4,5$. The conclusion follows by the arguments exposed in Remark 9. $\square$

Proposition 21

We consider the family of vector fields ${\mathcal {X}}_\lambda$ associated to (6) with parameter space $\Lambda$ defined in (7) and $V_{\mathcal {C}}$ the variety defined in Proposition  19. Then the center variety ${\mathbf {V}}({\mathcal {I}})$ intersects E as ${\mathbf {V}}({\mathcal {I}}) \cap E = (C_1 \cup C_2) \cap E$ with the irreducible components

$$\begin{aligned} C_1= & {} \{ (\alpha _0, \lambda ) \in {\mathbb {R}}^{8} : D := c_0 + c_1 + 2 c_2 \alpha _0^2 = a_1 = b_0 = b_1 = 0 \}, \\ C_2= & {} \{ (\alpha _0, \lambda ) \in {\mathbb {R}}^{8} : D := c_0 + c_1 + 2 c_2 \alpha _0^2 = a_0 = a_1 = b_1 = 0 \}. \end{aligned}$$

Moreover, for any $\nu \in {\mathbb {Z}}^+$, there are points $(\alpha _0, \lambda ^*) \in C_i \cap E$ with $i=1,2$, such that there are bifurcations of (6) producing 2 $\nu$-limit cycles from the origin for parameter values $\lambda$ sufficiently close to $\lambda ^*$. In particular $\mathrm{Cyc}_\nu ({\mathcal {X}}_\lambda , 0, \Lambda \cap V_{\mathcal {C}}) \ge 2$ where $V_{\mathcal {C}}$ the variety defined in Proposition  19.

Proof

Recall that ${\mathbf {V}}({\mathcal {I}}) \cap E = {\mathbf {V}}({\mathcal {I}}_{12}) \cap E$ is independent of $\nu$ and ${\mathcal {I}}_{12} = \langle D, P_4, P_6, P_8, P_{10}, P_{12} \rangle$. Using Singular we obtain the irreducible components of ${\mathbf {V}}({\mathcal {I}}_{12})$ following the lines of Remark 10. From the prime decomposition $\sqrt{{\mathcal {I}}_{12}} = \cap _{j=1}^{10} {\mathcal {P}}_j$ it follows that the irreducible components of ${\mathbf {V}}({\mathcal {I}}_{12})$ are $C_j = {\mathbf {V}}({\mathcal {P}}_j)$. We obtain by simple inspection that $C_j \cap E = \emptyset$ for $j \ge 3$ and that

$$\begin{aligned} {\mathcal {P}}_1 = \langle D, a_1, b_0, b_1 \rangle , \ \ {\mathcal {P}}_2 = \langle D, a_0, a_1, b_1 \rangle . \end{aligned}$$

We take an arbitrary point $P=(\alpha _0, \lambda ^*) \in C_1 \cap E$, that is with $\lambda ^* = (a_0, 0, 0, 0, c_0, c_1, c_2) \in {\mathbb {R}}^{7}$. The map (16) with $\kappa \leqslant p = 7$ for the values $\kappa \in \{5, 4, 3 \}$ does not satisfy the condition $\mathrm{rank}(\nabla _\lambda F_\kappa (P)) = \kappa + 1$ of Theorem 8. But with $\kappa = 2$ it does, that is, the map

$$\begin{aligned} (\alpha _0, \lambda ) \mapsto F_2(\alpha _0, \lambda ) = (D(\alpha _0; \lambda ), {{\hat{\delta }}}_{4}(\alpha _0, \lambda ), {{\hat{\delta }}}_{6}(\alpha _0, \lambda )) \end{aligned}$$

has $\mathrm{rank}(\nabla _\lambda F_2(P)) = 3$ when either $a_0 \ne 0$ or $c_1 + c_2 \alpha _0^2 \ne 0$. But $\mathrm{codim}(C_1) = 4 \ne 3$ because $C_1 = G^{-1}(0)$ where $G: {\mathbb {R}}^8 \rightarrow {\mathbb {R}}^4$ is the map $(\alpha _0, \lambda ) \mapsto G(\alpha _0, \lambda )$ whose components are the generators of the ideal ${\mathcal {P}}_1$ and it is a submersion at any point $q \in C_1 \cap E$ since $\mathrm{rank}(d_q G) = 4$. In short we cannot use statement (ii) of Theorem 8 and only statement (i) of that theorem works. In particular, since ${\mathbf {N}}(\Delta )$ has only one segment with two associated branches with the same index, we conclude that there are perturbations of (6) starting at some points in $C_1 \cap E$ producing $2(\kappa -1) = 2$ $\nu$-limit cycles.

Regarding the bifurcations from points $P=(\alpha _0, \lambda ^*) \in C_2 \cap E$ we get, in a similar way that in the previous component, that there are points P with $\mathrm{rank}(\nabla _\lambda F_2(P)) = 3$ and $\mathrm{codim}(C_2) = 4$. Hence the obtain the same consequences about the $\nu$-limit cycles bifurcations from the origin. $\square$

3.5 A Challenging Family: Remarks to Explain Some Technical Difficulties

In this section we consider a specific three-dimensional quadratic system where the $\nu$-cyclicity of the zero-Hopf singularity of this system is inconclusive. It is included because we wanted to illustrate technical difficulties that may appear when working with the cyclicity problem. We take a look at the following quadratic family

$$\begin{aligned} \begin{array}{lll} {\dot{x}} &{}=&{} -y + z (a_1 x + a_2 y), \\ {\dot{y}} &{}=&{} x + z (b_1 x + b_2 y) \\ {\dot{z}} &{}=&{} c_1 x^2 + c_2 y^2 + c_3 z^2, \end{array} \end{aligned}$$

(28)

with parameter space $\Lambda = \{ \lambda = (a_1, a_2, b_1, b_2, c_1, c_2, c_3) \in {\mathbb {R}}^{7} : c_3 (c_1 + c_2) \ne 0 \}$. Then the restriction $v_{20}^{[2]}|_\Lambda \ne 0$ and $v_{02}^{[2]}|_\Lambda \ne 0$ hold and we fall in the generic case.

This example is different from the other examples in this paper by several reasons. First there appears reduced coefficients ${{\hat{\delta }}}_{2i}$ (the first one is ${{\hat{\delta }}}_{10}$) that depend on $\nu$ not only as a multiplicative constant. This fact implies that the center variety ${\mathbf {V}}({\mathcal {I}})$ does depend on $\nu$. To see that, after some routine calculations we may verify that the determining polynomial is $D(\alpha _0; \nu , \lambda ) = \nu \pi [c_1 + c_2 +2 c_3 \alpha _0^2]$, and computing in the ring ${\mathcal {R}} = {\mathbb {R}}(\alpha _0, c_3)[\alpha _0, \lambda ]$ we obtain, up to positive constants, the first non-identically zero reduced coefficients ${{\hat{\delta }}}_{2i}$ with $i=2,3,4,10$. The vanishing of ${{\hat{\delta }}}_4$ gives $b_2 = -a_1$ and then ${{\hat{\delta }}}_6(\alpha _0(\lambda ), \lambda ) = (a_2 + b_1) (c_1 - c_2)$. We continue assuming the particular case $b_1 = -a_2$ so that ${{\hat{\delta }}}_6$ vanishes and ${{\hat{\delta }}}_8(\alpha _0(\lambda ), \lambda ) = -a_1 (a_1 c_1 + a_1 c_2 + c_1 c_3 - c_2 c_3) \alpha _0(\lambda )$. We take the simplest subcase that vanishes ${{\hat{\delta }}}_8$, that is, $a_1=0$. Now we get ${{\hat{\delta }}}_{10}(\alpha _0(\lambda ), \lambda ) = a_2 (a_2 - 2 c_2) (c_1 + c_2)^2 (3 a_2^2 - 6 a_2 c_2 - 18 c_2^2 + a_2^2 \nu ^2 \pi ^2 - 10 a_2 c_2 \nu ^2 \pi ^2 + 24 c_2^2 \nu ^2 \pi ^2)$. Let’s take the particular choice $a_2 = 2 c_2$ that vanishes ${{\hat{\delta }}}_{10}$.

Even in this very particular case, a new challenge appears: in order to check if the set $\{ (\alpha _0, \lambda ) \in E : b_2 = a_1 = 0, b_1 = -a_2 = -2 c_2, \}$ is a component of the center variety ${\mathbf {V}}({\mathcal {I}})$ we need to check under what parameter conditions family

$$\begin{aligned} {\dot{x}} = -y (1 - 2 c_2 z), \ \ {\dot{y}} = x (1 - 2 c_2 z), \ \ {\dot{z}} = c_1 x^2 + c_2 y^2 + c_3 z^2, \end{aligned}$$

(29)

possesses a periodic invariant manifold ${\mathcal {M}}$ through the origin. We are unsuccessful (in the interesting nontrivial case $c_2 \ne 0$).

Appendix: Some Degenerate Cases

The degenerate cases appear when $v_{20}^{[2]}(\lambda ) v_{02}^{[2]}(\lambda ) = 0$. Here we give the explicit formulas of the first degenerate cases: either $v_{20}^{[2]}(\lambda ) = 0$ and $v_{02}^{[2]}(\lambda ) \ne 0$ or $v_{20}^{[2]}(\lambda ) \ne 0$ and $v_{02}^{[2]}(\lambda ) = 0$ which where not explicitly stated in [10].

1.1 The First Degenerate Case

We assume that $\partial ^2_x F_3(0,0,0; \lambda ) + \partial ^2_y F_3(0,0,0; \lambda ) = 0$ but $\partial ^2_z F_3(0,0,0; \lambda ) \ne 0$ which corresponds to $v_{20}^{[2]}(\lambda ) = 0$ and $v_{02}^{[2]}(\lambda ) \ne 0$.

1.1.1 The Case $v_{30}^{[2]}(\lambda ) \ne 0$

The Newton’s diagram ${\mathbf {N}}(\Delta )$ has just one descending segments ${\mathfrak {s}}$. More specifically the endpoints of ${\mathfrak {s}}$ are (0, 3) and (2, 0) so that $(r,s) = (3,2)$. Therefore we consider the parameter space $\Lambda = {\mathbf {V}}(v_{20}^{[2]}) \backslash {\mathbf {V}}(v_{02}^{[2]} v_{30}^{[2]})$.

Computations give the determining equation $D(\alpha _0; \lambda ) = \alpha _0^2 v_{02}^{[2]}(\lambda ) + v_{30}^{[2]}(\lambda ) = 0$, and in consequence if $v_{02}^{[2]}(\lambda ) v_{30}^{[2]}(\lambda ) < 0$ then there is no branch associated to the segment while two branches appear when $v_{02}^{[2]} v_{30}^{[2]} > 0$ and both branches are simple. Further computations give the following first coefficients $\alpha _i$ of the branch and $\delta _i$ of the reduced displacement map:

$$\begin{aligned} \alpha _1 = -v_{21}^{[2]} /(2 v_{02}^{[2]}), \ \ \alpha _2 = -(4 \alpha _0^2 v_{02}^{[2]} v_{12}^{[2]} - (v_{21}^{[2]})^2 + 4 v_{02}^{[2]} v_{40}^{[2]})/(8 \alpha _0 (v_{02}^{[2]})^2), \end{aligned}$$

and $\delta _5 = \alpha _0 v_{01}^{[1]}$, $\delta _6 = \alpha _1 v_{01}^{[1]}+ v_{20}^{[1]}$, $\delta _7 = \alpha _2 v_{01}^{[1]} + \alpha _0 v_{11}^{[1]}$.

1.1.2 The Case $v_{30}^{[2]}(\lambda ) = 0$ and $v_{21}^{[2]}(\lambda ) = 0$

We define $\ell \ge 4$ as the subindex such that $v_{j 0}^{[2]}(\lambda ) = 0$ for $j=2, \ldots , \ell -1$ but $v_{\ell \, 0}^{[2]}(\lambda ) \ne 0$. If $\ell = 4$ then ${\mathbf {N}}(\Delta )$ has only one edge with endpoints $(\ell , 0)$ and (2, 0), hence the slope is given by $(r,s) = (4, 2)$. Otherwise, if $\ell > 4$ then ${\mathbf {N}}(\Delta )$ has two descending segments ${\mathfrak {s}}_1$ and ${\mathfrak {s}}_2$, where the endpoints of ${\mathfrak {s}}_1$ are $(\ell , 0)$ and (1, 2) so that $(r,s) = (\ell -2, 1)$ while the endpoints of ${\mathfrak {s}}_2$ are (1, 2) and (2, 0) and therefore $(r,s) = (2,1)$.

Therefore we consider either the parameter space $\Lambda = {\mathbf {V}}(v_{20}^{[2]}, v_{21}^{[2]}, v_{30}^{[2]}) \backslash {\mathbf {V}}(v_{02}^{[2]} v_{4 0}^{[2]})$ when $\ell = 4$ or $\Lambda = {\mathbf {V}}(v_{21}^{[2]}, v_{j0}^{[2]} : j=2, \ldots , \ell -1) \backslash {\mathbf {V}}(v_{02}^{[2]} v_{\ell 0}^{[2]})$ when $\ell > 4$ .

1.1.3 The Case $v_{30}^{[2]}(\lambda ) = 0$ and $v_{21}^{[2]}(\lambda ) \ne 0$

We do not specify the computations but they are similar to the previous one.

1.2 The Second Degenerate Case

We assume here that $\partial ^2_x F_3(0,0,0; \lambda ) + \partial ^2_y F_3(0,0,0; \lambda ) \ne 0$ but $\partial ^2_z F_3(0,0,0; \lambda ) = 0$ which corresponds to $v_{20}^{[2]}(\lambda ) \ne 0$ and $v_{02}^{[2]}(\lambda ) = 0$.

1.2.1 The Case $v_{03}^{[2]}(\lambda ) \ne 0$

The Newton’s diagram ${\mathbf {N}}(\Delta )$ has just one descending segment ${\mathfrak {s}}$ with endpoints (0, 2) and (3, 0) so that $(r,s) = (2,3)$. Therefore we consider the parameter space $\Lambda = {\mathbf {V}}(v_{02}^{[2]}(\lambda )) \backslash {\mathbf {V}}(v_{20}^{[2]}(\lambda ) v_{03}^{[2]})$.

In this case the determining equation is $D(\alpha _0; \lambda ) = \alpha _0^3 v_{03}^{[2]}(\lambda ) + v_{20}^{[2]}(\lambda ) = 0$, hence there is just one branch associated to the segment, which is simple. The first coefficients $\alpha _i$ of the branch and $\delta _i$ of the reduced displacement map are:

$$\begin{aligned} \alpha _1 = -v_{12}^{[2]} /(3 v_{03}^{[2]}), \ \ \alpha _2 = -(3 \alpha _0^3 v_{03}^{[2]} v_{04}^{[2]} - (v_{12}^{[2]})^2 + 3 v_{03}^{[2]} v_{21}^{[2]})/(9 \alpha _0 (v_{03}^{[2]})^2), \end{aligned}$$

and $\delta _5 = \alpha _0 v_{01}^{[1]}$, $\delta _6 = \alpha _1 v_{01}^{[1]}$, $\delta _7 = \alpha _2 v_{01}^{[1]} + \alpha _0^2 v_{02}^{[1]}$.

1.2.2 The Case $v_{03}^{[2]}(\lambda ) = 0$

We left to the reader these specific computations.