1 Introduction

In the context of continuous-time dynamical systems, period-doubling bifurcation is a qualitative change of dynamics characterized by a splitting of a limit cycle which leads to an emergence of a new limit cycle whose period is approximately two times bigger than the original one. This phenomenon can be studied by tools of the theory of discrete-time dynamical systems using Poincaré maps. If this discrete-time dynamical system overcomes a flip bifurcation, then period-doubling occurs in the original dynamical system, Kuznetsov [10].

The mentioned discretization has its limitations which is demonstrated, for example, by the fact that the method does not provide an easily computable formula for normal form coefficients, Kuznetsov et al. [11]. Calculation of normal form coefficients of codimension one bifurcations of limit cycles in ordinary differential equations (ODE) was done in Kuznetsov et al. [11]. Their procedure depend on the normal form of a vector field in a neighbourhood of a limit cycle deduced in Iooss [8]. Recently, the existence of a smooth periodic center manifold in a neighbourhood of a non-hyperbolic periodic orbit in retarded functional differential equations (RFDE) has been proved, Lentjes et al. [12, Preprint]. Moreover, the existence of a suitable coordinate system on the manifold enables calculation of normal form coefficients of codimension one bifurcations of limit cycles in RFDE. These results has been accomplished using sun-star calculus, Diekmann et al. [4].

In this paper an alternative approach is presented. The following text is inspired by the eighth chapter of Golubitsky and Schaeffer [6]. In fact, we have adjusted techniques applied to investigation of Hopf bifurcation so that they can be used in the case of period-doubling bifurcation in RFDE. The first step of our approach consists of regarding periodic solutions of the original RFDE as zeros of an operator \(\varPhi \) acting between suitable Banach spaces. Subsequently, Lyapunov–Schmidt reduction can be applied in order to obtain a reduced map \(\phi \) operating between finite-dimensional spaces with the property that solutions of the equation \(\phi =0\) are in one to one correspondence with solutions of the equation \(\varPhi =0\). According to singularity theory and the mechanism of the reduction, derivatives of the map \(\phi \) at the bifurcation point determines a shape of its solution set near the point and they can be computed from derivatives of the original operator \(\varPhi \).

Sun-star calculus studies bifurcations in RFDE from the classical viewpoint of dynamical systems theory. On the other hand, the approach used by us consists in implementing very simple and general results from functional analysis and singularity theory. The computation of normal form coefficients of period-doubling bifurcation in this text can be considered as a demonstration of the robustness of these tools and a prospect for the successful application of this method in other and more general contexts. Although Lentjes et al. [12, Preprint], announced the computation of these coefficients and Szalai and Stépán [16], derived them for the special case where delay and period are equal, we provide explicit formulae in full generality.

We consider an autonomous retarded functional differential equation of the form

$$\begin{aligned} {\dot{x}}(t)=F(x_t,\lambda ) \end{aligned}$$
(1)

where \(F:C\times {\mathbb {R}}\rightarrow {\mathbb {R}}^n\) is supposed to have enough continuous derivatives. The state space \(C=C([-r,0],{\mathbb {R}}^n)\) where r is a given non-negative real number is endowed with the supremum norm \(\left\Vert \cdot \right\Vert \). The history \(x_t\) at time t is an element of the space C given by \(x_t(\theta )=x(t+\theta )\), where \(\theta \in [-r,0]\). The symbol \(\lambda \) represents the bifurcation parameter. We assume that this equation has a p-periodic solution \(u_0\) for \(\lambda =\lambda _0\).

The rest of the paper is organized as follows. In Sect. 2 we reduce the problem of finding periodic solutions of (1) to a problem of finding zeros of a function of the type \({\mathbb {R}}^2\rightarrow {\mathbb {R}}\). We derive sufficient conditions for an occurrence of period-doubling bifurcation and expressions for its normal form coefficients. In Sect. 3, the exchange of stability of solutions is proved in the case of the non-degenerate period-doubling bifurcation. We should emphasize that the theory is presented in the terminology of germs. Especially, most of the equalities occurring after Proposition 4 are considered as relations between germs rather than functions.

2 Normal form Coefficients

First of all, we want to transform equation (1) to an algebraic equation. Let us define the following sets of functions:

$$\begin{aligned} C^1_{q}&{:=}\{f :{\mathbb {R}}\rightarrow {\mathbb {R}}^n \mid f \in C^1({\mathbb {R}}) \text { and }f(t+q)=f(t) \text { for all }t \in {\mathbb {R}}\}, \\ C_{q}&{:=}\{f :{\mathbb {R}}\rightarrow {\mathbb {R}}^n \mid f \in C({\mathbb {R}}) \text { and }f(t+q)=f(t) \text { for all }t \in {\mathbb {R}}\}. \end{aligned}$$

In words, the set \(C^1_{q}\) contains all continuously differentiable q-periodic functions, meanwhile the symbol \(C_{q}\) represents all continuous q-periodic functions. In order to simplify subsequent expressions, we reserve the adjective q-periodic for functions satisfying the common defining condition of sets \(C^1_{q}\) and \(C_{q}\) so q is not necessarily their period. Obviously, \(C^1_{q}\) is a subset of \(C_{q}\). It is easy to see, that the equations

$$\begin{aligned} \left\Vert u\right\Vert _1&=\max _{t \in {\mathbb {R}}}\left| u(t)\right| +\max _{t \in {\mathbb {R}}}\left| u'(t)\right| , \\ \left\Vert u\right\Vert&=\max _{t \in {\mathbb {R}}}\left| u(t)\right| , \end{aligned}$$

where \(\left| {\cdot }\right| \) is a fixed norm in \({\mathbb {R}}^n\), define Banach spaces \((C^1_{q},\left\Vert {\cdot }\right\Vert _1)\) and \((C_{q},\left\Vert {\cdot }\right\Vert )\). Next we omit to mention the norms explicitly.

Since we suppose that \(u_0\) is a p-periodic solution of equation (1) and we are going to study period-doubling bifurcation, it is natural to choose \(C^1_{2p}\) and \(C_{2p}\) as a domain and range of the operator \(\varPhi \), respectively. However, there is a little complication. Generically, a period of a solution varies with a change in a bifurcation parameter. Hence, if \(\lambda \) is close to \(\lambda _0\), in general there will be solutions with periods close to p or 2p but not equal to these values. For this, we introduce an extra parameter to the studied equation, which enable us to solve this problem. Let us introduce a new time variable \(s=(1+\tau )t\), where \(\tau \in {\mathbb {R}}\) is the mentioned parameter. This leads to the following equation:

$$\begin{aligned} (1+\tau ){\dot{u}}(s)=F(u_{s,\tau },\lambda ), \end{aligned}$$
(2)

where \(u_{s,\tau }(\theta )=u(s+(1+\tau )\theta )\). If u(s) is a 2p-periodic solution of (2), then \(u((1+\tau )t)\) is a \(2p/(1+\tau )\)-periodic solution of (1). On the other hand, if x(t) is a \(2p/(1+\tau )\)-periodic solution of (1), then \(x(s/(1+\tau ))\) is a 2p-periodic solution of (2). Now we can define the operator \(\varPhi :C^1_{2p}\times {\mathbb {R}}\times {\mathbb {R}}\rightarrow C_{2p}\) by the following equation:

$$\begin{aligned} \varPhi (u,\lambda ,\tau ){:=}(1+\tau ){\dot{u}}(s)-F(u_{s,\tau },\lambda ). \end{aligned}$$
(3)

According to the previous analysis, solutions of the equation \(\varPhi =0\) are in one-to-one correspondence with periodic solutions of (1) whose period is approximately 2p.

Now we briefly mention an action of the group \(({\mathbb {R}},+)\) on the set \(C_{2p}\) called a phase shift. An element \(\alpha \in {\mathbb {R}}\) acts on a function \(u \in C_{2p}\) in the following way:

$$\begin{aligned} (\alpha \cdot u)(s)=u(s-\alpha ). \end{aligned}$$

It is easy to see that the previous formula defines a group action correctly. Since

$$\begin{aligned} (\alpha \cdot u)_{s,\tau }(\theta )=(\alpha \cdot u)(s+(1+\tau )\theta )=u(s-\alpha +(1+\tau )\theta )=u_{s-\alpha ,\tau }(\theta ) \end{aligned}$$

and (1) is autonomous, the operator \(\varPhi \) commutes with a phase shift:

$$\begin{aligned} \varPhi (\alpha \cdot u,\lambda ,\tau )(s)&=(1+\tau )\dfrac{\textrm{d}(\alpha \cdot u)}{\textrm{d}s}(s)-F((\alpha \cdot u)_{s,\tau },\lambda ) \\&=(1+\tau ){\dot{u}}(s-\alpha )-F(u_{s-\alpha ,\tau },\lambda ) \\&=\varPhi (u,\lambda ,\tau )(s-\alpha )=(\alpha \cdot \varPhi (u,\lambda ,\tau ))(s). \end{aligned}$$

Note that each element of the subgroup \(2p{\mathbb {Z}}\subseteq {\mathbb {R}}\) is a symmetry of all functions in \(C_{2p}\) which means that these elements represent transformations acting trivially on \(C_{2p}\). Of course, there are functions, such as p-periodic functions, which have more symmetries. Every element of the subgroup \(p{\mathbb {Z}}\) is a symmetry of the solution \(u_0\), for example.

The Fréchet differential, Chow and Hale [1], of \(\varPhi :C^1_{2p}\times {\mathbb {R}}\times {\mathbb {R}}\rightarrow C_{2p}\) with respect to the first variable at \((u_0,\lambda _0,0)\) is given by

$$\begin{aligned} (Lv)(s)=((\textrm{d}\varPhi )_{u_0,\lambda _0,0}\cdot v)(s)={\dot{v}}(s)-(\textrm{d}F)_{(u_0)_s,\lambda _0}\cdot v_s, \end{aligned}$$
(4)

where \((\textrm{d}F)_{(u_0)_s,\lambda _0}\) is the derivative of F with respect to the first variable evaluated at \(((u_0)_s,\lambda _0)\). The operator \(K :{\mathbb {R}}\times C \rightarrow {\mathbb {R}}^n\) defined by \(K(s,\phi )=(\textrm{d}F)_{(u_0)_s,\lambda _0}\cdot \phi \) is p-periodic in the first variable and continuous and linear in the second variable. Hence it can be expressed as follows

$$\begin{aligned} K(s,\phi )=\int _{-r}^0\textrm{d}_\theta \eta (s,\theta )\phi (\theta ), \end{aligned}$$

where \(\eta \) is continuous from the left in \(\theta \) on \((-r,0)\), has bounded variation in \(\theta \) on \([-r,0]\) and is normalized so that \(\eta (s,\theta )=0\) for \(\theta \ge 0\) and \(\eta (s,\theta )=\eta (s,-r)\) for \(\theta \le -r\), see Hale [7].

Proposition 1

The linearization \(L :C_{2p}^1 \rightarrow C_{2p}\) of the operator \(\varPhi \) is a Fredholm operator of index zero.

Proof

It is easily seen that \(\dot{({\cdot })}:C_{2p}^1 \rightarrow C_{2p}\) is a Fredholm operator of index zero. According to Schechter [14], it is enough to justify compactness of the operator \(B :C_{2p}^1 \rightarrow C_{2p}\) defined by \((Bv)(s)=K(s,v_s)\). Let \(k=\max _{s\in {\mathbb {R}}}\left\Vert K(s,{\cdot })\right\Vert _{{\mathcal {L}}(C,{\mathbb {R}}^n)}\) which exists since \((\textrm{d}F)_{(u_0)_{\cdot },\lambda _0} :{\mathbb {R}} \rightarrow {\mathcal {L}}(C,{\mathbb {R}}^n)\) is continuous and periodic. If \(M \subseteq C_{2p}^1\) is bounded, then there is \(m \in {\mathbb {R}}\) such that \(\left\Vert v\right\Vert _1\le m\) for every \(v \in M\). Since

$$\begin{aligned} \left\Vert Bv\right\Vert =\max _{s\in {\mathbb {R}}}\left| Bv(s)\right| \le \max _{s\in {\mathbb {R}}}\left\Vert K(s,{\cdot })\right\Vert _{{\mathcal {L}}(C,{\mathbb {R}}^n)}\cdot \left\Vert v_s\right\Vert \le km, \end{aligned}$$

the set BM is uniformly bounded. It remains to verify uniform continuity. We start with the following calculations:

$$\begin{aligned}&\left| Bv(s)\right. -\left. Bv(t)\right| =\left| K(s,v_s)-K(t,v_t)\right| \\&\quad =\left| (\textrm{d}F)_{(u_0)_{s},\lambda _0}\cdot v_s-(\textrm{d}F)_{(u_0)_{t},\lambda _0}\cdot v_t\right| \\&\quad =\left| (\textrm{d}F)_{(u_0)_{s},\lambda _0}\cdot v_s-(\textrm{d}F)_{(u_0)_{s},\lambda _0}\cdot v_t+(\textrm{d}F)_{(u_0)_{s},\lambda _0}\cdot v_t-(\textrm{d}F)_{(u_0)_{t},\lambda _0}\cdot v_t\right| \\&\quad \le \left\Vert (\textrm{d}F)_{(u_0)_{s},\lambda _0}\right\Vert _{{\mathcal {L}}(C,{\mathbb {R}}^n)}\left\Vert v_s-v_t\right\Vert +\left\Vert (\textrm{d}F)_{(u_0)_{s},\lambda _0}-(\textrm{d}F)_{(u_0)_{t},\lambda _0}\right\Vert _{{\mathcal {L}}(C,{\mathbb {R}}^n)}\left\Vert v_t\right\Vert \\&\quad \le k \left\Vert v_s-v_t\right\Vert +m\left\Vert (\textrm{d}F)_{(u_0)_{s},\lambda _0}-(\textrm{d}F)_{(u_0)_{t},\lambda _0}\right\Vert _{{\mathcal {L}}(C,{\mathbb {R}}^n)}. \end{aligned}$$

There is \(\delta _1>0\) such that for every \(v \in M\) and \(s,t\in {\mathbb {R}}\) satisfying \(\left| s-t\right| <\delta _1\) the following inequalities hold:

$$\begin{aligned} \left| v(s)-v(t)\right|<\dfrac{\varepsilon }{2k} \quad \Rightarrow \quad \left| v_s(\theta )-v_t(\theta )\right|<\dfrac{\varepsilon }{2k} \quad \Rightarrow \quad \left\Vert v_s-v_t\right\Vert <\dfrac{\varepsilon }{2k}. \end{aligned}$$

On the other hand, continuity and periodicity of \((\textrm{d}F)_{(u_0)_{\cdot },\lambda _0} :{\mathbb {R}} \rightarrow {\mathcal {L}}(C,{\mathbb {R}}^n)\) implies its uniform continuity. Therefore, there is \(\delta _2>0\) such that for every \(s,t\in {\mathbb {R}}\) satisfying \(\left| s-t\right| <\delta _2\) we have

$$\begin{aligned} \left\Vert (\textrm{d}F)_{(u_0)_{s},\lambda _0}-(\textrm{d}F)_{(u_0)_{t},\lambda _0}\right\Vert _{{\mathcal {L}}(C,{\mathbb {R}}^n)}<\dfrac{\varepsilon }{2m}. \end{aligned}$$

We can conclude that for a given \(\varepsilon >0\) and \(v \in M\) an appropriate \(\delta =\min \{\delta _1,\delta _2\}\) can be found so that for every \(s,t\in {\mathbb {R}}\) satisfying \(\left| s-t\right| <\delta \) the following inequality hold:

$$\begin{aligned} \left| Bv(s)-Bv(t)\right| \le k\cdot \dfrac{\varepsilon }{2k}+m\cdot \dfrac{\varepsilon }{2m}=\varepsilon . \end{aligned}$$

The precompactness of the set BM has been proved. \(\square \)

Let us continue with solutions of the equation \(L=0\). According to Hale [7], for any \(\sigma \in {\mathbb {R}}\) and \(\phi \in C\) there is a unique solution \(x :[\sigma -r,\infty )\rightarrow {\mathbb {R}}^n\) of the equation \(L=0\) satisfying the initial condition \(x_\sigma =\phi \). The solution operator \(T(s,\sigma ):C\rightarrow C\) for \(s\ge \sigma \) is defined by \(T(s,\sigma )\phi =x_s\). Recall that non-zero eigenvalues of the monodromy operator \(T(\sigma +p,\sigma )\) do not depend on \(\sigma \) and they are called Floquet multipliers of \(u_0\). From now on, we will adopt the following assumption:

  1. (C1)

    The only Floquet multipliers of \(u_0\) satisfying \(\gamma ^2=1\) are \(\gamma _1=1\) and \(\gamma _2=-1\) and these two eigenvalues are simple.

Proposition 2

If \(u_0\) satisfies the condition (C1), then \(\dim \ker L=2\).

Proof

According to the assumption, 1 is an eigenvalue of the operator

$$\begin{aligned} T(2p,0)=T(2p,p)T(p,0)=T(p,0)T(p,0)=T(p,0)^2, \end{aligned}$$

of which both algebraic and geometric multiplicity are equal to two. Let \(({\dot{u}}_0)_0\) and \(\phi \) be corresponding eigenvectors such that \(T(p,0)\phi =-\phi \). Obviously, the 2p-periodic function \({\dot{u}}_0(s)\) and the 2p-periodic extension of the function \(v_0(s)=T(s,0)\phi (0)\) span the kernel of the operator L, which leads to \(\dim \ker L=2\). \(\square \)

It should be emphasized that the function \(v_0\) has the following property: \(v_0(s+p)=-v_0(s)\) where \(s\in {\mathbb {R}}\) is arbitrary.

Next, we are going to use the concept of the formal adjoint equation given by the following relation:

$$\begin{aligned} y(s)+\int _s^\infty y(\beta )\eta (\beta ,s-\beta )\,\textrm{d}\beta =\text {constant}, \end{aligned}$$
(5)

where \(y(s)\in {\mathbb {R}}^{n*}\). Let \(B_0\) denote the Banach space of functions \(\psi :[-r,0]\rightarrow {\mathbb {R}}^{n*}\) of bounded variation on \([-r,0]\), continuous from the left on \((-r,0)\) and vanishing at zero with norm \(\textrm{Var}_{[-r,0]}\psi \). According to Hale [7], for every \(t \in {\mathbb {R}}\) and \(\psi \in B_0\) there is a unique function \(y:{\mathbb {R}}\rightarrow {\mathbb {R}}^{n*}\) which vanishes on \([t,\infty )\), satisfies the equation (5) on \((-\infty ,t-r]\) and \(y_t=\psi \). The solution operator \({\widetilde{T}}(s,t):B_0\rightarrow B_0\) for \(s\le t\) is defined by \({\widetilde{T}}(s,t)\psi =y^0_s\) where \(y^0_s(0)=0\) and \(y^0_s(\theta )=y(s+\theta )\) for \(-r\le \theta <0\).

Recall that spectra of T(p, 0) and \(T(2p,0)=T(p,0)^2\) are the same as those of \({\widetilde{T}}(0,p)\) and \({\widetilde{T}}(0,2p)={\widetilde{T}}(0,p)^2\), respectively. Moreover, generalized eigenspaces of these operators can be used to decompose spaces C and \(B_0\) in the following way:

$$\begin{aligned} C&=\ker (I-T(p,0))\oplus {{\,\textrm{Im}\,}}(I-T(p,0))=\ker (I+T(p,0))\oplus {{\,\textrm{Im}\,}}(I+T(p,0)) \\&=\ker (I-T(2p,0))\oplus {{\,\textrm{Im}\,}}(I-T(2p,0)), \\ B_0&=\ker (I-{\widetilde{T}}(0,p))\oplus {{\,\textrm{Im}\,}}(I-{\widetilde{T}}(0,p))=\ker (I+{\widetilde{T}}(0,p))\oplus {{\,\textrm{Im}\,}}(I+{\widetilde{T}}(0,p)) \\&=\ker (I-{\widetilde{T}}(0,2p))\oplus {{\,\textrm{Im}\,}}(I-{\widetilde{T}}(0,2p)), \end{aligned}$$

where

$$\begin{aligned} \dim \ker (I-T(p,0))&=\dim \ker (I-{\widetilde{T}}(0,p))=1, \\ \dim \ker (I+T(p,0))&=\dim \ker (I+{\widetilde{T}}(0,p))=1, \\ \dim \ker (I-T(2p,0))&=\dim \ker (I-{\widetilde{T}}(0,2p))=2. \end{aligned}$$

Let \(\psi _1\) and \(\psi _2\) be eigenvectors of the operator \({\widetilde{T}}(0,p)\) corresponding to eigenvalues 1 and \(-1\), respectively. Obviously, these vectors span the eigenspace of the operator \({\widetilde{T}}(0,2p)\) corresponding to 1. Let \(v_1\) and \(v_2\) be 2p-periodic extensions of solutions of (5) satisfying the initial conditions \((v_1)_{2p}=\psi _1\) and \((v_2)_{2p}=\psi _2\). These functions span the space of 2p-periodic solutions of (5). Moreover, it can be easily seen that \(v_1\) and \(v_2\) are continuous.

The formal adjoint equation is necessary for decomposition of spaces \(C^1_{2p}\) and \(C_{2p}\) so that Lyapunov–Schimdt reduction can be used. This is the right time to recall an important theorem from Hale [7] (Corollary 5.1 from Chapter 6 or Theorem 1.2 from Chapter 9):

Theorem 1

If \(f \in C_{2p}\), then the equation \(Lv=f\) has a 2p-periodic solution v if and only if \(\langle u,f\rangle =0\) for all 2p-periodic solutions u of the formal adjoint equation.

An important consequence of this statement is the inclusion \({{\,\textrm{Im}\,}}L\subseteq [v_1,v_2]^\perp \). We can conclude the following proposition.

Proposition 3

If the solution \(u_0\) satisfies the condition (C1), then the spaces \(C^1_{2p}\) and \(C_{2p}\) can be expressed as the following sums:

$$\begin{aligned} C^1_{2p}=\ker L\oplus M,\qquad C_{2p}=N\oplus {{\,\textrm{Im}\,}}L, \end{aligned}$$

where \(M{:=}(\ker L)^\perp \) and \(N{:=}[L(s{\dot{u}}_0(s)),L(sv_0(s))]\). Moreover, spaces M and N are invariant under the action \(v({\cdot })\mapsto v({\cdot }+p)\).

Proof

The first splitting is obvious. According to Propositions 2 and 1 it is enough to show that \({{\,\textrm{Im}\,}}L\cap N=0\) in order to verify the second equality. However, it is a simple consequence of the previous theorem and the following relations

$$\begin{aligned} \langle v_2,L(s{\dot{u}}_0(s))\rangle =\langle v_1,L(sv_0(s))\rangle&=0, \nonumber \\ \langle v_1,L(s{\dot{u}}_0(s))\rangle&\ne 0\ne \langle v_2,L(sv_0(s))\rangle , \end{aligned}$$
(6)

which will be proven latter. \(\square \)

Before we proceed to Lyapunov–Schmidt reduction, we implement a little modification of the operator \(\varPhi \). Let us define an operator \({\widetilde{\varPhi }} :C_{2p}^1\times {\mathbb {R}}\times {\mathbb {R}}\rightarrow C_{2p}\times {\mathbb {R}}\) by the following rule:

$$\begin{aligned} {\widetilde{\varPhi }}(u,\lambda ,\tau ){:=}\left( \varPhi (u,\lambda ,\tau ),\int _{0}^{2p}\langle u(s),{\dot{u}}_{0}(s)\rangle \,\textrm{d}s\right) . \end{aligned}$$

The invariance of the operator \(\varPhi \) under a time shift adds one more unnecessary dimension to the reduced problem. The reduced map can be simplified by the second component of the operator \({\widetilde{\varPhi }}\) which fixes a phase of solutions, Kuznetsov [10]. The Fréchet differential of \({\widetilde{\varPhi }}\) with respect to the first variable at the point \((u_0,\lambda _0,0)\) can be expressed in the following way:

$$\begin{aligned} {\widetilde{L}}v=(\textrm{d}{\widetilde{\varPhi }})_{u_0,\lambda _0,0}\cdot v=\left( Lv,\int _{0}^{2p}\langle v(s),{\dot{u}}_{0}(s)\rangle \,\textrm{d}s\right) . \end{aligned}$$

Since \(\ker {\widetilde{L}}=[v_0]\) and \({{\,\textrm{Im}\,}}{\widetilde{L}}={{\,\textrm{Im}\,}}L\times {\mathbb {R}}\), spaces \(C_{2p}^1\) and \(C_{2p}\times {\mathbb {R}}\) can be decomposed in a way similar to the one in the last proposition:

$$\begin{aligned} C_{2p}^1=\ker {\widetilde{L}} \oplus {\widetilde{M}},\qquad C_{2p}\times {\mathbb {R}}={{\,\textrm{Im}\,}}{\widetilde{L}}\oplus (N\times \{0\}), \end{aligned}$$

where \({\widetilde{M}}{:=}M+[{\dot{u}}_0]\).

Lyapunov–Schmidt reduction is clearly described in Golubitsky and Schaeffer [6], so we mention only necessary aspects of the procedure. Let \(E:C_{2p}\rightarrow C_{2p}\) denote the projection of \(C_{2p}\) onto \({{\,\textrm{Im}\,}}L\) with \(\ker E=N\). The range and the kernel of the complementary projection \(I-E\) are N and \({{\,\textrm{Im}\,}}L\), respectively. Obviously, the equation \({\widetilde{\varPhi }}(u,\lambda ,\tau )=0\) is equivalent to the following set of equations:

$$\begin{aligned} E\varPhi (u,\lambda ,\tau )=0,\quad (I-E)\varPhi (u,\lambda ,\tau )=0, \quad \langle u,{\dot{u}}_0\rangle =0. \end{aligned}$$
(7)

The left-hand sides of the first and last equations can be considered as an operator \((E\varPhi ,\langle {\cdot },{\dot{u}}_0\rangle ) :\ker {\widetilde{L}} \times {\widetilde{M}}\times {\mathbb {R}}^2 \rightarrow {{\,\textrm{Im}\,}}{\widetilde{L}}\). The differential of this operator with respect to the second variable evaluated at \((u_0,\lambda _0,0)\) is invertible. The implicit function theorem in Banach spaces implies existence of a function \(W :\ker {\widetilde{L}} \times {\mathbb {R}}^2 \rightarrow {\widetilde{M}}\) defined in a neighbourhood of \((u_0,\lambda _0,0)\) which is uniquely determined by equations

$$\begin{aligned} E\varPhi (v+W(v,\lambda ,\tau ),\lambda ,\tau )=0,\,\, \langle v+W(v,\lambda ,\tau ),{\dot{u}}_0\rangle =0,\,\, W(0,\lambda _0,0)=u_0. \end{aligned}$$
(8)

where \(v \in \ker {\widetilde{L}}\). In the last equation we have used the fact that \(u_0\in {\widetilde{M}}\). If we substitute \(u=v+W(v,\lambda ,\tau )\) into the left-hand side of the second equation of (7), then we obtain the reduced mapping \(\phi :\ker {\widetilde{L}}\times {\mathbb {R}}^2\rightarrow N\) defined by

$$\begin{aligned} \phi (v,\lambda ,\tau ){:=}(I-E)\varPhi (v+W(v,\lambda ,\tau ),\lambda ,\tau ). \end{aligned}$$

An important consequence of the previous analysis is a one-to-one correspondence between solutions of the equations \({\widetilde{\varPhi }}=0\) and \(\phi =0\).

This is the right time to discuss how symmetry enters our problem. Since \(\varPhi \) commutes with the action of \(({\mathbb {R}},+)\), the equality \(\varPhi (\alpha u,\lambda ,\tau )=\alpha \varPhi (u,\lambda ,\tau )\) holds for every \(\alpha \in {\mathbb {R}}\) and \((u,\lambda ,\tau ) \in C^1_{2p}\times {\mathbb {R}}\times {\mathbb {R}}\). Differentiation with respect to \(C^1_{2p}\) and evaluation at \((u_0,\lambda _0,0)\) gives the following equation:

$$\begin{aligned} (\textrm{d}\varPhi )_{\alpha u_0,\lambda _0,0}\cdot \alpha =\alpha \cdot (\textrm{d}\varPhi )_{u_0,\lambda _0,0}. \end{aligned}$$

Generally, we can not deduce \(L\alpha =\alpha L\), because \(u_0\) is a non-trivial function which means that the equation \(\alpha u_0=u_0\) does not hold for every \(\alpha \in {\mathbb {R}}\). However, the function \(u_0\) is p-periodic, hence it has more symmetries than a general element of the space \(C_{2p}\). Particularly, the isotropy subgroup of \(u_0\) is \(p{\mathbb {Z}}\), unlike the isotropy subgroup \(2p{\mathbb {Z}}\) of a general function in \(C_{2p}\). We can factor \({\mathbb {R}}\) by its subgroup \(2p{\mathbb {Z}}\) and think about the action of the circle group \(S^1\cong {\mathbb {R}}/2p{\mathbb {Z}}\) on the space \(C_{2p}\). In this context, the isotropy subgroup of \(u_0\) is \(\{0,p\}\subseteq S^1\), which is isomorphic to \({\mathbb {Z}}_2\). We can conclude that the equation \(L\alpha =\alpha L\) holds for every \(\alpha \in {\mathbb {Z}}_2\), so the operator L commutes with the action of \({\mathbb {Z}}_2\) on the space \(C_{2p}\). This result has the following important consequences, whose proofs can be found in Golubitsky and Schaeffer [6]. Spaces \(\ker L\) and \({{\,\textrm{Im}\,}}L\) are \({\mathbb {Z}}_2\)-invariant and the projection E also commutes with the action of \({\mathbb {Z}}_2\). Moreover, \({\mathbb {Z}}_2\)-invariance of spaces \(\ker {{\widetilde{L}}}\) and \({\widetilde{M}}\) and the fact that W commutes with the action of \({\mathbb {Z}}_2\) can be also verified by a simple calculation. Finally, \({\mathbb {Z}}_2\)-equivariance of the reduced mapping \(\phi \) can be concluded:

$$\begin{aligned} \phi (\alpha v,\lambda ,\tau )=\alpha \phi (v,\lambda ,\tau ). \end{aligned}$$

This fact enforces a special form of the reduced mapping which will be seen in a moment.

In place of abstract subspaces of a Banach space, it is more comfortable to work with Euclidean spaces, hence we try to find an appropriate coordinate representation of \(\phi \). We can use ordered bases \((v_0)\) and \((L(s{\dot{u}}_0(s)),L(sv_0(s)))\) of spaces \(\ker {\widetilde{L}}\) and N, respectively, and consider W and \(\phi \) as mappings \(W:{\mathbb {R}}\times {\mathbb {R}}^2\rightarrow {\widetilde{M}}\) and \(\phi :{\mathbb {R}}\times {\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\), respectively: \(W(x,\lambda ,\tau )=W(xv_0,\lambda ,\tau )\),

$$\begin{aligned} \phi (x,\lambda ,\tau )&=\left( \begin{matrix}\phi _1(x,\lambda ,\tau ) \\ \phi _2(x,\lambda ,\tau )\end{matrix}\right) \\&=\left( \begin{matrix}\langle v_1,(I-E)\varPhi (xv_0+W(x,\lambda ,\tau ),\lambda ,\tau )\rangle /\langle v_1,L(s{\dot{u}}_0(s))\rangle \\ \langle v_2,(I-E)\varPhi (xv_0+W(x,\lambda ,\tau ),\lambda ,\tau )\rangle /\langle v_2,L(sv_0(s))\rangle \end{matrix}\right) \\&=\left( \begin{matrix}\langle v_1,\varPhi (xv_0+W(x,\lambda ,\tau ),\lambda ,\tau )\rangle /\langle v_1,L(s{\dot{u}}_0(s))\rangle \\ \langle v_2,\varPhi (xv_0+W(x,\lambda ,\tau ),\lambda ,\tau )\rangle /\langle v_2,L(sv_0(s))\rangle \end{matrix}\right) , \end{aligned}$$

where the penultimate equality can be derived from \(\phi =\phi _1\cdot L(s{\dot{u}}_0(s))+\phi _2\cdot L(sv_0(s))\) and (6) and the last equality is a consequence of \({{\,\textrm{Im}\,}}L \subseteq [v_1,v_2]^\perp \) and \({{\,\textrm{Im}\,}}E={{\,\textrm{Im}\,}}L\). The subsequent calculations can be simplified by involving the linear transformation

$$\begin{aligned} S=\left( \begin{matrix}\langle v_1,L(s{\dot{u}}_0(s))\rangle &{} 0 \\ 0 &{} \langle v_2,L(sv_0(s))\rangle \end{matrix}\right) , \end{aligned}$$

which leads to the mapping

$$\begin{aligned} g(x,\lambda ,\tau )=\left( \begin{matrix}g_1(x,\lambda ,\tau ) \\ g_2(x,\lambda ,\tau )\end{matrix}\right) =S\phi (x,\lambda ,\tau )=\left( \begin{matrix}\langle v_1,\varPhi (xv_0+W(x,\lambda ,\tau ),\lambda ,\tau )\rangle \\ \langle v_2,\varPhi (xv_0+W(x,\lambda ,\tau ),\lambda ,\tau )\rangle \end{matrix}\right) . \end{aligned}$$

This function has qualitatively the same solution set as \(\phi \) and, moreover, the transformation S has no influence on \({\mathbb {Z}}_2\)-equivariance. Therefore, we can deduce the following properties of components of the reduced map: \(g_1(-x,\lambda ,\tau )=g_1(x,\lambda ,\tau )\) and \(g_2(-x,\lambda ,\tau )=-g_2(x,\lambda ,\tau )\). This simply says that \(g_1\) and \(g_2\) are respectively even and odd functions of x. We can find germs \(a(z,\lambda ,\tau )\) and \(b(z,\lambda ,\tau )\) satisfying \(g_1(x,\lambda ,\tau )=a(x^2,\lambda ,\tau )\) and \(g_2(x,\lambda ,\tau )=xb(x^2,\lambda ,\tau )\). A proof of this statement can be found, for example, in Golubitsky and Schaeffer [6].

The following proposition enables us to leave out the parameter \(\tau \).

Proposition 4

The germ a satisfies the following relations:

$$\begin{aligned} a(0,\lambda _0,0)=0,\qquad a_\tau (0,\lambda _0,0)\ne 0. \end{aligned}$$

Proof

The first statement can be simply proved by substitution:

$$\begin{aligned} a(0,\lambda _0,0)=\langle v_1,\varPhi (W(0,\lambda _0,0),\lambda _0,0)\rangle =\langle v_1,\varPhi (u_0,\lambda _0,0)\rangle =\langle v_1,0\rangle =0. \end{aligned}$$

The inequality requires differentiation:

$$\begin{aligned} a_\tau (0,\lambda _0,0)&=\langle v_1, (\textrm{d}\varPhi )_{u_0,\lambda _0,0}\cdot W_\tau (0,\lambda _0,0)+\varPhi _\tau (u_0,\lambda _0,0)\rangle \\&=\langle v_1, \varPhi _\tau (u_0,\lambda _0,0)\rangle , \end{aligned}$$

where the inclusion \({{\,\textrm{Im}\,}}L \subseteq [v_1,v_2]^\perp \) is used in the last equality. The idea behind verification of the inequality \(\langle v_1, \varPhi _\tau (u_0,\lambda _0,0)\rangle \ne 0\) comes from the proof of Theorem 2.2 in Chapter 10 in Hale [7].

Firstly, we evaluate the derivative \(\varPhi _\tau \) at the point \((u_0,\lambda _0,0)\):

$$\begin{aligned} \varPhi _\tau (u_0,\lambda _0,0)(s)={\dot{u}}_0(s)-(\textrm{d}F)_{(u_0)_s}\cdot ({\cdot })({\dot{u}}_0)_s({\cdot })={\dot{u}}_0(s)-K(s,({\cdot })({\dot{u}}_0)_s({\cdot })). \end{aligned}$$

Note that \(\langle v_2, \varPhi _\tau (u_0,\lambda _0,0)\rangle =0\), since \(\varPhi _\tau (u_0,\lambda _0,0)\) is a p-periodic function. Let us suppose that \(\langle v_1,\varPhi _\tau (u_0,\lambda _0,0)\rangle = 0\). Consequently, the equation

$$\begin{aligned} {\dot{z}}(s)=K(s,z_s)+\varPhi _\tau (u_0,\lambda _0,0)(s) \end{aligned}$$
(9)

has a non-trivial 2p-periodic solution \(z_0(s)\). Let \(x_0(s)=z_0(s)-s{\dot{u}}_0(s)\). The following calculations show that \(x_0\) is a solution of the equation \(L=0\). Since

$$\begin{aligned} (z_0)_s(\theta )&=z_0(s+\theta )=x_0(s+\theta )+s{\dot{u}}_0(s+\theta )+\theta {\dot{u}}_0(s+\theta ) \\&=(x_0)_s(\theta )+s({\dot{u}}_0)_s(\theta )+\theta ({\dot{u}}_0)_s(\theta ), \end{aligned}$$

we can conclude that

$$\begin{aligned} {\dot{x}}_0(s)&={\dot{z}}_0(s)-\dot{u_0}(s)-s\ddot{u}_0(s) \\&=K(s,(z_0)_s)+\varPhi _\tau (u_0,\lambda _0,0)(s)-\dot{u_0}(s)-sK(s,({\dot{u}}_0)_s) \\&=K(s,(z_0)_s)+{\dot{u}}_0(s)-K(s,({\cdot })({\dot{u}}_0)_s({\cdot }))-\dot{u_0}(s)-sK(s,({\dot{u}}_0)_s) \\&=K(s,(x_0)_s)+sK(s,({\dot{u}}_0)_s)+K(s,({\cdot })({\dot{u}}_0)_s({\cdot }))-K(s,({\cdot })({\dot{u}}_0)_s({\cdot })) \\&-sK(s,({\dot{u}}_0)_s) \\&=K(s,(x_0)_s). \end{aligned}$$

This fact together with the equation

$$\begin{aligned} T(s+2p,s)(x_0)_s&=(x_0)_{s+2p}=(z_0)_s-({\cdot })({\dot{u}}_0)_s-s({\dot{u}}_0)_s-2p({\dot{u}}_0)_s \\&=(x_0)_s-2p({\dot{u}}_0)_s \end{aligned}$$

implies that \((T(2p,0)-I)(x_0)_0=-2p({\dot{u}}_0)_0\), which contradicts the fact that the geometric multiplicity of 1 is equal to its algebraic multiplicity as an eigenvalue of the operator T(2p, 0). \(\square \)

We can conclude that the equation \(a(x^2,\lambda ,\tau )=0\) implicitly defines a function \(\tau (x^2,\lambda )\) in a neighbourhood of the point \((0,\lambda _0,0)\). Consequently, a solution set of the equation \(g=0\) coincides with a solution set of the equation \(g_2(x,\lambda ,\tau (x^2,\lambda ))=0\).

Heretofore, we have discussed only the hypothesis (C1) which is an essential part of the set of sufficient conditions for an occurrence of period-doubling bifurcation. However, we need more information about the equation \(\varPhi =0\) in order to describe situation around the bifurcation point. Particularly, we should focus on the so called non-degeneracy conditions. For this purpose, we look at derivatives of \(g_2\) at \((0,\lambda _0)\) which can be computed from derivatives of \(\varPhi \). Singularity theory enables us to use this data to determine the type of a bifurcation up to an equivalence of germs. This is the aim of the following proposition taken from Golubitsky and Schaeffer [6] (Proposition 2.14 from Chapter 6):

Proposition 5

A germ \(xr(x^2,\lambda )\) is strongly \({\mathbb {Z}}_2\)-equivalent to the germ \(x(\varepsilon x^2+\delta (\lambda -\lambda _0))\) if and only if \(r(0,\lambda _0)=0\), \({{\,\textrm{sgn}\,}}r_z(0,\lambda _0)=\varepsilon \) and \({{\,\textrm{sgn}\,}}r_\lambda (0,\lambda _0)=\delta \), where \(r=r(z,\lambda )\) and \(\varepsilon ,\delta \ne 0\).

It is useful to introduce the following notation: \(r(x^2,\lambda ){:=}b(x^2,\lambda ,\tau (x^2,\lambda ))\), hence \(g_2(x,\lambda ,\tau (x^2,\lambda ))=xr(x^2,\lambda )\). The following identities can be achieved:

$$\begin{aligned}&r_\lambda (0,\lambda _0)=\langle v_2, (\textrm{d}\varPhi _\lambda )_{u_0,\lambda _0,0}\cdot v_0+(\textrm{d}^2\varPhi )_{u_0,\lambda _0,0}(W_\lambda (0,\lambda _0,0),v_0)\rangle \end{aligned}$$
(10)
$$\begin{aligned}&-\dfrac{\langle v_1,\varPhi _\lambda (u_0,\lambda _0,0)\rangle }{\langle v_1,\varPhi _\tau (u_0,\lambda _0,0)\rangle }\langle v_2, (\textrm{d}\varPhi _\tau )_{u_0,\lambda _0,0}\cdot v_0+(\textrm{d}^2\varPhi )_{u_0,\lambda _0,0}(W_\tau (0,\lambda _0,0),v_0)\rangle , \nonumber \\&r_z(0,\lambda _0)=\dfrac{\langle v_2, (\textrm{d}^3\varPhi )_{u_0,\lambda _0,0}(v_0,v_0,v_0)+3(\textrm{d}^2\varPhi )_{u_0,\lambda _0,0}(W_{xx}(0,\lambda _0,0),v_0)\rangle }{6} \nonumber \\&-\dfrac{\langle v_1, (\textrm{d}^2\varPhi )_{u_0,\lambda _0,0}(v_0,v_0)\rangle }{2\langle v_1,\varPhi _\tau (u_0,\lambda _0,0)\rangle }\langle v_2, (\textrm{d}\varPhi _\tau )_{u_0,\lambda _0,0}\cdot v_0 \nonumber \\&+(\textrm{d}^2\varPhi )_{u_0,\lambda _0,0}(W_\tau (0,\lambda _0,0),v_0)\rangle . \end{aligned}$$
(11)

These values are in fact normal form coefficients. All calculations which are necessary for derivation of the above formulae together with calculations of derivatives of the implicitly defined function W are included in the Appendix. Sufficient conditions for an occurrence of a non-degenerate period-doubling bifurcation are simple consequences of these formulae and Proposition 5:

Theorem 2

If \(u_0\) is a periodic solution of the equation (1) for \(\lambda =\lambda _0\) with period p such that the condition (C1) is satisfied and values (10) and (11) are non-zero, then the system (1) overcomes a non-degenerate period-doubling bifurcation for \(\lambda =\lambda _0\).

It remains to uncover the relation between the derivative \(r_\lambda (0,\lambda _0)\) and the derivative of the critical multiplier. We need to construct a family of periodic solutions parametrized by \(\lambda \) to which a family \(\rho (\lambda )\) of Floquet multipliers can be assigned. A possible choice is \(W(0,\lambda ,\tau (0,\lambda ))\). The definition of W implies annihilation of \(E\varPhi \). The germ a is also annihilated, according to the definition of the function \(\tau \). Finally, \(g_2=0\) also holds because \(x=0\). Consequently, \(\varPhi (W(0,\lambda ,\tau (0,\lambda )),\lambda ,\tau (0,\lambda ))=0\) which implies that \(W(0,\lambda ,\tau (0,\lambda ))\) is a family of solutions.

Lemma 1

Functions \(W(0,\lambda ,\tau (0,\lambda ))\) are p-periodic.

Proof

If we restrict to p-periodic functions only, then the space \(\ker L\) would be one-dimensional. This degeneracy is caused by the action of \({\mathbb {R}}\) defined by a phase shift. Hence, for every \(\lambda \approx \lambda _0\) there should be a p-periodic solution of \(\varPhi =0\) which is unique up to a phase shift. Since the variable x corresponds to a direction leading into the space \(C^1_{2p}\), it is intuitive to choose \(x=0\) in order to reach these solutions.

Recall, that \(W:{\mathbb {R}}\times {\mathbb {R}}^2\rightarrow {\widetilde{M}}\) is defined implicitly by the equation

$$\begin{aligned} E\varPhi (xv_0+W(x,\lambda ,\tau ),\lambda ,\tau )=0, \qquad \langle xv_0+W(x,\lambda ,\tau ),{\dot{u}}_0\rangle =0 \end{aligned}$$
(12)

in a neighbourhood of \((0,\lambda _0,0)\). Let us fix \(x=0\) and consider the following equation:

$$\begin{aligned} E\varPhi (w,\lambda ,\tau )=0, \qquad \langle w,{\dot{u}}_0\rangle =0, \end{aligned}$$

where \(w \in {\widetilde{M}}\). The differential of left-hand sides with respect to w evaluated at \((u_0,\lambda _0,0)\) is again invertible, hence the equation uniquely determines a function \(w(\lambda ,\tau )\). However, since \(W(0,\lambda ,\tau )\) also satisfies the equation, we can conclude \(w(\lambda ,\tau )=W(0,\lambda ,\tau )\). Let us consider \(\varPhi \) as an operator \(\varPhi :C^1_{p} \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow C_{p}\) for a moment. The differential of left-hand sides of the equations

$$\begin{aligned} {\widetilde{E}}\varPhi ({\widetilde{w}},\lambda ,\tau )=0,\qquad \dfrac{1}{p}\int _{0}^{p}\langle {\widetilde{w}}(s),{\dot{u}}_{0}(s) \rangle \,\textrm{d}s=0 \end{aligned}$$

with respect to \({\widetilde{w}}\in C_p^1\) evaluated at \((u_0,\lambda _0,0)\) is also invertible. The symbol \({\widetilde{E}}\) denotes the projection of \(C_p\) onto the space \({{\,\textrm{Im}\,}}\left. L\right| _{C^1_{p}}\) with the kernel \([L(s{\dot{u}}_0(s))]\). Therefore, the equation uniquely determines a function

$$\begin{aligned} {\widetilde{w}}(\lambda ,\tau ):{\mathbb {R}}^2\rightarrow C^1_{p}\subseteq {\widetilde{M}}. \end{aligned}$$

The uniqueness of the function w implies \({\widetilde{w}}(\lambda ,\tau )=w(\lambda ,\tau )=W(0,\lambda ,\tau )\). Consequently, \(W(0,\lambda ,\tau )\) is p-periodic. \(\square \)

Since a multiplier of \(W(0,\lambda ,\tau (0,\lambda ))\) is an eigenvalue of the monodromy operator \(T(\lambda ,p,0)\), where \(T(\lambda ,s,\sigma )\) is the solution operator of the equation \((\textrm{d}\varPhi )_{W(0,\lambda ,\tau (0,\lambda )),\lambda ,\tau (0,\lambda )}=0\), it can be defined implicitly in the following way. Let \(\varSigma :{\mathbb {R}}\times {{\,\textrm{Im}\,}}(I+T(p,0))\times {\mathbb {R}}\rightarrow C\) be defined by

$$\begin{aligned} \varSigma (\rho ,\upsilon ,\lambda ){:=}T(\lambda ,p,0)(\phi _0+\upsilon )-\rho (\phi _0+\upsilon ). \end{aligned}$$

Obviously, the point \((-1,0,\lambda _0)\) satisfies the equation \(\varSigma =0\). The differential of \(\varSigma \) with respect to the first two variables evaluated at \((-1,0,\lambda _0)\) is given by

$$\begin{aligned} \varSigma _\rho =-\phi _0,\qquad \varSigma _\upsilon =T(p,0)+I. \end{aligned}$$

Since \(C=\ker (I+T(p,0))\oplus {{\,\textrm{Im}\,}}(I+T(p,0))\), the differential is invertible. Consequently, the equation \(\varSigma =0\) defines smooth functions \(\rho (\lambda )\) and \(\upsilon (\lambda )\) in a neighbourhood of the point \((-1,0,\lambda _0)\). Therefore, \(\rho (\lambda )\) is the critical Floquet multiplier of the solution \(W(0,\lambda ,\tau (0,\lambda ))\) with the corresponding eigenvector \(\phi _0+\upsilon (\lambda )\). We can formulate the following theorem:

Theorem 3

The following equality holds:

$$\begin{aligned} r_\lambda (0,\lambda _0)=\dfrac{\rho '(\lambda _0)}{p}\langle v_2,(\textrm{d}\varPhi )_{u_0,\lambda _0,0}(sv_0(s))\rangle , \end{aligned}$$

where \(\langle v_2,(\textrm{d}\varPhi )_{u_0,\lambda _0,0}(sv_0(s))\rangle \ne 0\).

Proof

Obviously, the function

$$\begin{aligned} u(s)=\textrm{e}^{\frac{\log \rho (\lambda )^2}{2p}s}\textrm{e}^{-\frac{\log \rho (\lambda )^2}{2p}s}T(\lambda ,s,0)(\phi _0+\upsilon (\lambda ))(0)=\textrm{e}^{\frac{\log \rho (\lambda )^2}{2p}s}h(\lambda ,s) \end{aligned}$$

is a solution of the equation \((\textrm{d}\varPhi )_{W(0,\lambda ,\tau (0,\lambda )),\lambda ,\tau (0,\lambda )}=0\). Moreover, the function \(h(\lambda ,s)\) is 2p-periodic:

$$\begin{aligned} h(\lambda ,s+2p)&=\textrm{e}^{-\frac{\log \rho (\lambda )^2}{2p}(s+2p)}T(\lambda ,s+2p,0)(\phi _0+\upsilon (\lambda ))(0) \\&=\textrm{e}^{-\frac{\log \rho (\lambda )^2}{2p}s}\dfrac{1}{\rho (\lambda )^2}T(\lambda ,s+2p,2p)T(\lambda ,p,0)^2(\phi _0+\upsilon (\lambda ))(0) \\&=\textrm{e}^{-\frac{\log \rho (\lambda )^2}{2p}s}\dfrac{1}{\rho (\lambda )^2}T(\lambda ,s,0)\rho (\lambda )^2(\phi _0+\upsilon (\lambda ))(0) \\&=\textrm{e}^{-\frac{\log \rho (\lambda )^2}{2p}s}T(\lambda ,s,0)(\phi _0+\upsilon (\lambda ))(0)=h(\lambda ,s). \end{aligned}$$

Differentiation of the equation

$$\begin{aligned} (\textrm{d}\varPhi )_{W(0,\lambda ,\tau (0,\lambda )),\lambda ,\tau (0,\lambda )}\exp (s\log \rho (\lambda )^2/2p)h(\lambda ,s)=0 \end{aligned}$$

with respect to \(\lambda \) and evaluation at \(\lambda _0\) leads to the following equation:

$$\begin{aligned} (\textrm{d}^2\varPhi )_{u_0,\lambda _0,0}(W_\lambda&+W_\tau \cdot \tau _\lambda ,v_0)+(\textrm{d}\varPhi _\lambda )_{u_0,\lambda _0,0}(v_0)+(\textrm{d}\varPhi _\tau )_{u_0,\lambda _0,0}(v_0)\cdot \tau _\lambda \\&+(\textrm{d}\varPhi )_{u_0,\lambda _0,0}\left( -\dfrac{\rho '(\lambda _0)}{p}sv_0(s)+h_\lambda (\lambda _0,s)\right) =0. \end{aligned}$$

Multiplication by the function \(v_2\) gives us the desired result.

The inequality \(\langle v_2,(\textrm{d}\varPhi )_{u_0,\lambda _0,0}(sv_0(s))\rangle \ne 0\) can be deduced in the same way as it was done for \(\langle v_1,\varPhi _\tau (u_0,\lambda _0,0)\rangle \ne 0\). The following lines are spent by verification of this inequality. Firstly, we show that \(L(sv_0(s))\) is periodic:

$$\begin{aligned} L(sv_0(s))&=v_0(s)+s{\dot{v}}_0(s)-K(s,(({\cdot })v_0({\cdot }))_s) \\&=v_0(s)+s{\dot{v}}_0(s)-sK(s,(v_0)_s)-K(s,({\cdot })(v_0)_s({\cdot })) \\&=v_0(s)+sLv_0(s)-K(s,({\cdot })(v_0)_s({\cdot }))=v_0(s)-K(s,({\cdot })(v_0)_s({\cdot })). \end{aligned}$$

Moreover, this function has the following property: \(L((s+p)v_0(s+p))=-L(sv_0(s))\) for every \(s \in {\mathbb {R}}\). Consequently, the scalar product \(\langle v_1, L(sv_0(s))\rangle \) is equal to zero. Suppose that \(\langle v_2, L(sv_0(s))\rangle =0\). According to Theorem 1, there is a 2p-periodic solution z(s) of the equation \({\dot{z}}(s)=K(s,z_s)+L(sv_0(s))\). The function \(x(s)=z(s)-sv_0(s)\) satisfies the following equation:

$$\begin{aligned} {\dot{x}}(s)&=K(s,z_s)+L(sv_0(s))-v_0(s)-s{\dot{v}}_0(s) \\&=K(s,x_s)+sK(s,(v_0)_s)+K(s,({\cdot })(v_0)_s({\cdot }))-K(s,({\cdot })(v_0)_s({\cdot }))-s{\dot{v}}_0(s) \\&=K(s,x_s). \end{aligned}$$

This fact together with the equation

$$\begin{aligned} T(s+2p,s)(x)_s=(x)_{s+2p}=(z)_s-({\cdot })(v_0)_s({\cdot })-s(v_0)_s-2p(v_0)_s=(x)_s-2p(v_0)_s \end{aligned}$$

implies that \((T(2p,0)-I)(x)_0=-2p(v_0)_0\), which contradicts the fact that the geometric multiplicity of 1 is equal to its algebraic multiplicity as an eigenvalue of the operator T(2p, 0). \(\square \)

Note that the condition \(\rho '(\lambda _0)\ne 0\) together with the last theorem implies existence of doubled periodic orbits. If \(r_\lambda (0,\lambda _0)\ne 0\), then the equation \(r(x^2,\lambda )=0\) implicitly defines a function \(\lambda (x^2)\) satisfying \(r(x^2,\lambda (x^2))=0\) in a neighbourhood of the point \((0,\lambda _0)\). However, if there are no other non-degeneracy conditions, we are not able to tell anything about this branch of solutions. The non-degenerate period-doubling bifurcation which resembles pitchfork bifurcation is achieved by including the condition \(r_z(0,\lambda _0)\ne 0\).

3 Stability

The aim of this section is verification of exchange of stability in the case of a non-degenerate period-doubling bifurcation. Alongside the hypothesis (C1) we will assume that

  1. (C2)

    All Floquet multipliers of the 2p-periodic solution \(u_0\) lies inside the unit circle except of \(\gamma _1=1\) and \(\gamma _2=(-1)^2\).

Since the only possible accumulation point of the set of Floquet multipliers is zero, there is \(0<\delta <1\) such that all Floquet multipliers \(\gamma \) except of \(\gamma _1\) and \(\gamma _2\) satisfy \(\left| \gamma \right| \le \delta \).

Let us introduce the idea behind the following procedure shortly. Firstly, we define an auxiliary function \(\mu (x,\lambda )\) implicitly in virtue of the following proposition taken from Hale [7].

Proposition 6

A linear q-periodic RFDE has a characteristic multiplier \(\textrm{e}^{\mu q}\) if and only if it has a solution of the form \(u(s)\textrm{e}^{\mu s}\) where \(u(s+q)=u(s)\).

The function \(\mu \) enable us to connect derivatives of the reduced mapping with derivatives of the critical multiplier which is going to be defined implicitly as an eigenvalue of the monodromy operator. We adopt the following notation: \(\varOmega (x,\lambda ){:=}xv_0+W(x,\lambda ,\tau (x^2,\lambda ))\) and \(f(x,\lambda ){:=}g_2(x,\lambda ,\tau (x^2,\lambda ))\).

According to Proposition 6, if \(\varOmega (x,\lambda )\) is a solution of the equation \(\varPhi =0\), then its Floquet exponents satisfy the equation

$$\begin{aligned} (\textrm{d}\varPhi )_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}(\textrm{e}^{\mu s}u(s))=0 \end{aligned}$$

for a suitable 2p-periodic function u. In order to use this equation for an implicit definition of a function, it is necessary to extend the operator defined by the left-hand side of this equation and to find an appropriate range for it. The purpose of the following lemma is just to motivate the choice of the range. Since it can be easily verified by tools built in Hale [7] (Chapter 8), we state the lemma without a proof.

Lemma 2

Let \(\varLambda u(s)={\dot{u}}(s)-K(s,u_s)\) be a linear q-periodic RFDE. If \(\mu \in {\mathbb {R}}\) and \(u\in C^1_{q}\), then the equation \(\varLambda (\textrm{e}^{\mu {\cdot }}u({\cdot }))(s)=0\) holds for every \(s \in {\mathbb {R}}\) if and only if it holds for every \(s \in [0,q]\).

Note that the space \(C([0,2p],{\mathbb {R}}^n)\) has the following obvious property:

$$\begin{aligned} C([0,2p],{\mathbb {R}}^n)\cong C_{2p}\oplus [se_1,\ldots ,se_n], \end{aligned}$$
(13)

where \(e_i=(0,\ldots ,0,1,0,\ldots ,0)\) and 1 is on the i-th position. Now we can define an operator \( \varPsi :{\mathbb {R}}^n\times {\mathbb {R}}^2\times {\widetilde{M}}\times {\mathbb {R}}^2\rightarrow C([0,2p],{\mathbb {R}}^n)\) by the following equation:

$$\begin{aligned} \varPsi (&c,\mu ,\eta ,w,x,\lambda ) \\&{:=}(\textrm{d}\varPhi )_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}(\textrm{e}^{\mu s}(v_0(s)+w(s)))+\eta \varPhi _\tau (\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda ))+cs, \end{aligned}$$

where the domain of the function defined by the right-hand side is reduced to the interval [0, 2p]. Finally, in order to reduce the kernel of the linearization of the previous operator, we define an operator \({\widetilde{\varPsi }} :{\mathbb {R}}^n\times {\mathbb {R}}^2\times {\widetilde{M}}\times {\mathbb {R}}^2\rightarrow C([0,2p],{\mathbb {R}}^n)\times {\mathbb {R}}\) with the larger range:

$$\begin{aligned} {\widetilde{\varPsi }}(c,\mu ,\eta ,w,x,\lambda ){:=}(\varPsi (c,\mu ,\eta ,w,x,\lambda ),\langle w,{\dot{u}}_0\rangle ). \end{aligned}$$

It is easily seen that \({\widetilde{\varPsi }}(0,0,0,0,0,\lambda _0)=(0,0)\).

Lemma 3

The equation \({\widetilde{\varPsi }}=0\) implicitly defines a smooth function \(\mu (x,\lambda )\).

Proof

The derivatives of \({\widetilde{\varPsi }}\) with respect to the first four variables are given by

$$\begin{aligned} \begin{aligned} {\widetilde{\varPsi }}_{c_i}&=(se_i,0),&{\widetilde{\varPsi }}_\mu&=(L(sv_0(s)),0), \\ {\widetilde{\varPsi }}_\eta&=(\varPhi _\tau (u_0,\lambda _0,0),0),&\qquad \textrm{d}_w{\widetilde{\varPsi }}&=(L,\langle {\cdot },{\dot{u}}_0\rangle ). \end{aligned} \end{aligned}$$

The relations \(\langle v_1,L(sv_0(s))\rangle =\langle v_2,\varPhi _\tau (u_0,\lambda _0,0)\rangle =0\) and \(\langle v_1,\varPhi _\tau (u_0,\lambda _0,0)\rangle \ne 0\ne \langle v_2,L(sv_0(s))\rangle \) together with (13) implies invertibility of the differential of \({\widetilde{\varPsi }}\) with respect to the first four variables. According to the implicit function theorem, the equation \({\widetilde{\varPsi }}=0\) defines smooth functions \(c(x,\lambda )\), \(\mu (x,\lambda )\), \(\eta (x,\lambda )\) and \(w(x,\lambda )\) in a neighbourhood of the point \((0,0,0,0,0,\lambda _0)\). \(\square \)

Now we are going to define a function which coincides with the critical Floquet multiplier. Let \(T(x,\lambda ,t,s)\) be the solution operator of the equation \((\textrm{d}\varPhi )_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}=0\), where \(t\ge s\) and the corresponding monodromy operator will be denoted by \(U(x,\lambda ){:=}T(x,\lambda ,2p,0)\). Finally, let \(\phi =({\dot{u}}_0)_0\) and \(\phi _0=(v_0)_0\) be eigenvectors of the monodromy operator \(U{:=}U(0,\lambda _0)\) corresponding to the eigenvalue 1. We define the operator \(\varXi :{\mathbb {R}}^2\times {{\,\textrm{Im}\,}}(I-U)\times {\mathbb {R}}^2\rightarrow C\) by the following expression:

$$\begin{aligned} \varXi (\sigma ,\nu ,\omega ,x,\lambda ){:=}U(x,\lambda )(\phi _0+\omega )+\nu ({\dot{\varOmega }}(x,\lambda ))_0-\sigma (\phi _0+\omega ). \end{aligned}$$

Obviously, the point \((1,0,0,0,\lambda _0)\) satisfies the equation \(\varXi =0\).

Lemma 4

The equation \(\varXi =0\) implicitly defines a smooth function \(\sigma (x,\lambda )\).

Proof

The differential of \(\varXi \) with respect to the first three variables is given by

$$\begin{aligned} \varXi _\sigma =-\phi _0,\qquad \varXi _\nu =({\dot{\varOmega }}(0,\lambda _0))_0=({\dot{u}}_0)_0=\phi ,\qquad \varXi _\omega =U(0,\lambda _0)-I=U-I. \end{aligned}$$

Since \(C=\ker (I-U)\oplus {{\,\textrm{Im}\,}}(I-U)\), the differential is invertible. Consequently, the equation \(\varXi =0\) defines smooth functions \(\sigma (x,\lambda )\), \(\nu (x,\lambda )\) and \(\omega (x,\lambda )\) in a neighbourhood of the point \((1,0,0,0,\lambda _0)\). \(\square \)

Lemma 5

If \(f(x,\lambda )=0\), then \(\sigma (x,\lambda )\) is the critical Floquet multiplier of the solution \(\varOmega (x,\lambda )\) of the equation \(\varPhi =0\).

Proof

The assumption implies \((\textrm{d}\varPhi )_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}{\dot{\varOmega }}(x,\lambda )=0\) which leads to the equality \(U(x,\lambda )({\dot{\varOmega }}(x,\lambda ))_0=({\dot{\varOmega }}(x,\lambda ))_{2p}=({\dot{\varOmega }}(x,\lambda ))_0\). Let us use the notation \(V(x,\lambda )=\phi _0+\omega (x,\lambda )\). If the arguments are omitted, then the equations have the following forms:

$$\begin{aligned} (U-I){\dot{\varOmega }}_0=0,\qquad (U-\sigma I)V=-\nu {\dot{\varOmega }}_0. \end{aligned}$$

Suppose that \(\sigma (x,\lambda )\ne 1\). The first equation can be expressed in the following way:

$$\begin{aligned} (U-I){\dot{\varOmega }}_0=0\,\Leftrightarrow \, (U-\sigma I){\dot{\varOmega }}_0=(1-\sigma ){\dot{\varOmega }}_0\,\Leftrightarrow \, (U-\sigma I)\left( \dfrac{\nu }{1-\sigma }{\dot{\varOmega }}_0\right) =\nu {\dot{\varOmega }}_0. \end{aligned}$$

Summing of the two equations gives

$$\begin{aligned} (U-\sigma I)\left( V+\dfrac{\nu }{1-\sigma }{\dot{\varOmega }}_0\right) =0, \end{aligned}$$

so \(\sigma (x,\lambda )\) is a Floquet multiplier of \(\varOmega (x,\lambda )\). If \(\sigma (x,\lambda )=1\), then the equation \(\varXi =0\) has the form \((U-I)V=-\nu {\dot{\varOmega }}_0\). Consequently, \((U-I)^2V=0\) which implies that \(\sigma (x,\lambda )=1\) is a Floquet multiplier. It remains to connect the eigenvalue \(\sigma \) with the critical multiplier \(\gamma _2\). According to (C2) and upper semicontinuity of separated parts of a spectrum of a continuous linear operator, Kato [9], all multipliers except \(\gamma _1\) and \(\gamma _2\) satisfy \(\left| \gamma \right| \le \delta <1\) for some \(\delta >0\) so these multipliers do not come into play. Moreover, the sum of generalized eigenspaces corresponding to \(\gamma _1\) and \(\gamma _2\) is two-dimensional and the multiplier \(\gamma _1\) is associated with the eigenfunction \({\dot{\varOmega }}\). Thus, the second multiplier is the only possibility. \(\square \)

The proof of the previous lemma gives us more than just information about the critical Floquet multiplier. If \(\varOmega (x,\lambda )\) is a solution of the equation \(\varPhi =0\), then the pair \({\mathcal {B}}=({\dot{\varOmega }}(x,\lambda ),\phi _0+\omega (x,\lambda ))\) forms a basis of the sum of generalized eigenspaces corresponding to eigenvalues of \(U(x,\lambda )\) which are close to 1. Moreover, \(U(x,\lambda ){\mathcal {B}}={\mathcal {B}}M\) where

$$\begin{aligned} M=\left( \begin{array}{cc} 1 &{} -\nu (x,\lambda ) \\ 0 &{} \sigma (x,\lambda ) \end{array}\right) . \end{aligned}$$

Since M is invertible, we can calculate its logarithm:

$$\begin{aligned} \log M=\left( \begin{array}{cc} 0 &{} -\dfrac{\nu (x,\lambda )\log \sigma (x,\lambda )}{\sigma (x,\lambda )-1} \\ 0 &{} \log \sigma (x,\lambda ) \end{array}\right) , \end{aligned}$$

where it is assumed that \(\log \sigma /(\sigma -1)=1\) for \(\sigma =1\). If we define \(B=\log M/2p\), then

$$\begin{aligned} \textrm{e}^{Bs}=\left( \begin{array}{cc} 1 &{} -\dfrac{\nu (x,\lambda )}{\sigma (x,\lambda )-1}\left( \exp \left( \dfrac{s}{2p}\log \sigma (x,\lambda )\right) -1\right) \\ 0 &{} \exp \left( \dfrac{s}{2p}\log \sigma (x,\lambda )\right) \end{array}\right) . \end{aligned}$$

The vector \(P(s)=T(x,\lambda ,s,0){\mathcal {B}}\textrm{e}^{-Bs}\) with elements in C is 2p-periodic, Hale [7]:

$$\begin{aligned} P(s+2p)&=T(x,\lambda ,s+2p,0){\mathcal {B}}\textrm{e}^{-B(s+2p)} \nonumber \\&=T(x,\lambda ,s+2p,2p)T(x,\lambda ,2p,0){\mathcal {B}}\textrm{e}^{-2pB}\textrm{e}^{-Bs} \nonumber \\&=T(x,\lambda ,s,0){\mathcal {B}}MM^{-1}\textrm{e}^{-Bs}=T(x,\lambda ,s,0){\mathcal {B}}\textrm{e}^{-Bs}=P(s). \end{aligned}$$
(14)

This is going to be used in a moment.

Before we accept fulfillment of non-degeneracy conditions, we formulate one more statement giving us a useful identity.

Lemma 6

If \(\varOmega (x,\lambda )\) is a solution of the equation \(\varPhi =0\), then

$$\begin{aligned} (\textrm{d}\varPhi )_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}(s{\dot{\varOmega }}(x,\lambda )(s))=(1+\tau (x^2,\lambda ))\varPhi _\tau (\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )). \end{aligned}$$

Proof

We start with evaluation of the following expression:

$$\begin{aligned} (({\cdot })u({\cdot }))_{s,\tau }(\theta )&=(s+(1+\tau )\theta )u(s+(1+\tau )\theta ) \\&=su(s+(1+\tau )\theta )+(1+\tau )\theta u(s+(1+\tau )\theta ) \\&=su_{s,\tau }(\theta )+(1+\tau )(({\cdot })u_{s,\tau }({\cdot }))(\theta ). \end{aligned}$$

This together with \((\textrm{d}\varPhi )_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}{\dot{\varOmega }}(x,\lambda )=0\) give us the following set of equalities:

$$\begin{aligned} (\textrm{d}\varPhi )&_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}(s{\dot{\varOmega }}(x,\lambda )(s)) \\&=(1+\tau (x^2,\lambda ))\left( {\dot{\varOmega }}(x,\lambda )(s)+s\ddot{\varOmega }(x,\lambda )(s)\right) \\&\quad -(\textrm{d}F)_{\varOmega (x,\lambda )_{s,\tau (x^2,\lambda )},\lambda }(({\cdot }){\dot{\varOmega }}(x,\lambda )({\cdot }))_{s,\tau (x^2,\lambda )} \\&=(1+\tau (x^2,\lambda ))\left( {\dot{\varOmega }}(x,\lambda )(s)+s\ddot{\varOmega }(x,\lambda )(s)\right) \\&\quad -s(\textrm{d}F)_{\varOmega (x,\lambda )_{s,\tau (x^2,\lambda )},\lambda }{\dot{\varOmega }}(x,\lambda )_{s,\tau (x^2,\lambda )} \\&\quad -(1+\tau (x^2,\lambda ))(\textrm{d}F)_{\varOmega (x,\lambda )_{s,\tau (x^2,\lambda )},\lambda }(({\cdot }){\dot{\varOmega }}(x,\lambda )_{s,\tau (x^2,\lambda )}({\cdot })) \\&=s(\textrm{d}\varPhi )_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}{\dot{\varOmega }}(x,\lambda )+(1+\tau (x^2,\lambda ))\varPhi _\tau (\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )) \\&=(1+\tau (x^2,\lambda ))\varPhi _\tau (\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )) \end{aligned}$$

which finishes the proof. \(\square \)

Henceforth, we assume that \(r_\lambda (0,\lambda _0)\ne 0\). This implies existence of a function \(\lambda (x^2)\) satisfying \(r(x^2,\lambda (x^2))=0\). Especially, \(\varOmega (x,\lambda (x^2))\) is a family of solutions bifurcating from the point \((u_0,\lambda _0)\). Let us denote the second component of \(P(s)(0)=T(x,\lambda (x^2),s,0){\mathcal {B}}(0)\textrm{e}^{-Bs}\) by \({\widetilde{w}}(x)(s)\) and \({\widetilde{\mu }}(x){:=}\log (\sigma (x,\lambda (x^2))/2p)\):

$$\begin{aligned} {\widetilde{w}}{:=}\dfrac{\nu (x,\lambda (x^2))}{\textrm{e}^{2p{\widetilde{\mu }}(x)}-1}(1-\textrm{e}^{-{\widetilde{\mu }}(x)s})&{\dot{\varOmega }}(x,\lambda (x^2))(s) \\&+\textrm{e}^{-{\widetilde{\mu }}(x)s}T(x,\lambda (x^2),s,0)(\phi _0+\omega (x,\lambda (x^2)))(0). \end{aligned}$$

According to (14), the function \({\widetilde{w}}(x)(s)\) is periodic. Obviously, both components of \(T(x,\lambda ,s,0){\mathcal {B}}(0)\) are solutions of the equation \((\textrm{d}\varPhi )_{\varOmega (x,\lambda ),\lambda ,\tau (x^2,\lambda )}=0\) so we can conclude the following equations:

$$\begin{aligned} (\textrm{d}\varPhi )_{\varOmega (x,\lambda (x^2)),\lambda (x^2),\tau (x^2,\lambda (x^2))}\left( T(x,\lambda (x^2),s,0){\mathcal {B}}(0)\textrm{e}^{-Bs}\textrm{e}^{Bs}\left( \begin{array}{c}0 \\ 1\end{array}\right) \right)&=0, \\ (\textrm{d}\varPhi )_{\varOmega (x,\lambda (x^2)),\lambda (x^2),\tau (x^2,\lambda (x^2))}\left( ({\dot{\varOmega }}(x,\lambda (x^2))(s),{\widetilde{w}}(x)(s))\textrm{e}^{Bs}\left( \begin{array}{c}0 \\ 1\end{array}\right) \right)&=0, \\ (\textrm{d}\varPhi )_{\ldots }\left( \dfrac{-\nu (x,\lambda (x^2))}{\textrm{e}^{2p{\widetilde{\mu }}(x)}-1}(\textrm{e}^{{\widetilde{\mu }}(x)s}-1){\dot{\varOmega }}(x,\lambda (x^2))(s)+\textrm{e}^{{\widetilde{\mu }}(x)s}{\widetilde{w}}(x)(s)\right)&=0. \end{aligned}$$

The last equation can be used for calculation of derivatives of the function \({\widetilde{\mu }}(x)\) which is the critical Floquet exponent of the solution \(\varOmega (x,\lambda (x^2))\).

The whole theoretical background has been presented so it remains to perform necessary calculations the results of which are summarized in the following statements. Since their proofs are quite technical, they are postponed to the Appendix.

Lemma 7

\(\mu _{xx}(0,\lambda _0)=-\dfrac{f_{xxx}(0,\lambda _0)}{\langle v_2,(\textrm{d}\varPhi )_{u_0,\lambda _0,0}(sv_0(s))\rangle }\).

In order to simplify notation, we write \(m(x){:=}\mu (x,\lambda (x^2))\).

Lemma 8

\(m_{xx}(0)={\widetilde{\mu }}_{xx}(0)\).

Theorem 4

\({\widetilde{\mu }}_{xx}(0)=-\dfrac{4r_z(0,\lambda _0)}{\langle v_2, L(sv_0(s))\rangle }\).

Now we are prepared to verify the main result of this section.

Theorem 5

In the case of a non-degenerate period-doubling bifurcation the exchange of stability occurs.

Proof

Let us suppose that \(\langle v_2, L(sv_0(s))\rangle >0\), the opposite case can be solved analogously. If

$$\begin{aligned} 0<r_\lambda (0,\lambda _0)=\dfrac{\rho '(\lambda _0)}{p}\langle v_2,(\textrm{d}\varPhi )_{u_0,\lambda _0,0}(sv_0(s))\rangle , \end{aligned}$$

then \(\rho (\lambda )\) is increasing which means that p-periodic solutions are stable for \(\lambda >\lambda _0\) and unstable for \(\lambda <\lambda _0\). According to Proposition 5, in the case \(r_z(0,\lambda _0)>0\) the doubled solutions appear for \(\lambda <\lambda _0\) and they are stable because \({\widetilde{\mu }}\) is negative in a neighbourhood of 0. Otherwise, these solutions appear for \(\lambda >\lambda _0\) and they are unstable. The case \(r_\lambda (0,\lambda _0)<0\) can be investigated in the same way. This proves the exchange of stability for non-degenerate period-doubling bifurcation. \(\square \)

4 Discussion and Conclusion

Generally, normal form coefficients are considered to be the quantities a and c in the normal form of a period-doubling bifurcation:

$$\begin{aligned} \dfrac{\textrm{d}\tau }{\textrm{d}t}&=1+a\xi ^2+{\mathcal {O}}(\xi ^4), \\ \dfrac{\textrm{d}\xi }{\textrm{d}t}&=c\xi ^3+{\mathcal {O}}(\xi ^4), \end{aligned}$$

where \(\tau \) and \(\xi \) are coordinates on the center manifold with \(\xi \) being transverse to the orbit of \({\dot{u}}_0\), Lentjes et al. [12, Preprint]. On the other hand, normal form coefficients derived in this work correspond to Taylor coefficients of the reduced mapping \(\phi \). The definition of a and c together with our analysis imply that a, c and \(r_\lambda (0,\lambda _0)\), \(r_z(0,\lambda _0)\) have the same sign, respectively. Since formulae for these coefficients coincide in the case of ODE, see Kuznetsov et al. [11], we can expect their agreement also in the case of RFDE.

A next step would be to actually implement the obtained expressions and see how well they behave numerically. Additionally, the following equations could potentially be used for the continuation of period-doubling bifurcations of limit cycles in RFDE:

$$\begin{aligned} {\widetilde{\varPhi }}(u,\lambda ,\tau )=0,\quad (\textrm{d}\varPhi )_{u,\lambda ,\tau }\cdot v=0,\quad \langle v,v\rangle =1,\quad v(p)=-v(0). \end{aligned}$$

This should then be compared with the current implementation in DDE-BifTool and Knut, see Engelborghs et al. [5] and Szalai [15]. Moreover, the left-hand sides of (10) and (11) could play the role of a test function for detection of singularities, for the case of ODE see Dhooge et al. [2], Dhooge et al. [3] and Kuznetsov [10]. The development of appropriate numerical methods can lead to a strong tool for investigation of real-life models incorporating delays which overcome period-doubling bifurcation. Examples of such models include, for instance, the system of coupled Fitzhugh-Nagumo oscillators with two delays studied by Saha and Feudel [13], and the time-periodic model of machining analyzed by Szalai and Stépán [16], where authors used sun-star calculus.

The existence of a smooth periodic center manifold in a neighbourhood of a limit cycle together with a suitable coordinate system is a base of the investigation of period-doubling bifurcation in Iooss [8], Kuznetsov et al. [11] and Lentjes et al. [12, Preprint]. We have not needed such information, since the main idea of the presented theory lies in considering a given differential equation as an algebraic equation. This can be expressed rigorously as the possibility to find suitable Banach spaces and an operator between them with the property that its zero set coincides with solutions of the original equation. The key to success is an appropriate choice of an operator of which linearization satisfies conditions imposed on Fredholm operators so that Lyapunov–Schmidt reduction can be used. In general, this approach could be applied to another bifurcations and to another types of differential equations, for example, bifurcations in neutral functional differential equations or Arnold tongues. The second area would probably lead to equivariant singularity theory.