Computation of fold and cusp bifurcation points in a system of ordinary differential equations using the Lagrange multiplier method

Abstract

In this paper, an alternative is proposed for computing fold and cusp bifurcation in a system of two ordinary differential equations depending on one or two parameters. In particular, it is proven that the fold bifurcation point in that system corresponds to a local maximum or minimum of a constrained optimization problem, which can be computed using the classical Lagrange Multiplier Method. Conversely, a sufficient condition is provided so that the solution of a particular constrained optimization problem using the Lagrange Multiplier Method corresponds to a fold bifurcation of two ordinary differential equations. Similarly, for system with two parameters, some sufficient conditions for cusp bifurcation points are also provided. These results are applied to three examples: the Bazykin’s system, a two-dimensional predator–prey system with nonmonotonic response function, and a subsystem of the so-called tritrophic food-chain model. The results are compared with those in the literature and also with the results using the numerical continuation software AUTO. Furthermore, a swallowtail bifurcation is detected in the latter model.

Appendices

Proof of the Theorem 2

Without loss of generality, we assume that $x_0 = 0$, $y_0 = 0$, and $\beta _0 = 0$. Using the Center Manifold Theorem (see [14, 23]) we can reduce System (2) to a single equation on the center manifold. Let us consider the coordinate transformation $\varvec{\xi } = P \varvec{\nu }$ with:

$$\begin{aligned} P = \left( \begin{matrix} -F_y(0,0) &{} 1 \\ F_x(0,0) &{} \lambda _0 \end{matrix}\right) , \end{aligned}$$

$\varvec{\xi } = \left( x, y \right) ^{\scriptscriptstyle T}$, and $\varvec{\nu } = \left( u, v \right) ^{\scriptscriptstyle T}$.

Applying the transformation to System (2), we derive:

$$\begin{aligned} \left\{ \begin{array}{l} {\dot{u}} = \frac{1}{\eta } \beta + f(u,v) \\ {\dot{v}} = \frac{1}{\eta } F_y(0,0) \beta + \eta \, v + g(u,v). \\ \end{array} \right. \end{aligned}$$

(21)

Here:

$$\begin{aligned} f(u,v) = {\textstyle \frac{1}{2\eta }} \; \varvec{\nu }^{\scriptscriptstyle T} \left( P^{\scriptscriptstyle T} \left( B - \lambda _0 A \right) P \right) \varvec{\nu } + O(\parallel \varvec{\nu } \parallel ^3), \end{aligned}$$

(22)

and

$$\begin{aligned} g(u,v) = {\textstyle \frac{1}{2\eta }} \; \varvec{\nu }^{\scriptscriptstyle T} \left( P^{\scriptscriptstyle T} \left( F_y(0,0) A + F_x(0,0) B \right) P \right) \varvec{\nu } + O(\parallel \varvec{\nu } \parallel ^3). \end{aligned}$$

See Eq. (4) for the definition of A and B.

From the Center Manifold Theorem, for sufficiently small $\delta$ there exists a function:

$$\begin{aligned} \begin{array}{lccc} V: &{} (-\delta , \delta ) \times {\mathbb {R}} &{} \longrightarrow &{} {\mathbb {R}}\\ &{} (u,\beta ) &{} \longmapsto &{} V(u,\beta ) \end{array} \end{aligned}$$

with $V(0,0) = 0$, $V_u (0,0) = 0$, and $V_{\beta }(0,0) = 0$, so that System (21) can be written as:

$$\begin{aligned}\ {\dot{u}} = \textstyle \frac{1}{\eta } \beta + f(u,V) = h(u,\beta ), \end{aligned}$$

where $V = V(u, \beta )$ to shorten the notation.

All that remains is to check if these two conditions:

($C_1$):

$h_{\beta }(0,0) \ne 0$, and

($C_2$):

$h_{uu}(0,0) \ne 0$,

are satisfied. In that case, there exists a time-orientation preserving homeomorphism which maps its orbits of System (3.1) to the orbits of

$$\begin{aligned} w' = \gamma \pm w^2. \end{aligned}$$

This last equation is a normal form for fold bifurcation with unfolding parameter: $\gamma$ (see Theorem 3.2 in [14]).

It is easy to see that

$$\begin{aligned} h_\beta (u, \beta ) = \textstyle \frac{1}{\eta } + f_v(u, V(u,\beta )) V_{\beta }(u, \beta ), \end{aligned}$$

which implies that $h_\beta (0,0) \ne 0$. Thus $(C_1)$ is satisfied.

Next, by applying the chain rule we have:

$$\begin{aligned} \textstyle h_u(u,\beta ) = f_{u}(u,V) + f_v(u,V) V_u, \end{aligned}$$

and

$$\begin{aligned} h_{uu}(u,\beta ) = f_{uu}(u,V) + 2 f_{uv}(u,V) V_u + f_v (u,V) V_{uu}. \end{aligned}$$

Let $\varvec{e_1}$ and $\varvec{e_2}$ be the unit vectors on x- and y- axis, respectively. Then, since

$$\begin{aligned} f_u(u,v) = \textstyle \frac{1}{\eta } \varvec{e_1}^{\scriptscriptstyle T} \left( P^{\scriptscriptstyle T} \left( B - \lambda _0 A \right) P \right) \varvec{\nu }, \end{aligned}$$

and

$$\begin{aligned} f_v(u,v) = \textstyle \frac{1}{\eta } \varvec{e_2}^{\scriptscriptstyle T} \left( P^{\scriptscriptstyle T} \left( B - \lambda _0 A \right) P \right) \varvec{\nu }, \end{aligned}$$

we have: $f_u(0,0) = 0 = f_v(0,0)$. Since $V(0,0) = 0$, we conclude that: $h_u(0,0) = 0$ and

$$\begin{aligned} h_{uu}(u,\beta ) = f_{uu}(u,V). \end{aligned}$$

Lastly, from the definition of f in Eq. (22) we derive:

$$\begin{aligned} f_{uu}(0,0)= & {} \frac{1}{\eta } \varvec{e_1}^{\scriptscriptstyle T} \left( P^{\scriptscriptstyle T} \left( B - \lambda _0 A \right) P \right) \varvec{e_1}\\= & {} -\frac{1}{\eta } \left( J \nabla F(0, 0) \right) ^T \left( \lambda _0 A - B \right) J \nabla F(0, 0). \end{aligned}$$

Thus, by the condition in Eq. (6), the nondegeneracy condition $(C_2)$ is satisfied.

Proof of Theorem 3

Recall that we have assumed that

• 1. $(x_0, y_0, \lambda _0)$ is a solution of System (9), with $\lambda _0 \not =0$,

• 2. $\alpha = \alpha _0$ is a solution of Eq. (11), and

• 3. we have set: $\beta _0 = G(x_0, y_0,\alpha _0)$.

Without loss of generality, we assume that $x_0 = 0$, $y_0= 0$, $\alpha _0 = 0$ and $\beta _0 = 0$. We will use the same notation for the Hessians (A and B), and for the matrix P as in the previous section; however, they are now $\alpha$-dependent.

Premise 1 in Theorem 3 guarantees that: $\displaystyle \eta = F_x(0,0,0) + \lambda _0 F_y(0,0,0) \not =0$. Using a similar argument as in the previous section, we apply the coordinate transformation $\varvec{\xi } = P \varvec{\nu }$ to System (8). Using the Center Manifold Theorem, we conclude that for a sufficiently small $\delta$, there exists a function:

$$\begin{aligned} \begin{array}{lccc} V: &{} (-\delta , \delta )\times {\mathbb {R}} \times {\mathbb {R}} &{} \longrightarrow &{}{\mathbb {R}}, \\ &{} (u, \alpha , \beta ) &{} \longmapsto &{} V(u,\alpha , \beta ) \end{array} \end{aligned}$$

with

$$\begin{aligned}&V(0,0,0) = 0, V_u (0,0,0) = 0, V_\alpha (0,0,0)\nonumber \\&= 0, V_\beta (0,0,0) = 0, \end{aligned}$$

(23)

such that on the center manifold, the dynamics of System (8) is determined by:

$$\begin{aligned} {\dot{u}} = \textstyle \frac{1}{\eta } \beta + f(u,V(u,\alpha ,\beta ),\alpha ) = h(u,\alpha ,\beta ). \end{aligned}$$

(24)

where f as is defined in Eq. (22) but adjusted to include the second parameter.

If these four conditions, i.e.

$(C_1)$:

$h_{u} = 0$,

$(C_2)$:

$h_{uu} = 0$,

$(C_3)$:

$h_{uuu} \ne 0$, and

$(C_4)$:

$h_{\beta } h_{u \alpha } - h_{\alpha } h_{u\beta } \ne 0$,

are satisfied, then System (24) is topologically equivalent to the normal form of cusp bifurcation:

$$\begin{aligned} w' = \gamma _1 + \gamma _2 w \pm w^3.\\ \end{aligned}$$

To simplify the notation, we will write V for $V(u, \alpha , \beta )$, h for $h(u,\alpha , \beta )$ and f for $f(u,V(u,\alpha ,\beta ),\alpha )$. Using the chain rule, we compute:

$$\begin{aligned} h_u = f_u + f_v V_u. \end{aligned}$$

(25)

Using a similar argument as in the previous subsection, $f_u$ and $f_v$ are both vanishing. Thus, we conclude $(C_1)$.

Next, we compute the derivative of Eq. (25) with respect to u, i.e.:

$$\begin{aligned} h_{uu} = f_{uu} + 2f_{uv} V_u + f_{uv} V_{uu} + f_{vv}\left( V_u \right) ^2. \end{aligned}$$

(26)

From Eq. (23), at $(u,\alpha ,\beta ) = (0,0,0)$ we derive: $h_{uu}(0,0,0) = f_{uu}(0,0,0)$. By premise 2 in Theorem (3) we have:

$$\begin{aligned}&h_{uu}(0,0,0) = -\frac{1}{\eta } \left( J \nabla F(0, 0, 0) \right) ^T\\&\qquad \left( \lambda _0 A - B \right) J \nabla F(0, 0, 0) =0, \end{aligned}$$

where A and B are the Hessians of F and G respectively.

Let us carry on with computing the derivative of Eq. (26) with respect to u:

$$\begin{aligned} h_{uuu}= & {} \displaystyle f_{uuu} + 2\left( \left( f_{uuv} + f_{uvv} V_u \right) V_u\right. \\&\left. + \, f_{uv} V_{uu} \right) + \left( f_{uuv} + f_{uvv} V_u \right) V_{uu} \\&\displaystyle + f_{uv} V_{uuu} + \left( f_{uvv} + f_{vvv}V_u \right) {V_{u} }^2 + 2 f_{vv}V_u V_{uu}. \end{aligned}$$

Applying Eq. (23), we have:

$$\begin{aligned}&h_{uuu}(0,0,0) = \displaystyle f_{uuu} (0,0,0) + 2 f_{uv}(0,0,0) V_{uu}(0,0,0)\\&\quad + f_{uuv}(0,0,0) V_{uu} + f_{uv}(0,0,0) V_{uuu}(0,0,0). \end{aligned}$$

Writing,

$$\begin{aligned} A_2= & {} \left( \begin{matrix} F_y F_{xxx} &{} 3 F_y F_{xxy}\\ -3F_x F_{xyy} &{} -F_x F_{yyy} \end{matrix}\right) ,\\ B_2= & {} \left( \begin{matrix} F_y G_{xxx} &{} 3 F_yG_{xxy} \\ -3 F_x G_{xyy} &{} -F_x G_{yyy} \end{matrix}\right) , \text { and } {\varvec{p}} = \left( \begin{matrix} 1\\ \lambda _0 \end{matrix}\right) , \end{aligned}$$

we have:

$$\begin{aligned} h_{uuu}(0,0,0) = c_1 - 3c_2 c_3, \end{aligned}$$

where:

$$\begin{aligned} \begin{array}{l} c_1 = \eta ^2 \left( \left( J\nabla F \right) ^{\scriptscriptstyle T} \left( \lambda _0 A_2 - B_2 \right) J\nabla F \right) ,\\ c_2 = \left( \left( J\nabla F\right) ^{\scriptscriptstyle T} \left( \lambda _0 A - B \right) \; {\varvec{p}} \right) , \\ c_3 = \left( \left( J \nabla F \right) ^{\scriptscriptstyle T} \left( F_{x} A + F_{y} B \right) J \nabla F \right) ; \end{array} \end{aligned}$$

all evaluated at (0, 0, 0). Thus, $(C_3)$ follows from premise 3 of Theorem 3.

Lastly, since:

$$\begin{aligned} \begin{array}{l} h_\beta = \frac{1}{\eta } + f_{v} V_{\beta }, \\ h_\alpha = {f}_{\alpha } + {f}_{ v} {V}_{ \alpha }, \\ h_{u\beta } = f_{uv} V_\beta + f_{vv} V_u V_\beta + f_v V_{u\beta } , \\ h_{u\alpha } = f_{u\alpha } + f_{uv} V_{\alpha } + \left( f_{v \alpha } + {f}_{vv} V_\alpha \right) V_{u} + f_{v} V_{u \alpha }, \end{array} \end{aligned}$$

and furthermore:

$$\begin{aligned} f_{u}(0,0,0) = 0 = f_{v}(0,0,0), \text { and } V_{u}(0,0,0) = 0, \end{aligned}$$

then, at (0, 0, 0):

$$\begin{aligned}&h_{\beta } h_{u \alpha } - h_{\alpha } h_{u\beta } = \frac{1}{\eta } \left( f_{u\alpha } + f_{uv} V_{\alpha } \right) - f_\alpha f_{uv} V_\beta \\&\quad \displaystyle = \left( J \nabla F(0, 0, 0) \right) ^T \left( \left( \begin{matrix} G_{xx\alpha }(0, 0, 0)\\ G_{yy\alpha }(0, 0, 0) \end{matrix}\right) - \lambda _0 \left( \begin{matrix} F_{xx\alpha }(0, 0, 0)\\ F_{yy\alpha }(0, 0, 0) \end{matrix}\right) \right) . \end{aligned}$$

Thus, condition $(C_4)$ follows from premise 4 of Theorem 3.