Analysis of Degenerate Fold–Hopf Bifurcation in a Three-Dimensional Differential System

Abstract

In this work we study degenerate fold–Hopf bifurcation of codimension three in a differential system of the form

$$\begin{aligned} {\dot{x}}=-x-y-z,\ {\dot{y}}=x+ay+bxy,\ {\dot{z}}=bx-cz. \end{aligned}$$

We prove that the classical theory of the fold–Hopf bifurcation cannot be applied for studying the system’s behavior. However, using tools from averaging theory we prove the existence of periodic orbits generated by this bifurcation.

Appendix

In this section we recall the main results concerning non-degenerate fold–Hopf bifurcations needed to deal with degenerate cases. For more details we refer the reader at [6].

Consider a system in the form

$$\begin{aligned} {\dot{x}}=f(x,\alpha ), \end{aligned}$$

(31)

$x\in {\mathbb {R}}^{3},\alpha \in {\mathbb {R}}^{2},$ f smooth, which has an equilibrium $x=0$ with the eigenvalues $\lambda _{1}=\nu (\alpha ),$ $\lambda =\mu (\alpha )+i\omega (\alpha )$ and ${\bar{\lambda }}$ for $||\alpha ||=\sqrt{\alpha _{1}^{2}+\alpha _{2}^{2}}$ small enough, such that $\nu (0)=\mu (0)=0$ and $\omega (0)=\omega _{0}>0,$ i.e. for $\alpha =0$ the equilibrium $x=0$ has a zero and two purely complex eigenvalues. The reader will notice that we write $x=0$ and $\alpha =0$ for $x=\left( 0,0,0\right)$ and $\alpha =\left( 0,0\right) .$

Expanding $f(x,\alpha )$ with respect to x at $x=0,$ the system (31) reads

$$\begin{aligned} {\dot{x}}=a(\alpha )+J(\alpha )x+F(x,\alpha ), \end{aligned}$$

(32)

where $a(0)=0$ and $F(x,\alpha )=O(||x||^{2}).$ The system can be further put in the form

$$\begin{aligned} \begin{array}{ccc} {\dot{u}} &{} = &{} \Gamma \left( \alpha \right) +\nu \left( \alpha \right) u+g(u,z,{\bar{z}},\alpha ), \\ {\dot{z}} &{} = &{} \Omega \left( \alpha \right) +\lambda \left( \alpha \right) z+h(u,z,{\bar{z}},\alpha ), \end{array} \end{aligned}$$

(33)

where $u=\left\langle p_{0}(\alpha ),x\right\rangle ,z=\left\langle p_{1}(\alpha ),x\right\rangle$ and $p_{0}(\alpha )\in {\mathbb {R}}^{3},p_{1}(\alpha )\in {\mathbb {C}}^{3}$ are two adjoint eigenvectors given by

$$\begin{aligned} J^{T}(\alpha )p_{0}(\alpha )=\nu \left( \alpha \right) p_{0}(\alpha )\ \ \text {and}\ \ J^{T}(\alpha )p_{1}(\alpha )={\bar{\lambda }}(\alpha )p_{1}(\alpha ) \end{aligned}$$

(34)

such that

$$\begin{aligned} \left\langle p_{0}(\alpha ),q_{0}(\alpha )\right\rangle =\left\langle p_{1}(\alpha ),q_{1}(\alpha )\right\rangle =1\ \ \text {and}\ \ \left\langle p_{1}(\alpha ),q_{0}(\alpha )\right\rangle =\left\langle p_{0}(\alpha ),q_{1}(\alpha )\right\rangle =0, \end{aligned}$$

(35)

for all $||\alpha ||$ small enough. As usual, for any two vectors we have $\left\langle u,v\right\rangle =\overline{u}_{1}v_{1}+\overline{u}_{2}v_{2}+ \overline{u}_{3}v_{3}.$

The vectors $q_{0}(\alpha )$ and $q_{1}(\alpha )$ are two eigenvectors corresponding to the eigenvalues $\nu \left( \alpha \right)$ and $\lambda =\mu (\alpha )+i\omega (\alpha ),$ i.e. $J(\alpha )q_{0}(\alpha )=\nu \left( \alpha \right) q_{0}(\alpha )$ and $J(\alpha )q_{1}(\alpha )=\lambda (\alpha )q_{1}(\alpha ).$ We can write

$$\begin{aligned} x=uq_{0}(\alpha )+zq_{1}(\alpha )+{\bar{z}}\bar{q}_{1}(\alpha ). \end{aligned}$$

(36)

Here

$$\begin{aligned} \Gamma \left( \alpha \right) =\left\langle p_{0}(\alpha ),a\left( \alpha \right) \right\rangle ,\Omega \left( \alpha \right) =\left\langle p_{1}(\alpha ),a\left( \alpha \right) \right\rangle \end{aligned}$$

(37)

are smooth functions of $\alpha$ with $\Gamma \left( 0\right) =\Omega \left( 0\right) =0$ and

$$\begin{aligned} \begin{array}{ccc} g(u,z,{\bar{z}},\alpha ) &{} = &{} \left\langle p_{0}(\alpha ),F(uq_{0}(\alpha )+zq_{1}(\alpha )+{\bar{z}}\bar{q}_{1}(\alpha ),\alpha )\right\rangle , \\ h(u,z,{\bar{z}},\alpha ) &{} = &{} \left\langle p_{1}(\alpha ),F(uq_{0}(\alpha )+zq_{1}(\alpha )+{\bar{z}}\bar{q}_{1}(\alpha ),\alpha )\right\rangle , \end{array} \end{aligned}$$

(38)

are smooth functions of their variables whose Taylor expansions in $u,z,\bar{ z}$ start with quadratic terms:

$$\begin{aligned} \begin{array}{ccc} g(u,z,{\bar{z}},\alpha ) &{} = &{} \sum _{j+k+l\ge 2}\frac{1}{j!k!l!} g_{jkl}(\alpha )u^{j}z^{k}{\bar{z}}^{l}, \\ h(u,z,{\bar{z}},\alpha ) &{} = &{} \sum _{j+k+l\ge 2}\frac{1}{j!k!l!} h_{jkl}(\alpha )u^{j}z^{k}{\bar{z}}^{l}. \end{array} \end{aligned}$$

(39)

Using the changes

$$\begin{aligned} \begin{array}{lll} v &{} = &{} u+\delta _{0}+\delta _{1}u+\delta _{2}z+\delta _{3}{\bar{z}}+\frac{1}{2 }V_{020}z^{2}+\frac{1}{2}V_{002}{\bar{z}}^{2}+V_{110}uz+V_{101}u{\bar{z}}, \\ w &{} = &{} z+\Delta _{0}+\Delta _{1}u\!+\!\Delta _{2}z\!+\!\Delta _{3}{\bar{z}}\!+\!\frac{1}{2 }W_{200}u^{2}\!+\!\frac{1}{2}W_{020}z^{2}+\frac{1}{2}W_{002}{\bar{z}}^{2}+W_{101}u {\bar{z}}\!+\!W_{011}z{\bar{z}}, \end{array}\nonumber \\ \end{aligned}$$

(40)

where $\delta _{i}\left( \alpha \right) ,\Delta _{i}\left( \alpha \right)$ are smooth functions, $\delta _{i}\left( 0\right) =\Delta _{i}\left( 0\right) =0,i=0,1,2,3,$ the system (33) can be further brought to Poincaré normal form. More exactly we have the following Theorem [6].

Theorem 4.1

If (G.1) $g_{200}\left( 0\right) \ne 0,$ then there exists a locally defined smooth, invertible variable transformation of the form (40), smoothly depending on the parameters, that for $||\alpha ||$ small enough brings the system (33) in the form

$$\begin{aligned} \begin{array}{lll} \dot{v} &{} = &{} \gamma (\alpha )+\frac{1}{2}G_{200}(\alpha )v^{2}+G_{011}(\alpha )|w|^{2}+\frac{1}{6}G_{300}(\alpha )v^{3}+G_{111}(\alpha )v|w|^{2}\\ &{}&{}+\,O\left( ||(v,w,\bar{w})||^{4}\right) , \\ \dot{w} &{} = &{} R\left( \alpha \right) w+H_{110}(\alpha )vw+\frac{1}{2} H_{210}(\alpha )v^{2}w+\frac{1}{2}H_{021}(\alpha )w|w|^{2}\\ &{}&{}+\,O\left( ||(v,w, \bar{w})||^{4}\right) , \end{array} \end{aligned}$$

(41)

where $v\in {\mathbb {R}} ,w\in {\mathbb {C}}$ and $\gamma \left( \alpha \right) ,G_{jkl}\left( \alpha \right)$ are real-valued smooth functions while $R(\alpha ),H_{jkl}\left( \alpha \right)$ are complex-valued smooth functions such that $\gamma (0)=0,R(0)=i\omega _{0}.$ Their expressions at $\alpha =0$ are given below.

If in addition (G.2) $G_{011}\left( 0\right) \ne 0,$ the Poincaré form (41) can be further reduced to (44) by means of time reparametrization

$$\begin{aligned} dt=\left( 1+e_{1}\left( \alpha \right) v+e_{2}\left( \alpha \right) |w|^{2}\right) d\tau \end{aligned}$$

(42)

and transformations

$$\begin{aligned} u=v+e_{4}\left( \alpha \right) v+\frac{1}{2}e_{3}\left( \alpha \right) v^{2} \text { and }z=w+K\left( \alpha \right) vw, \end{aligned}$$

(43)

where $e_{i}\in {\mathbb {R}} ,K\in {\mathbb {C}}$ are smooth functions and $e_{4}\left( 0\right) =0.$ More exactly, the system (41) is locally smoothly orbitally equivalent near the origin to the system

$$\begin{aligned} \begin{array}{ccc} {\dot{u}} &{} = &{} \delta (\alpha )+B(\alpha )u^{2}+C(\alpha )|z|^{2}+O(||u,z, {\bar{z}}||^{4}), \\ {\dot{z}} &{} = &{} \Sigma (\alpha )z+D(\alpha )uz+E(\alpha )u^{2}z+O(||u,z,{\bar{z}} ||^{4}), \end{array} \end{aligned}$$

(44)

where $u\in {\mathbb {R}} ,z\in {\mathbb {C}}$ and $\delta (\alpha ),B\left( \alpha \right) ,C(\alpha ),E(\alpha )$ are real-valued smooth functions while $\Sigma (\alpha ),D(\alpha )$ are complex-valued smooth functions (given below for $\alpha =0$) such that $\delta (0)=0,\Sigma (0)=i\omega _{0}.$

Finally, if (G.3) $E\left( 0\right) \ne 0$ is satisfied, using the linear scaling $u=\frac{B\left( \alpha \right) }{E\left( \alpha \right) }\xi ,z= \frac{B^{3}\left( \alpha \right) }{C\left( \alpha \right) E^{2}\left( \alpha \right) }\zeta$ and the time-reparametrization, $t=\frac{E\left( \alpha \right) }{B^{2}\left( \alpha \right) }\tau ,$ the system (44) leads to normal form:

$$\begin{aligned} \begin{array}{lll} {\dot{\xi }} &{} = &{} \beta _{1}(\alpha )+\xi ^{2}+s|\zeta |^{2}+O(||\left( \xi ,\zeta ,\bar{\zeta }\right) ||^{4}), \\ {\dot{\zeta }} &{} = &{} (\beta _{2}(\alpha )+i\omega _{1}(\alpha ))\zeta +(\theta (\alpha )+i\omega _{2}(\alpha ))\xi \zeta +\xi ^{2}\zeta +O(||\left( \xi ,\zeta ,{\bar{\zeta }}\right) ||^{4}), \end{array} \end{aligned}$$

(45)

where $s=sign\left[ B(0)C(0)\right] =\pm 1$ and

$$\begin{aligned} \beta _{1}(\alpha )= & {} \frac{E^{2}(\alpha )}{B^{3}(\alpha )}\delta (\alpha ), \ \ \beta _{2}(\alpha )=\frac{E(\alpha )}{B^{2}(\alpha )}{Re}(\Sigma (\alpha )), \nonumber \\ \theta (\alpha )+i\omega _{2}(\alpha )= & {} \frac{D(\alpha )}{B(\alpha )}, \ \ \ \omega _{1}(\alpha )=\frac{E(\alpha )}{B^{2}(\alpha )}{Im} (\Sigma (\alpha )), \end{aligned}$$

(46)

with $||\left( \xi ,\zeta ,\bar{\zeta }\right) ||^{4}=\left( \xi ^{2}+|\zeta |^{2}\right) ^{2}.$

This form (45), for (G.4) $\theta _{0}=\theta \left( 0\right) \ne 0$ and (G.5) the map $\alpha \longmapsto \beta \left( \alpha \right)$ is regular at $\alpha =0,$ leads to the truncated normal form

$$\begin{aligned} {\dot{\xi }}= & {} \beta _{1}+\xi ^{2}+sr^{2}, \nonumber \\ \dot{r}= & {} r\left( \beta _{2}+\theta \left( \alpha \right) \xi +\xi ^{2}\right) , \nonumber \\ {\dot{\varphi }}= & {} \omega _{1}+\omega _{2}\xi . \end{aligned}$$

(47)

$\square$

The coefficients needed above in Theorem 4.1 are:

$$\begin{aligned} G_{200}(0)= & {} g_{200},G_{011}(0)=g_{011},H_{110}(0)=h_{110}, \nonumber \\ G_{300}(0)= & {} g_{300}-\frac{6}{\omega _{0}}Im\left( g_{110}h_{200}\right) , \nonumber \\ G_{111}(0)= & {} g_{111}-\frac{1}{\omega _{0}}Im\left( 2g_{110}h_{011}+g_{020}h_{101}\right) ,\nonumber \\ H_{210}(0)= & {} h_{210}+\frac{i}{\omega _{0}}\left[ h_{200}\left( h_{020}-2g_{110}\right) -\left| h_{101}\right| ^{2}-h_{011}\bar{h} _{200}\right] , \nonumber \\ H_{021}(0)= & {} h_{021}+\frac{i}{\omega _{0}}\left( h_{011}h_{020}-\frac{1}{2} g_{020}h_{101}-2\left| h_{011}\right| ^{2}-\frac{1}{3}\left| h_{002}\right| ^{2}\right) ,\qquad \end{aligned}$$

(48)

respectively,

$$\begin{aligned} B(0)= & {} \frac{1}{2}G_{200}(0),C(0)=G_{011}(0), \nonumber \\ D(0)= & {} H_{110}(0)-i\omega _{0}\frac{G_{300}(0)}{3G_{200}(0)}, \nonumber \\ E(0)= & {} \frac{1}{2}Re\left( H_{210}(0)+H_{110}(0)\left( \frac{Re\left( H_{021}(0)\right) }{G_{011}(0)}-\frac{G_{300}(0)}{G_{200}(0)}+\frac{ G_{111}(0)}{G_{011}(0)}\right) \right. \nonumber \\&\left. -\frac{H_{021}(0)G_{200}(0)}{2G_{011}(0)} \right) . \end{aligned}$$

(49)

The needed coefficients for our system (2) are:

$$\begin{aligned} G_{200}&=-\frac{2\left( a-1\right) ^{2}}{a^{2}+a-1},G_{011}=-\frac{ 2a^{3}\left( 2a-1\right) \left( a-1\right) ^{2}}{a^{2}+a-1},\nonumber \\ G_{111}&=a^{2}\left( 2a-1\right) \left( a-1\right) ^{3}\frac{ 2a^{3}+5a-1}{\left( a+a^{2}-1\right) ^{2}}, \nonumber \\ G_{300}&=6\left( a^{2}+1\right) \frac{\left( a-1\right) ^{3}}{\left( a^{2}+a-1\right) ^{2}},H_{110}=\frac{ \left( a-1\right) \left( a+i\omega _{0}-1\right) }{2\left( 2a-1\right) \left( a^{2}+a-1\right) }\left( a^{3}+i\omega _{0}a^{2}+2a-1\right) ,\nonumber \\ H_{210}&=\frac{1}{a\omega _{0}}\frac{\left( a-1\right) ^{3}}{\left( a^{2}+a-1\right) ^{2}}\left( -a\omega _{0}\left( a+4a^{2}+2\right) +i\left( \frac{11}{4}a^{3}-2a^{2}-\frac{3}{2}a+4a^{4}-\frac{1}{4}\right) \right) , \nonumber \\ H_{021}&=-\frac{2a\left( a-1\right) ^{3}\left( 2a-1\right) }{\omega _{0}\left( a^{2}+a-1\right) ^{2}}\left( \frac{1}{4}a\omega _{0}\left( 3a+2a^{2}+4a^{3}-1\right) \right. \nonumber \\&\quad \left. +\,i\left( \frac{2}{3}a+\frac{1}{6}a^{2}+\frac{1}{4} a^{3}-\frac{3}{2}a^{4}-a^{5}-\frac{1}{4}\right) \right) , \end{aligned}$$

(50)

respectively,

$$\begin{aligned}&B(0)=-\frac{\left( a-1\right) ^{2}}{\left( a^{2}+a-1\right) },\ \ C(0)=- \frac{2a^{3}\left( 2a-1\right) \left( a-1\right) ^{2}}{a^{2}+a-1}, \nonumber \\&D\left( 0\right) =\frac{1}{2}\frac{a-1}{a^{2}+a-1}\left( a^{3}+3i\omega _{0}a^{2}+3i\omega _{0}-1\right) ,\nonumber \\&E\left( 0\right) =\frac{\left( a-1\right) ^{3}}{16a\left( a+a^{2}-1\right) }\left( 12a^{3}+2a^{2}+5a+1\right) . \end{aligned}$$

(51)

We have also $\left| h_{101}\right| ^{2}=-\frac{1}{4} \left( a\!-\!1\right) ^{3}\frac{2a+3a^{3}\!-\!1}{a\left( a+a^{2}\!-\!1\right) ^{2}},$ $\left| h_{011}\right| ^{2}\!=\!-a^{5}\left( 2a-1\right) \frac{\left( a-1\right) ^{3}}{\left( a+a^{2}-1\right) ^{2}}$ and $\left| h_{002}\right| ^{2}=-a^{2}\left( a-1\right) ^{3}\frac{\left( 2a-1\right) ^{2}}{\left( a+a^{2}-1\right) ^{2}}.$