The heteroclinic and codimension-4 bifurcations of a triple SD oscillator

Abstract

In this paper, the complicated heteroclinic and codimension-four bifurcations of a triple SD (smooth and discontinuous) oscillator are investigated by analyzing the bifurcation sets in three-dimensional parameter space. The structure of the transition set including the equilibrium bifurcation set and a special kind of heteroclinic orbit bifurcation set is constructed comprising of a catastrophe point of the fifth order, the catastrophe curves of third order and also the catastrophe surfaces of the first order, respectively, according to the restoring forces and also the potentials, respectively. Also, a theorem of structural stability of heteroclinic orbit in 2-dimensional Hamilton system is introduced to find the heteroclinic bifurcation set. The equilibria and the phase structures are classified and shown in details on the transition set and the enclosed structurally stable areas for smooth and discontinuous cases, respectively. The normal forms for each bifurcation surface are built up showing the complex supercritical subcritical pitchfork bifurcations and also the double saddle-node bifurcations, along with the bifurcations of homoclinic and heteroclinic orbit. Taken one of the bifurcation surfaces as an example, the complicated bifurcation is investigated by employing subharmonic Melnikov functions including Hopf, double Hopf, the closed orbit and also the homoclinic/heteroclinic bifurcations. The results presented herein this paper enriched the complex dynamic behavior for the geometrical nonlinear systems.

Appendices

Phase structures

1.1 Smooth cases

Phase portraits of smooth case for $\alpha >0$ and $\gamma >0$ are plotted in Fig. 11. Orange lines represent small periodic orbits, green and blue lines represent large periodic orbits which encircle heteroclinic or homoclinic orbits, and black lines represent heteroclinic or homoclinic orbits.

Fig. 11

Phase portraits for $\alpha >0$ and $\gamma >0$ with a catastrophe point, 5 catastrophe curves, 7 bifurcation surfaces and 5 areas (The value of parameters and the belonged set are shown in each figure legend)

Fig. 12

Bifurcation diagrams for $\alpha =0$: a. equilibrium bifurcation surfaces; b. bifurcation sets on $\beta$-$\gamma$ plane. (Bifurcation sets divide the $(\beta ,\gamma )$ plane into three persistent regions, marked $\varOmega _3$, $\varOmega _4$ and $\varOmega _5$, for which the corresponding phase portraits are persistent, on the boundaries $\textrm{B}_{3-1}$, $\textrm{B}_{3-2}$, $\textrm{B}_5$ $\textrm{B}_7$, the portraits are nonpersistent)

1.2 Discontinuous cases

For $\alpha =0$ and $\gamma >0$, only $x=0$ is discontinuous. The restoring force can be written as $f(x,0,\beta ,\gamma )=3x-\textrm{sgn}x-\dfrac{x+\beta }{\sqrt{(x+\beta )^2+\gamma ^2}}-\dfrac{x-\beta }{\sqrt{(x-\beta )^2+\gamma ^2}}$. The bifurcation diagram along with the bifurcation sets is shown in Fig. 12. And the phase portraits corresponding to each set and each area are shown in Fig. 13.

Fig. 13

Phase portraits for $\alpha =0$ (The value of parameters and the belonged set are shown in each figure legend)

For $\alpha >0$, $\beta >0$ and $\gamma =0$, $x=\beta$ and $x=-\beta$ are discontinuous. The restoring force can be written as $f(x,\alpha ,\beta ,0)=-\dfrac{x}{\sqrt{x^2+\alpha ^2}}-\textrm{sgn}(x+\beta )-\textrm{sgn}(x-\beta )$. The bifurcation diagram along with the bifurcation sets is shown in Fig. 14. The corresponding phase portraits are shown in Fig. 15.

For $\alpha =0$, $\beta >0$ and $\gamma =0$, $x=0$, $x=\beta$ and $x=-\beta$ are all discontinuous. The restoring force can be written as $f(0,\beta ,0)=3x-\textrm{sgn}x-\textrm{sgn}(x+\beta )-\textrm{sgn}(x-\beta )$. The bifurcation diagram is shown in Fig. 16 for x versus $\beta$. The phase portraits are plotted in Fig. 17.

Relationship between bifurcation surfaces and catastrophe curves

To prove the relationship between bifurcation surfaces and catastrophe curves shown in Eq. (8), set $\textrm{B}_1$ to $\textrm{B}_7$ are divided into a series of sets as shown in the following

Fig. 14

Bifurcation diagrams for $\gamma =0$: a. equilibrium bifurcation surfaces; b. bifurcation sets on $\alpha$-$\beta$ plane. (Bifurcation sets divide the $(\alpha ,\beta )$ plane into five persistent regions, marked $\varOmega _1$ to $\varOmega _5$, for which the corresponding phase portraits are persistent, on the boundaries $\textrm{B}_1$ to $\textrm{B}_7$, the portraits are nonpersistent)

Fig. 15

Phase portraits for $\gamma =0$ (The value of parameters and the belonged set are shown in each figure legend)

$$\begin{aligned}&{\mathcal {M}}_+=\{ (\alpha ,\beta ,\gamma )\vert \exists x_1<0<x_2,\\&f(x_i)=f'(x_i)=0,i=1,2, f''(x_1)<0<f''(x_2) \},\\&{\mathcal {M}}_-=\{ (\alpha ,\beta ,\gamma )\vert \exists x_1<0<x_2,\\&f(x_i)=f'(x_i)=0,i=1,2, f''(x_1)>0>f''(x_2) \},\\&{\mathcal {N}}_-=\{ (\alpha ,\beta ,\gamma )\vert \exists x>0, f(x)<0 \},\\&{\mathcal {N}}_+=\{ (\alpha ,\beta ,\gamma )\vert \forall x>0, f(x)>0 \},\\&{\mathcal {E}}_-^i=\{ (\alpha ,\beta ,\gamma )\vert C_i(\alpha ,\beta ,\gamma )<0 \},\\&{\mathcal {E}}_+^i=\{ (\alpha ,\beta ,\gamma )\vert C_i(\alpha ,\beta ,\gamma )>0 \},\quad (i=1,3,5)\\&{\mathcal {G}}=\{ (\alpha ,\beta ,\gamma )\vert \exists x_i,\\&V(x_i)=V(0), V'(x_i)=0, V''(x_i)<0,i=1,2 \},\\&{\mathcal {K}}=\{ (\alpha ,\beta ,\gamma )\vert \exists x_1<0<x_2,\\&f(x_i)=f'(x_i)=f''(x)=0,i=1,2 \}.\\ \end{aligned}$$

It is obvious that ${\mathcal {N}}_\pm$ and ${\mathcal {E}}_\pm ^i$ are areas, ${\mathcal {M}}_\pm$ and ${\mathcal {G}}$ are surfaces, ${\mathcal {K}}$ is curve. And we have

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {M}}_+\cap {\mathcal {M}}_-={\mathcal {N}}_+\cap {\mathcal {N}}_-={\mathcal {E}}_+^i\cap {\mathcal {E}}_-^i=\varnothing ,\quad \partial ^2{\mathcal {E}}_\pm ^i=\varnothing \\ \partial {\mathcal {E}}_-^i=\partial {\mathcal {E}}_+^i=\{ (\alpha ,\beta ,\gamma )\vert C_i(\alpha ,\beta ,\gamma )=0 \},\\ \partial {\mathcal {N}}_-=\partial {\mathcal {N}}_+={\mathcal {M}}_+, \quad \partial {\mathcal {M}}_-=\partial {\mathcal {M}}_+={\mathcal {K}}. \end{array}\right. } \end{aligned}$$

(B1)

Also, we give some lemmas about the characteristics of restoring force f(x) without proving:

Lemma 1

$f(-x)=-f(x)$, $\lim \limits _{x\rightarrow \pm \infty }f(x)=\pm \infty$ and $f^{(2n)}(0)=0, n\in {\mathbb {N}}$;

Lemma 2

f(x) has at most 3 zero points for $x\in (0,+\infty )$;

Lemma 3

f(x) has at most 3 extreme points and 2 inflection points for $x\in (0,+\infty )$;

Lemma 4

if $f'(0)=f'''(0)=0$, then $f^{(5)}(0)<0\iff (\alpha ,\beta ,\gamma )\in {\mathcal {N}}_-$;

Lemma 5

if $f'(0)=f'''(0)=0$, then $f^{(5)}(0)>0\iff (\alpha ,\beta ,\gamma )\in {\mathcal {N}}_+$;

Lemma 6

if $f'(0)=0$ and $(\alpha ,\beta ,\gamma )\in {\mathcal {M}}_+$, then $C_3(\alpha ,\beta ,\gamma )>0$.

(1)

Set $\textrm{B}_1$ can be divided into $\textrm{B}_1={\mathcal {E}}_+^1\cap {\mathcal {M}}_+$ and $\partial \textrm{B}_1$ can be divided into $\partial \textrm{B}_1=(\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {M}}_+)\cup ({\mathcal {E}}_+^1\cap \partial {\mathcal {M}}_\pm )\cup (\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {M}}_\pm )$.

We have $\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {M}}_+=\textrm{L}_4$, $\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {M}}_\pm =\textrm{P}$ and

$$\begin{aligned}{} & {} {\mathcal {E}}_+^1\cap \partial {\mathcal {M}}_\pm =\lim \limits _{x_i\rightarrow 0}\{ (\alpha ,\beta ,\gamma )\vert C_1(\alpha ,\beta ,\gamma )>0;\nonumber \\ {}{} & {} \exists x_1<0<x_2, f(x_i)=f'(x_i)=0,i=1,2, f''(x_1)\nonumber \\ {}{} & {} <0<f''(x_2) \}. \end{aligned}$$

(B2)

Fig. 16

Bifurcation diagrams for $\alpha =0$ and $\gamma =0$

Fig. 17

Phase portraits for $\alpha =0$ and $\gamma =0$ (The value of parameters and the belonged set are shown in each figure legend)

Assume that $f(x)=ax(x^2-x_i^2)^2+bx^3(x^2-x_i^2)^3+cx^5(x^2-x_i^2)^4+o(x^7)$, which is

$$\begin{aligned}{} & {} f(x)=ax_i^4x-(2ax_i^2+bx_i^6)x^3+(a+3bx_i^4+cx_i^8)\nonumber \\ {}{} & {} x^5+o(x^7), \end{aligned}$$

(B3)

and let Eq. (B3) be the Taylor series of the restoring force, so that $C_1(\alpha ,\beta ,\gamma )=ax_i^4$, $C_3(\alpha ,\beta ,\gamma )=-(2ax_i^2+bx_i^6)$ and $C_5(\alpha ,\beta ,\gamma )=(a+3bx_i^4+cx_i^8)$, where $x_i=x_i(\alpha ,\beta ,\gamma )$ and $a>0$. Let $x_i\rightarrow 0$ and we can obtain $\lim \limits _{x_i\rightarrow 0}C_1(\alpha ,\beta ,\gamma )=0$, $\lim \limits _{x_i\rightarrow 0}C_3(\alpha ,\beta ,\gamma )=0$ and $\lim \limits _{x_i\rightarrow 0}C_5(\alpha ,\beta ,\gamma )=a>0$.

Therefore, we have ${\mathcal {E}}_+^1\cap \partial {\mathcal {M}}_\pm =\textrm{L}_5$, and $\partial \textrm{B}_1=\textrm{L}_4\cup \textrm{L}_5\cup \textrm{P}$.

(2)

Set $\textrm{B}_2$ can be divided into $\textrm{B}_2=\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {E}}_+^3\cap {\mathcal {N}}_-$ and $\partial \textrm{B}_2$ can be divided into $\partial \textrm{B}_2=(\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {E}}_\pm ^3\cap {\mathcal {N}}_-)\cup (\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {E}}_+^3\cap \partial {\mathcal {N}}_\pm )\cup (\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {E}}_\pm ^3\cap \partial {\mathcal {N}}_\pm )$.

From lemma 4 we have $\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {E}}_\pm ^3\cap {\mathcal {N}}_-=\textrm{L}_3$, and form lemma 6 we have $\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {E}}_+^3\cap \partial {\mathcal {N}}_\pm =\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {M}}_+={\mathcal {L}}_4$, we also have $\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {E}}_\pm ^3\cap \partial {\mathcal {N}}_\pm =\textrm{P}$. Therefore, $\partial {B}_2=\textrm{L}_3\cup \textrm{L}_4\cup \textrm{P}$.

(3)

Set $\textrm{B}_3$ can be divided into $\textrm{B}_3={\mathcal {E}}_-^1\cap {\mathcal {M}}_+$ and $\partial \textrm{B}_3$ can be divided into $\partial \textrm{B}_3=(\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {M}}_+)\cup ({\mathcal {E}}_-^1\cap \partial {\mathcal {M}}_\pm )\cup (\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {M}}_\pm )$.

It is obvious that $\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {M}}_+=\textrm{L}_4$, ${\mathcal {E}}_-^1\cap \partial {\mathcal {M}}_\pm ={\mathcal {E}}_-^1\cap {\mathcal {K}}=\textrm{L}_1$ and $\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {M}}_\pm =\textrm{P}$. Therefore, $\partial \textrm{B}_3=\textrm{L}_1\cup \textrm{L}_4\cup \textrm{P}$.

(4)

Set $\textrm{B}_4$ can be divided into $\textrm{B}_4=\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {E}}_+^3\cap {\mathcal {N}}_+$ and $\partial \textrm{B}_4$ can be divided into $\partial \textrm{B}_4=(\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {E}}_\pm ^3\cap {\mathcal {N}}_+)\cup (\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {E}}_+^3\cap \partial {\mathcal {N}}_\pm )\cup (\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {E}}_\pm ^3\cap \partial {\mathcal {N}}_\pm )$.

From lemma 5 we have $\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {E}}_\pm ^3\cap {\mathcal {N}}_+=\textrm{L}_5$. We also have $\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {E}}_+^3\cap \partial {\mathcal {N}}_\pm =\textrm{L}_4$ and $\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {E}}_\pm ^3\cap \partial {\mathcal {N}}_\pm =\textrm{P}$. Therefore, $\partial {B}_4=\textrm{L}_4\cup \textrm{L}_5\cup \textrm{P}$.

(5)

Set $\textrm{B}_5$ can be divided into $\textrm{B}_5={\mathcal {E}}_-^1\cap {\mathcal {M}}_-$ and $\partial \textrm{B}_5$ can be divided into $\partial \textrm{B}_5=(\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {M}}_-)\cup ({\mathcal {E}}_-^1\cap \partial {\mathcal {M}}_\pm )\cup (\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {M}}_\pm )$.

It is obvious that ${\mathcal {E}}_\pm ^1\cap \partial {\mathcal {M}}_\pm =\textrm{L}_1$, $\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {M}}_\pm =\textrm{P}$ and

$$\begin{aligned}{} & {} \partial {\mathcal {E}}_\pm ^1\cap {\mathcal {M}}_-=\lim \limits _{C_1\rightarrow 0^-}\{ (\alpha ,\beta ,\gamma )\vert C_1(\alpha ,\beta ,\gamma )<0;\nonumber \\ {}{} & {} \exists x_1<0<x_2,\nonumber \\{} & {} \quad f(x_i)=f'(x_i)=0,i=1,2, f''(x_1)>0>f''(x_2) \}.\nonumber \\ \end{aligned}$$

(B4)

Considering $f(0)=f(x_i)=0$, $f'(0)<0$, $f'(x_i)=0$, $f''(0)=0$, $f''(x_i)<0$ and lemma 3, we have

• (a). there exists minimum points $x_\textrm{I}\in (0,x_i)$ and $\exists x_\textrm{II}\in (x_i,+\infty )$, $f(x_\textrm{I})<0$, $f(x_\textrm{II})<0$, $f'(x_\textrm{I})=f'(x_\textrm{II})=0$, $f''(x_\textrm{I})>0$, $f''(x_\textrm{II})>0$;

• (b). $\exists x_a\in (x_\textrm{I},x_i)$, $\exists x_b\in (x_i,x_\textrm{II})$, $f''(x_a)=0$, $f''(x_b)=0$;

• (c). $f'(x)$ monotone increases in $(0,x_a)$, monotone decreases in $(x_a,x_b)$ and monotone increases in $(x_b,+\infty )$.

So it is clear that

$$\begin{aligned} f(x_\textrm{I})=\int _0^{x_\textrm{I}}f'(x)dx>f'(0)x_\textrm{I}, \end{aligned}$$

let $f'(0)\rightarrow 0^-$ and we have $f(x_\textrm{I})\rightarrow 0^-$, then we have $f(x)\rightarrow 0^-$ for $\forall x\in (0,x_i)$ because $f(x_\textrm{I})\le f(x)<0$ in $(0,x_i)$. From lemma 2, it can be derived that

$$\begin{aligned} \lim \limits _{f'(0)\rightarrow 0^-}x_i=0. \end{aligned}$$

(B5)

Assume that $f(x)=ax(x^2-x_i^2)^2+bx^3(x^2-x_i^2)^3+cx^5(x^2-x_i^2)^4+o(x^7)$ and $a<0$, which is the same form as Eq. (B3). Then let it be the Taylor series of the restoring force, so that $C_1(\alpha ,\beta ,\gamma )=ax_i^4$, $C_3(\alpha ,\beta ,\gamma )=-(2ax_i^2+bx_i^6)$ and $C_5(\alpha ,\beta ,\gamma )=(a+3bx_i^4+cx_i^8)$, where $x_i=x_i(\alpha ,\beta ,\gamma )$ and $a>0$. Let $x_i\rightarrow 0$ and we can obtain $\lim \limits _{C_1\rightarrow 0^-}C_3(\alpha ,\beta ,\gamma )=0$ and $\lim \limits _{C_1\rightarrow 0^-}C_5(\alpha ,\beta ,\gamma )=a<0$.

Therefore, we have $\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {M}}_-=\textrm{L}_3$, and $\partial \textrm{B}_5=\textrm{L}_1\cup \textrm{L}_3\cup \textrm{P}$.

(6)

Set $\textrm{B}_6$ can be divided into $\textrm{B}_6=\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {E}}_-^3$ and $\partial \textrm{B}_6$ can be divided into $\partial \textrm{B}_6=\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {E}}_\pm ^3$, which is

$$\begin{aligned} \partial \textrm{B}_6=\{ (\alpha ,\beta ,\gamma )\vert C_1(\alpha ,\beta ,\gamma )=0;C_3(\alpha ,\beta ,\gamma )=0 \}. \end{aligned}$$

(B6)

It is obvious that $\partial \textrm{B}_6=\textrm{L}_3\cup \textrm{L}_5\cup \textrm{P}$.

(7)

Set $\textrm{B}_7$ can be divided into $\textrm{B}_7={\mathcal {E}}_-^1\cap {\mathcal {G}}$ and $\partial \textrm{B}_7$ can be divided into $\partial \textrm{B}_7=(\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {G}})\cup ({\mathcal {E}}_-^1\cap \partial {\mathcal {G}})\cup (\partial {\mathcal {E}}_\pm ^1\cap \partial {\mathcal {G}})$, where

$$\begin{aligned}&\partial {\mathcal {G}}=\{ (\alpha ,\beta ,\gamma )\vert \exists x_1<0<x_2, \\&V(x_i)=V(0), V'(x_i)=V''(x_i)=0, i=1,2 \}. \end{aligned}$$

(B6)

Fig. 18

Heteroclinic orbits for Hamilton system: a $a=0.95$, $b=-1$; b $a=1$, $b=-1$; c $a=1.05$, $b=-1$

Fig. 19

Heteroclinic orbits for Hamilton system: a $a=\frac{1}{2}$, $b=-1$, $c=-4$, $q=3.98$; b $a=\frac{1}{2}$, $b=-1$, $c=-4$, $q=4$; c $a=\frac{1}{2}$, $b=-1$, $c=-4$, $q=4.02$

It is obvious that ${\mathcal {E}}_-^1\cap \partial {\mathcal {G}}=\textrm{L}_2\subseteq \textrm{B}_3$. And

$$\begin{aligned}{} & {} \partial {\mathcal {E}}_\pm ^1\cap {\mathcal {G}}=\lim \limits _{C_1\rightarrow 0^-}\{ (\alpha ,\beta ,\gamma )\vert C_1(\alpha ,\beta ,\gamma )<0;\nonumber \\ {}{} & {} \exists x_1<0<x_2,\nonumber \\{} & {} \quad V(x_i)=V(0), V'(x_i)=0, V''(x_i)<0, i=1,2 \}.\nonumber \\ \end{aligned}$$

(B7)

Considering $\int _0^{x_i}f(x)dx=V(x_i)-V(0)=0$, $f(x_i)=0$, $f'(x_i)<0$, lemma 1, lemma 2 and lemma 3, we have

• (a). $\exists x_k\in (0,x_i), x_j\in (x_i,+\infty ), f(x_k)=f(x_i)=f(x_j)=0$;

• (b). there exists minimum $x_\textrm{I}\in (0,x_k)$, maximum $x_\textrm{II}\in (x_k,x_i)$ and minimum $x_\textrm{III}\in (x_i,x_j)$, $f'(x_\textrm{I})=f'(x_\textrm{II})=f'(x_\textrm{III})=0$;

• (c). $f'(x)$ monotone increases in $(0,x_\textrm{I})$.

So it is clear that

$$\begin{aligned} f(x_\textrm{I})=\int _0^{x_\textrm{I}}f'(x)dx>f'(0)x_\textrm{I}, \end{aligned}$$

let $f'(0)\rightarrow 0^-$ and we have $f(x_\textrm{I})\rightarrow 0^-$.

Considering $f(x_\textrm{I})x_k<\int _0^{x_k}f(x)dx<0$, we have $\int _0^{x_k}f(x)dx\rightarrow 0^-$ and $\int _{x_k}^{x_i}f(x)dx=-\int _0^{x_k}f(x)dx\rightarrow 0^+$. So it can be derived that $f(x)\rightarrow 0^-$ for $\forall x\in (0,x_k)$ and $f(x)\rightarrow 0^+$ for $\forall x\in (x_k,x_i)$.

From lemma 2, it can be derived that

$$\begin{aligned} \lim \limits _{f'(0)\rightarrow 0^-}x_i=\lim \limits _{f'(0)\rightarrow 0^-}x_k=0. \end{aligned}$$

(B8)

Assume that $V(x)=ax^2(x^2-x_i^2)^2+bx^4(x^2-x_i^2)^3+cx^6(x^2-x_i^2)^4+o(x^7)$, and $f(x)=V'(x)$, which is

$$\begin{aligned} f(x)= & {} 2ax_i^4x-4(2ax_i^2+bx_i^6)x^3\nonumber \\{} & {} +6(a+3bx_i^4+cx_i^8)x^5+o(x^7), \end{aligned}$$

(B9)

and let Eq. (B9) be the Taylor series of the restoring force, so that $C_1(\alpha ,\beta ,\gamma )=2ax_i^4$, $C_3(\alpha ,\beta ,\gamma )=-4(2ax_i^2+bx_i^6)$ and $C_5(\alpha ,\beta ,\gamma )=6(a+3bx_i^4+cx_i^8)$, where $x_i=x_i(\alpha ,\beta ,\gamma )$ and $a<0$. Let $x_i\rightarrow 0$ and we can obtain $\lim \limits _{C_1\rightarrow 0}C_3(\alpha ,\beta ,\gamma )=0$ and $\lim \limits _{C_1\rightarrow 0}C_5(\alpha ,\beta ,\gamma )=6a<0$.

Therefore, we have $\partial {\mathcal {E}}_\pm ^1\cap {\mathcal {G}}=\textrm{L}_3$, and $\partial \textrm{B}_7=\textrm{L}_2\cup \textrm{L}_3\cup \textrm{P}$.

Structural stability of Hamilton system with heteroclinic orbits

Without loss of generality, a 2-dimensional Hamilton system is considered, whose Hamilton function can be written in the following form

$$\begin{aligned} H(x,y)=\sum \limits _{i=1}^{\infty }\sum \limits _{j=1}^{\infty }a_{i,j}x^iy^j. \end{aligned}$$

(C1)

two examples are given to illustrate the condition of heteroclinic bifurcation of 2-dimensional Hamilton system, which is shown in theorem 1.

Example 1

Let $a_{2,0}=1, a_{1,1}=a, a_{1,2}=b, a_{2,3}=-1, a,b\in {\mathbb {R}}$ and $a_{i,j}\equiv 0$(else), the Hamilton function is $H(x,y,\varvec{p})=x^2+axy+bxy^2-x^2y^3$, where $\varvec{p}=(a,b)^\textrm{T}$, which leads to the following Hamilton system

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=ax+2bxy-3x^2y^2, \\ y'=-2x-ay-by^2+2xy^3. \end{array}\right. } \end{aligned}$$

(C2)

Three saddles $\textrm{A}(0,0)$, $\textrm{B}(0,-\frac{a}{b})$ and $C(x_c(\varvec{p}),y_c(\varvec{p}))$ can be found. When $\varvec{p}=(1,-1)^\textrm{T}$, we have $H\vert _\textrm{A}=H\vert _\textrm{B}=H\vert _\textrm{C}=0$, thus three heteroclinic orbits can be found, as shown in Fig. 18b: $\varGamma _1$ connecting $\textrm{A}(0,0)$ and $\textrm{B}(0,1)$; $\varGamma _2$ connecting $\textrm{B}(0,1)$ and $\textrm{C}(-\frac{1}{3},1)$; $\varGamma _3$ connecting $\textrm{A}(0,0)$ and $\textrm{C}(-\frac{1}{3},1)$. Calculation shows that

$$\begin{aligned}&\frac{\partial H}{\partial a}\vert _B-\frac{\partial H}{\partial a}\vert _A=0,\\&\frac{\partial H}{\partial b}\vert _B-\frac{\partial H}{\partial b}\vert _A=0,\\&\frac{\partial H}{\partial a}\vert _C-\frac{\partial H}{\partial a}\vert _B=x_c(\varvec{p})y_c(\varvec{p})\ne 0,\\&\frac{\partial H}{\partial b}\vert _C-\frac{\partial H}{\partial b}\vert _B=x_c(\varvec{p})y_c^2(\varvec{p})\ne 0, \\&\frac{\partial H}{\partial a}\vert _C-\frac{\partial H}{\partial a}\vert _A=x_c(\varvec{p})y_c(\varvec{p})\ne 0,\\&\frac{\partial H}{\partial b}\vert _C-\frac{\partial H}{\partial b}\vert _A=x_c(\varvec{p})y_c^2(\varvec{p})\ne 0, \end{aligned}$$

which means heteroclinic orbit $\varGamma _1$ is structurally stable, but $\varGamma _2$ and $\varGamma _3$ are structurally unstable. The neighborhood systems are shown in Fig. 18a and c. Heteroclinic orbit $\varGamma _2$ splits into orbits $\varGamma _{1,0}\cup \varGamma _{1,1}$ or $\varGamma _{2,0}\cup \varGamma _{2,1}$, while heteroclinic orbit $\varGamma _3$ splits into orbits $\varGamma _{1,0}\cup \varGamma _{1,2}$ or $\varGamma _{2,0}\cup \varGamma _{2,2}$.

Example 2

Consider the Hamilton function $H(x,y,\varvec{p})=\frac{1}{2}y^2-(x-a)^2(x-b)^2(x^2+cx+q)$, where $\varvec{p}=(a,b,c,q)^\textrm{T}$, which leads to the following Hamilton system

$$\begin{aligned} {\left\{ \begin{array}{ll} x'=y, \\ y'=2(x-a)(x-b)(2x-a-b)(x^2+cx+q)\\ \qquad +(x-a)^2(x-b)^2(2x+c). \end{array}\right. } \end{aligned}$$

(C3)

Three saddles $\textrm{A}(a,0)$, $\textrm{B}(b,0)$ and $C(x_c(\varvec{p}),0)$ can be found. When $\varvec{p}=(\frac{1}{2},-1,-4,4)^\textrm{T}$, we have $H\vert _\textrm{A}=H\vert _\textrm{B}=H\vert _\textrm{C}=0$, and two heteroclinic orbits can be found, as shown in Fig. 19b: $\varGamma _1$ connecting $\textrm{A}(\frac{1}{2},0)$ and $\textrm{B}(-1,0)$; $\varGamma _2$ connecting $\textrm{A}(\frac{1}{2},0)$ and $\textrm{C}(x_c(\varvec{p}),0)$. Calculation shows that

$$\begin{aligned}&\frac{\partial H}{\partial p_i}\vert _A-\frac{\partial H}{\partial p_i}\vert _B=0\ (p_i=a,b,c,q), \\&\frac{\partial H}{\partial c}\vert _C-\frac{\partial H}{\partial c}\vert _A=-x_c(\varvec{p})(x_c(\varvec{p})-a)^2(x_c(\varvec{p})-b)^2\ne 0,\\&\frac{\partial H}{\partial q}\vert _C-\frac{\partial H}{\partial q}\vert _A=-(x_c(\varvec{p})-a)^2(x_c(\varvec{p})-b)^2\ne 0, \end{aligned}$$

which means heteroclinic orbit $\varGamma _1$ is structurally stable, but $\varGamma _2$ is structurally unstable. The neighborhood systems are shown in Fig. 19a and c. Heteroclinic orbit $\varGamma _2$ splits into orbits $\varGamma _{1,0}\cup \varGamma _{1,1}$ or $\varGamma _{2,0}\cup \varGamma _{2,1}$.