Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)

Hebai Chen; Xingwu Chen

doi:10.3934/dcdsb.2018130

Article Contents

2018, Volume 23, Issue 10: 4141-4170. Doi: 10.3934/dcdsb.2018130

This issue Previous Article On a free boundary problem for a nonlocal reaction-diffusion model Next Article On the Cauchy problem for the XFEL Schrödinger equation

Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)

Hebai Chen^1, and
Xingwu Chen^2, ,

1.
College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, China
2.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Xingwu Chen(xingwu.chen@hotmail.com)
* Corresponding author: Xingwu Chen(xingwu.chen@hotmail.com)

Received: July 31, 2017

Revised: November 30, 2017

Early access: April 2018

Published: September 2018

The first author is supported by NSFC grant 11572263. The second author is supported by NSFC grant 11471228.

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Abstract

The degenerate Bogdanov-Takens system $\dot x = y-(a_1x+a_2x^3),~\dot y = a_3x+a_4x^3$ has two normal forms, one of which is investigated in [Disc. Cont. Dyn. Syst. B (22)2017,1273-1293] and global behavior is analyzed for general parameters. To continue this work, in this paper we study the other normal form and perform all global phase portraits on the Poincaré disc. Since the parameters are not restricted to be sufficiently small, some classic bifurcation methods for small parameters, such as the Melnikov method, are no longer valid. We find necessary and sufficient conditions for existences of limit cycles and homoclinic loops respectively by constructing a distance function among orbits on the vertical isocline curve and further give the number of limit cycles for parameters in different regions. Finally we not only give the global bifurcation diagram, where global existences and monotonicities of the homoclinic bifurcation curve and the double limit cycle bifurcation curve are proved, but also classify all global phase portraits.

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Mathematics Subject Classification: Primary: 34C07, 34C23, 34C37, 34K18.

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Figure 1. Equilibria at infinity

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Figure 2. Limit cycles surrounding $O$

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Figure 3. Discussion in the case that $-a\leq b<0$

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Figure 4. Two large limit cycles

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Figure 5. $a$, $b$ satisfy condition (c4)

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Figure 6. Location of $x_A, x_B$

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Figure 7. Discussion on existence of limit cycles

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Figure 8. Discussion about the perturbation system

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Figure 9. Discussion about $\Pi(\rho, a, b)$

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Figure 10. Bifurcation diagram of (5) if $N_2^+$ does not intersect $DL, HL$

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Figure 11. Global phase portraits for $a\le 0$

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Figure 12. Global phase portraits for $a>0$

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Figure 13. Bifurcation diagram of (5) if $N_2^+$ intersects $DL, HL$

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Figure 14. More global phase portraits if $N_2^+$ intersects $DL, HL$

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Figure 15. More global phase portraits for special parameters

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Figure 16. The numerical phase portraits when $a = 1$

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Table 1. Equilibria in finite planes

possibilities of $(a, b)$		location of equilibria	types and stability
$a<0$	$b<-2\sqrt{-a}$	$E_0$	$E_0$ unstable bidirectional node
	$b=-2\sqrt{-a}$	$E_0$	$E_0$ unstable proper node
	$-2\sqrt{-a}<b<0$	$E_0$	$E_0$ unstable focus
	$b=0$	$E_0$	$E_0$ stable weak focus of order one
	$0<b<2\sqrt{-a}$	$E_0$	$E_0$ stable focus
	$b=2\sqrt{-a}$	$E_0$	$E_0$ stable proper node
	$b>2\sqrt{-a}$	$E_0$	$E_0$ stable bidirectional node
$a=0$	$b\geq0$	$E_0$	$E_0$ stable degenerate node
	$b<0$	$E_0$	$E_0$ unstable degenerate node
$a>0$	$b<-3a-2\sqrt{2a}$	$E_R$, $E_0$, $E_L$	$E_0$ saddle; $E_R$, $E_L$ unstable nodes
	$b=-3a-2\sqrt{2a}$	$E_R$, $E_0$, $E_L$	$E_0$ saddle; $E_R$, $E_L$ unstable proper nodes
	$-3a-2\sqrt{2a}<b<-3a$	$E_R$, $E_0$, $E_L$	$E_0$ saddle; $E_R$, $E_L$ unstable foci
	$b=-3a$	$E_R$, $E_0$, $E_L$	$E_0$ saddle;
			$E_R$, $E_L$ unstable weak foci of order one
	$-3a<b<-3a+2\sqrt{2a}$	$E_R$, $E_0$, $E_L$	$E_0$ saddle; $E_R$, $E_L$ stable foci
	$b=-3a+2\sqrt{2a}$	$E_R$, $E_0$, $E_L$	$E_0$ saddle; $E_R$, $E_L$ stable proper nodes
	$b>-3a+2\sqrt{2a}$	$E_R$, $E_0$, $E_L$	$E_0$ saddle; $E_R$, $E_L$ stable nodes

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Table 2. Limit cycles and homoclinic loops to Figure 10

$(a, b)\in $	limit cycles	homoclinic loops
$I, N_1^-, II$	one stable, small, surrounding $E_0$	no
$III, N_1^+, IV$	no	no
$V, N_{2}^+, VI$	no	no
$DL$	one semi-stable, large	no
	two, large,	no
$X$	the inner one is unstable, the outer one is stable
$HL$	one stable, large	one unstable figure-eight type
	one stable, large;
$IX$	one unstable, small, surrounding $E_L$;	no
	one unstable, small, surrounding $E_R$;
$VII, VIII, N_2^-$	one stable, large	no

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Table 3. Limit cycles and homoclinic loops

$(a, b)\in $	limit cycles	homoclinic loops
$\tilde V, N_{21}^+$	no	no
$DL_1, DL_2, DL_3$	one semi-stable, large	no
	two, large,	no
$\tilde X, N_{22}^+, XI$	inner one is unstable, outer one is stable
$HL_1, HL_2, HL_3$	one stable, large	one unstable figure-eight type
	one stable, large; two unstable, small;
$\widetilde{IX}, N_{23}^+, XII$	surrounding $E_L, E_R$ separately;	no

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