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

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December 2018, 23(10): 4141-4170. doi: 10.3934/dcdsb.2018130

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)

Received August 2017 Revised December 2017 Published December 2018 Early access April 2018

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

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to be sufficiently small, some classic bifurcation methods for small parameters, such as the Melnikov method, are no longer valid.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.

Keywords: Bifurcation, degenerate Bogdanov-Takens system, figure-eight loop, global phase portrait, limit cycle.

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

Citation: Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130

References:

[1]	S. M. Baer, B. W. Kooi, Yu. A. Kuznetsov and H. R. Thieme, Multiparametric bifurcation analysis of a basic two-stage population model, SIAM J. Appl. Math., 66 (2006), 1339-1365. doi: 10.1137/050627757.
[2]	J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981.
[3]	C. Castillo-Chavez, Z. Feng and W. Huang, Global dynamics of a Plant-Herbivore model with toxin-determined functional response, SIAM J. Appl. Math., 72 (2012), 1002-1020. doi: 10.1137/110851614.
[4]	H. Chen, X. Chen and J. Xie, Global phase portrait of a degenerate Bogdanov-Takens system with symmetry, Discrete Cont. Dyn. Syst. (Ser. B), 22 (2017), 1273-1293. doi: 10.3934/dcdsb.2017062.
[5]	S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, London, 1994.
[6]	F. Dumortier and C. Li, On the uniqueness of limit cycles surrounding one or more singularities in Liénard equations, Nonlinearity, 9 (1996), 1489-1500. doi: 10.1088/0951-7715/9/6/006.
[7]	F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations, 139 (1997), 41-59. doi: 10.1006/jdeq.1997.3291.
[8]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅰ) Saddle Loop and Two saddle Cycle, J. Differential Equations, 176 (2001), 114-157. doi: 10.1006/jdeq.2000.3977.
[9]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅱ)Cuspidal Loop, J. Differential Equations, 175 (2001), 209-243. doi: 10.1006/jdeq.2000.3978.
[10]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅲ)global centre, J. Differential Equations, 188 (2003), 473-511. doi: 10.1016/S0022-0396(02)00110-9.
[11]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅳ)figure eight-loop, J. Differential Equations, 188 (2003), 512-554. doi: 10.1016/S0022-0396(02)00111-0.
[12]	F. Dumortier, J. Llibre and J. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006.
[13]	F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian integrals, Springer-Verlag, Berlin, 1991.
[14]	F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039. doi: 10.1088/0951-7715/3/4/004.
[15]	A. Gasull, H. Giacomini, S. Pérez-González and J. Torregrosa, A proof of Perko's conjectures for the Bogdanov-Takens system, J. Diff. Equa., 255 (2013), 2655-2671. doi: 10.1016/j.jde.2013.07.006.
[16]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.
[17]	J. Hale, Ordinary Differential Equations, Krieger Publishing Company, Florida, 1980.
[18]	P. Holmes and D. A. Rand, Phase portraits and bifurcations of the nonlinear oscillator $\ddot x+(α+γ x^2)\dot x+β x+δ x^3 = 0$, Int. J. Non-linear Mech., 15 (1980), 449-458.
[19]	E. Horozov, Versal deformations of equivariant vector fields for cases of symmetry of order 2 and 3(In Russian), Trusdy Sem. Petrov., 5 (1979), 163-192.
[20]	A. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations, Nonlinearity, 11 (1998), 1505-1519. doi: 10.1088/0951-7715/11/6/005.
[21]	Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory(Third Edition), Springer-Verlag, New York, 2004.
[22]	Yu. A. Kuznetsov, Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations, Int. J. Bifurc. Chaos, 15 (2005), 3535-3546. doi: 10.1142/S0218127405014209.
[23]	C. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.
[24]	A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597 (1977), 1172-1192.
[25]	L. M. Perko, A global analysis of the Bogdanov-Takens system, SIAM J. Appl. Math., 52 (1992), 1172-1192. doi: 10.1137/0152069.
[26]	L. A. F. Roberto, P. R. da Silva and J. Torregrosa, Asymptotic expansion of the heteroclinic bifurcation for the planar normal form of the 1: 2 resonance, Int. J. Bifurc. Chaos, 26 (2016), 1650017, 8 pp. doi: 10.1142/S0218127416500176.
[27]	S. Ruan and D. Xiao, Global analysis in a Predator-Prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.
[28]	G. Sansone and R. Conti, Non-linear Differential Equations, Pergamon Press, Oxford City, 1964.
[29]	Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1992.

show all references

References:

[1]	S. M. Baer, B. W. Kooi, Yu. A. Kuznetsov and H. R. Thieme, Multiparametric bifurcation analysis of a basic two-stage population model, SIAM J. Appl. Math., 66 (2006), 1339-1365. doi: 10.1137/050627757.
[2]	J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981.
[3]	C. Castillo-Chavez, Z. Feng and W. Huang, Global dynamics of a Plant-Herbivore model with toxin-determined functional response, SIAM J. Appl. Math., 72 (2012), 1002-1020. doi: 10.1137/110851614.
[4]	H. Chen, X. Chen and J. Xie, Global phase portrait of a degenerate Bogdanov-Takens system with symmetry, Discrete Cont. Dyn. Syst. (Ser. B), 22 (2017), 1273-1293. doi: 10.3934/dcdsb.2017062.
[5]	S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, London, 1994.
[6]	F. Dumortier and C. Li, On the uniqueness of limit cycles surrounding one or more singularities in Liénard equations, Nonlinearity, 9 (1996), 1489-1500. doi: 10.1088/0951-7715/9/6/006.
[7]	F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations, 139 (1997), 41-59. doi: 10.1006/jdeq.1997.3291.
[8]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅰ) Saddle Loop and Two saddle Cycle, J. Differential Equations, 176 (2001), 114-157. doi: 10.1006/jdeq.2000.3977.
[9]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅱ)Cuspidal Loop, J. Differential Equations, 175 (2001), 209-243. doi: 10.1006/jdeq.2000.3978.
[10]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅲ)global centre, J. Differential Equations, 188 (2003), 473-511. doi: 10.1016/S0022-0396(02)00110-9.
[11]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅳ)figure eight-loop, J. Differential Equations, 188 (2003), 512-554. doi: 10.1016/S0022-0396(02)00111-0.
[12]	F. Dumortier, J. Llibre and J. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006.
[13]	F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian integrals, Springer-Verlag, Berlin, 1991.
[14]	F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039. doi: 10.1088/0951-7715/3/4/004.
[15]	A. Gasull, H. Giacomini, S. Pérez-González and J. Torregrosa, A proof of Perko's conjectures for the Bogdanov-Takens system, J. Diff. Equa., 255 (2013), 2655-2671. doi: 10.1016/j.jde.2013.07.006.
[16]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990.
[17]	J. Hale, Ordinary Differential Equations, Krieger Publishing Company, Florida, 1980.
[18]	P. Holmes and D. A. Rand, Phase portraits and bifurcations of the nonlinear oscillator $\ddot x+(α+γ x^2)\dot x+β x+δ x^3 = 0$, Int. J. Non-linear Mech., 15 (1980), 449-458.
[19]	E. Horozov, Versal deformations of equivariant vector fields for cases of symmetry of order 2 and 3(In Russian), Trusdy Sem. Petrov., 5 (1979), 163-192.
[20]	A. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations, Nonlinearity, 11 (1998), 1505-1519. doi: 10.1088/0951-7715/11/6/005.
[21]	Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory(Third Edition), Springer-Verlag, New York, 2004.
[22]	Yu. A. Kuznetsov, Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations, Int. J. Bifurc. Chaos, 15 (2005), 3535-3546. doi: 10.1142/S0218127405014209.
[23]	C. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.
[24]	A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597 (1977), 1172-1192.
[25]	L. M. Perko, A global analysis of the Bogdanov-Takens system, SIAM J. Appl. Math., 52 (1992), 1172-1192. doi: 10.1137/0152069.
[26]	L. A. F. Roberto, P. R. da Silva and J. Torregrosa, Asymptotic expansion of the heteroclinic bifurcation for the planar normal form of the 1: 2 resonance, Int. J. Bifurc. Chaos, 26 (2016), 1650017, 8 pp. doi: 10.1142/S0218127416500176.
[27]	S. Ruan and D. Xiao, Global analysis in a Predator-Prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.
[28]	G. Sansone and R. Conti, Non-linear Differential Equations, Pergamon Press, Oxford City, 1964.
[29]	Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1992.

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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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

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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$(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

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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$(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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