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

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

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to be sufficiently small, some classic bifurcation methods for small parameters, such as the Melnikov method, are no longer valid.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 & 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. Google Scholar
[2]	J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, London, 1994. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	F. Dumortier, J. Llibre and J. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006. Google Scholar
[13]	F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian integrals, Springer-Verlag, Berlin, 1991. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990. Google Scholar
[17]	J. Hale, Ordinary Differential Equations, Krieger Publishing Company, Florida, 1980. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory(Third Edition), Springer-Verlag, New York, 2004. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597 (1977), 1172-1192. Google Scholar
[25]	L. M. Perko, A global analysis of the Bogdanov-Takens system, SIAM J. Appl. Math., 52 (1992), 1172-1192. doi: 10.1137/0152069. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	G. Sansone and R. Conti, Non-linear Differential Equations, Pergamon Press, Oxford City, 1964. Google Scholar
[29]	Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1992. Google Scholar

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. Google Scholar
[2]	J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, London, 1994. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	F. Dumortier, J. Llibre and J. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006. Google Scholar
[13]	F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian integrals, Springer-Verlag, Berlin, 1991. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1990. Google Scholar
[17]	J. Hale, Ordinary Differential Equations, Krieger Publishing Company, Florida, 1980. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory(Third Edition), Springer-Verlag, New York, 2004. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	A. Lins, W. de Melo and C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597 (1977), 1172-1192. Google Scholar
[25]	L. M. Perko, A global analysis of the Bogdanov-Takens system, SIAM J. Appl. Math., 52 (1992), 1172-1192. doi: 10.1137/0152069. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	G. Sansone and R. Conti, Non-linear Differential Equations, Pergamon Press, Oxford City, 1964. Google Scholar
[29]	Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1992. Google Scholar

Figure 1. Equilibria at infinity

Download full-size image

Download as PowerPoint slide

Figure 2. Limit cycles surrounding $O$

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 4. Two large limit cycles

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 6. Location of $x_A, x_B$

Download full-size image

Download as PowerPoint slide

Figure 7. Discussion on existence of limit cycles

Download full-size image

Download as PowerPoint slide

Figure 8. Discussion about the perturbation system

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 15. More global phase portraits for special parameters

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download as excel

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

Download as excel

$(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

Download as excel

$(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

[1]	Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062
[2]	Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130
[3]	Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
[4]	Min Lu, Chuang Xiang, Jicai Huang. Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3125-3138. doi: 10.3934/dcdss.2020115
[5]	Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1499-1514. doi: 10.3934/mbe.2017078
[6]	Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[7]	Antonio Garijo, Armengol Gasull, Xavier Jarque. Local and global phase portrait of equation $\dot z=f(z)$. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 309-329. doi: 10.3934/dcds.2007.17.309
[8]	Zhihua Liu, Rong Yuan. Takens–Bogdanov singularity for age structured models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2041-2056. doi: 10.3934/dcdsb.2019201
[9]	Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[10]	Fátima Drubi, Santiago Ibáñez, David Rivela. Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 599-615. doi: 10.3934/dcdsb.2019256
[11]	Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[12]	Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893
[13]	Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846
[14]	Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[15]	Chun Huang. Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop. Electronic Research Archive, , () : -. doi: 10.3934/era.2021037
[16]	Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
[17]	Yasuhito Miyamoto. Global bifurcation and stable two-phase separation for a phase field model in a disk. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 791-806. doi: 10.3934/dcds.2011.30.791
[18]	Hayato Chiba. Continuous limit and the moments system for the globally coupled phase oscillators. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1891-1903. doi: 10.3934/dcds.2013.33.1891
[19]	Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations & Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257
[20]	Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Hebai Chen Xingwu Chen
on Google Scholar
Hebai Chen Xingwu Chen

2020 Impact Factor: 1.327

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.327

Tools

Figures and Tables

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Hebai Chen Xingwu Chen
on Google Scholar
Hebai Chen Xingwu Chen

[Back to Top]