June 2017, 22(4): 1273-1293. doi: 10.3934/dcdsb.2017062

Global phase portrait of a degenerate Bogdanov-Takens system with symmetry

1. 

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

3. 

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

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

Received  February 2016 Revised  November 2016 Published  February 2017

In this paper we study the global phase portrait of the normal form of a degenerate Bogdanov-Takens system with symmetry, i.e., a class of van der Pol-Duffing oscillators. This normal form is two-parametric and its parameters are considered in the whole parameter space, i.e., not viewed as a perturbation of some Hamiltonian system. We discuss the existence of limit cycles and prove its uniqueness if it exists. Moreover, by constructing a distance function we not only give the necessary and sufficient condition for the existence of heteroclinic loops connecting two saddles, but also prove its monotonicity and smoothness. Finally, we obtain a complete classification on the global phase portraits in the Poincaré disc as well as the complete global bifurcation diagram in the parameter space and find more plentiful phase portraits than the case that parameters are just sufficiently small.

Citation: 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
References:
[1]

A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcations of Dynamic Systems on a Plane John Wiley & Sons and Jerusalem: Israel Program for Scientific Translations, New York, 1973.

[2]

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equation 2$^nd$ edition, Springer-Verlag, Berlin, 1987.

[3]

S. M. BaerB. W. KooiYu. 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.

[4]

J. Carr, Applications of Center Manifold Theory Springer-Verlag, New York, 1981.

[5]

C. Castillo-ChavezZ. 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.

[6]

S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields Cambridge University Press, London, 1994. doi: 10.1017/CBO9780511665639.

[7]

W. A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics Reported, 2 (1989), 61-88.

[8]

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.

[9]

F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations, 139 (1997), 41-59. doi: 10.1006/jdeq.1997.3291.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian integrals Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0098353.

[15]

F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039. doi: 10.1088/0951-7715/3/4/004.

[16]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-1140-2.

[17]

C. Hayashi, Non-linear Oscillations in Physical Systems McGraw Hill Construction, New York, 1964.

[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]

L. S. Jacobsen and R. S. Ayre, Engineering Vibrations McGraw Hill Construction, New York, 1958.

[21]

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.

[22]

N. Levinson and O. K. Smith, A general equation for relaxation oscillations, J. Duke Math., 9 (1942), 382-403. doi: 10.1215/S0012-7094-42-00928-1.

[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. LinsW. de Melo and C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597 (1977), 335-357.

[25]

N. Minorsky, Nonlinear Oscillations Van Nostrand's Scientific Encyclopedia, Chicago, 1962.

[26]

L. M. Perko, A global analysis of the Bogdanov-Takens system, SIAM J. Appl. Math., 52 (1992), 1172-1192. doi: 10.1137/0152069.

[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. doi: 10.1137/S0036139999361896.

[28]

G. Sansone, Sopra lequazione di Liénard delle oscillazioni di rilassamento (In Italian), Ann. Mat. Pura Appl., 28 (1949), 153-181. doi: 10.1007/BF02411124.

[29]

G. Sansone and R. Conti, Non-linear Differential Equations Pergamon Press, Oxford City, 1964.

[30]

S. Timoshenko, Vibration Problems in Engineering 24$^{th}$ edition, John Wiley & Sons, Inc. , New Jersey, 1974.

[31]

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]

A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcations of Dynamic Systems on a Plane John Wiley & Sons and Jerusalem: Israel Program for Scientific Translations, New York, 1973.

[2]

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equation 2$^nd$ edition, Springer-Verlag, Berlin, 1987.

[3]

S. M. BaerB. W. KooiYu. 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.

[4]

J. Carr, Applications of Center Manifold Theory Springer-Verlag, New York, 1981.

[5]

C. Castillo-ChavezZ. 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.

[6]

S. -N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields Cambridge University Press, London, 1994. doi: 10.1017/CBO9780511665639.

[7]

W. A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics Reported, 2 (1989), 61-88.

[8]

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.

[9]

F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations, 139 (1997), 41-59. doi: 10.1006/jdeq.1997.3291.

[10]

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.

[11]

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.

[12]

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.

[13]

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.

[14]

F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian integrals Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0098353.

[15]

F. Dumortier and C. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity, 3 (1990), 1015-1039. doi: 10.1088/0951-7715/3/4/004.

[16]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-1140-2.

[17]

C. Hayashi, Non-linear Oscillations in Physical Systems McGraw Hill Construction, New York, 1964.

[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]

L. S. Jacobsen and R. S. Ayre, Engineering Vibrations McGraw Hill Construction, New York, 1958.

[21]

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.

[22]

N. Levinson and O. K. Smith, A general equation for relaxation oscillations, J. Duke Math., 9 (1942), 382-403. doi: 10.1215/S0012-7094-42-00928-1.

[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. LinsW. de Melo and C. C. Pugh, On Liénard's equation, Lecture Notes in Math., 597 (1977), 335-357.

[25]

N. Minorsky, Nonlinear Oscillations Van Nostrand's Scientific Encyclopedia, Chicago, 1962.

[26]

L. M. Perko, A global analysis of the Bogdanov-Takens system, SIAM J. Appl. Math., 52 (1992), 1172-1192. doi: 10.1137/0152069.

[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. doi: 10.1137/S0036139999361896.

[28]

G. Sansone, Sopra lequazione di Liénard delle oscillazioni di rilassamento (In Italian), Ann. Mat. Pura Appl., 28 (1949), 153-181. doi: 10.1007/BF02411124.

[29]

G. Sansone and R. Conti, Non-linear Differential Equations Pergamon Press, Oxford City, 1964.

[30]

S. Timoshenko, Vibration Problems in Engineering 24$^{th}$ edition, John Wiley & Sons, Inc. , New Jersey, 1974.

[31]

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.  The equilibria at infinity
Figure 2.  Orbits changing under Briot-Bouquet transformations
Figure 3.  Discussion about the uniqueness of limit cycle for (c4)
Figure 4.  $y^u(0)+y^s(0)>0$ for $b=0$
Figure 5.  $y^u(0)+y^s(0)>0$ for sufficiently small $b$
Figure 6.  Poincaré-Bendixson Annular region
Figure 7.  The bifurcation diagram of system (5)
Figure 8.  Global phase portraits of system (5)
Figure 9.  A heteroclinic loop for $(a, b)=(-1, 0.642054499)$
Figure 10.  A limit cycle surrounding a node for $(a, b)=(-16, -8)$
Figure 11.  A limit cycle crossing small neighborhoods of $E_L$ and $E_R$ for $(a, b)=(-8.87,-5.9554)$
Table 1.  Qualitative properties of equilibria
possibilities of (a, b) positions of equilibria types and Stability
a ≥ 0 E0 E0 is a saddle
a < 0 $b<-2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable bidirectional node
$b=-2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable unidirectional node
$-2\sqrt{-a}<b<0$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable rough focus
$b=0$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable weak focus of order $1$
$\!\!0<b<2\sqrt{-a}\!\!$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable rough focus
$b=2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable unidirectional node
$b>2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable bidirectional node
possibilities of (a, b) positions of equilibria types and Stability
a ≥ 0 E0 E0 is a saddle
a < 0 $b<-2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable bidirectional node
$b=-2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable unidirectional node
$-2\sqrt{-a}<b<0$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is an unstable rough focus
$b=0$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable weak focus of order $1$
$\!\!0<b<2\sqrt{-a}\!\!$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable rough focus
$b=2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable unidirectional node
$b>2\sqrt{-a}$ $E_L$, $E_0$, $E_R$ $E_R, E_L$ are saddles; $E_0$ is a stable bidirectional node
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