Controlling limit cycles in planar dynamical systems: a nonsmooth bifurcation approach

By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z2-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.

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BIFURCATIONS OF LIMIT CYCLES CREATED BY A MULTIPLE NILPOTENT CRITICAL POINT OF PLANAR DYNAMICAL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127411028544 ◽

2011 ◽

Vol 21 (02) ◽

pp. 497-504 ◽

Cited By ~ 6

Author(s):

YIRONG LIU ◽

JIBIN LI

Keyword(s):

Dynamical Systems ◽

Critical Point ◽

Limit Cycles ◽

Small Amplitude ◽

Hopf Bifurcations ◽

Bifurcation Phenomena ◽

Cubic System ◽

The Stability ◽

Nilpotent Critical Point ◽

Planar Dynamical Systems

Bifurcations of limit cycles created from a multiple critical point of planar dynamical systems are studied. It is different from the usual Hopf bifurcations of limit cycles created from an elementary critical point. This bifurcation phenomena depends on the stability of the multiple critical point and the multiple number of the critical point. As an example, a cubic system which can created four small amplitude limit cycles from the origin (a multiple critical point) is given.

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Double Bifurcation of Nilpotent Focus

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415500364 ◽

2015 ◽

Vol 25 (03) ◽

pp. 1550036 ◽

Cited By ~ 9

Author(s):

Yirong Liu ◽

Feng Li

Keyword(s):

Dynamical Systems ◽

Limit Cycles ◽

Weak Focus ◽

Bifurcation Phenomenon ◽

Cubic System ◽

Cubic Systems ◽

Planar Dynamical Systems

In this paper, an interesting bifurcation phenomenon is investigated — a 3-multiple nilpotent focus of the planar dynamical systems could be broken into two element focuses and an element saddle, and the limit cycles could bifurcate out from two element focuses. As an example, a class of cubic systems with 3-multiple nilpotent focus O(0, 0) is investigated, we prove that nine limit cycles with the scheme 7 ⊃ (1 ∪ 1) could bifurcate out from the origin when the origin is a weak focus of order 8. At the end of this paper, the double bifurcations of a class of Z2 equivalent cubic system with 3-multiple nilpotent focus or center O(0, 0) are investigated.

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BIFURCATIONS OF LIMIT CYCLES IN A Z3-EQUIVARIANT VECTOR FIELD OF DEGREE 5

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401003267 ◽

2001 ◽

Vol 11 (08) ◽

pp. 2287-2298 ◽

Cited By ~ 24

Author(s):

H. S. Y. CHAN ◽

K. W. CHUNG ◽

JIBIN LI

Keyword(s):

Vector Field ◽

Dynamical Systems ◽

Limit Cycles ◽

Bifurcation Theory ◽

Compound Eyes ◽

Hamiltonian Vector Field ◽

Numerical Example ◽

Hilbert Number ◽

Planar Dynamical Systems ◽

Equivariant Vector Field

A concrete numerical example of Z3-equivariant planar perturbed Hamiltonian vector field of fifth degree having 23 limit cycles and a configuration of compound eyes are given, by using the bifurcation theory of planar dynamical systems and the method of detection functions. It gives rise to the conclusion: the Hilbert number H(5) ≥ 23 for the second part of Hilbert's 16th problem.

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INVESTIGATIONS OF BIFURCATIONS OF LIMIT CYCLES IN Z2-EQUIVARIANT PLANAR VECTOR FIELDS OF DEGREE 5

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127402005698 ◽

2002 ◽

Vol 12 (10) ◽

pp. 2137-2157 ◽

Cited By ~ 36

Author(s):

JIBIN LI ◽

H. S. Y. CHAN ◽

K. W. CHUNG

Keyword(s):

Dynamical Systems ◽

Limit Cycles ◽

Bifurcation Theory ◽

Vector Fields ◽

Compound Eyes ◽

Polynomial Vector ◽

Polynomial Vector Fields ◽

Planar Vector Fields ◽

Planar Vector ◽

Planar Dynamical Systems

Some distributions of limit cycles of Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 are investigated. These include examples of specific Z2-equivariant fields and Z4-equivariant fields having up to 23 limit cycles. The configurations of compound eyes are also obtained by using the bifurcation theory of planar dynamical systems and the method of detection functions.

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