Global bifurcation structure of a Bonhoeffer-van der Pol oscillator driven by periodic pulse trains

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

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The global bifurcation characteristics of the forced van der Pol oscillator

Chaos Solitons & Fractals ◽

10.1016/0960-0779(95)00045-3 ◽

1996 ◽

Vol 7 (1) ◽

pp. 3-19 ◽

Cited By ~ 22

Author(s):

Jian-Xue Xu ◽

Jun Jiang

Keyword(s):

Global Bifurcation ◽

Van Der Pol Oscillator ◽

Van Der Pol

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ANALYSIS OF TORUS BREAKDOWN INTO CHAOS IN A CONSTRAINT DUFFING VAN DER POL OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408020835 ◽

2008 ◽

Vol 18 (04) ◽

pp. 1051-1068 ◽

Cited By ~ 12

Author(s):

MUNEHISA SEKIKAWA ◽

NAOHIKO INABA ◽

TAKASHI TSUBOUCHI ◽

KAZUYUKI AIHARA

Keyword(s):

Bifurcation Diagram ◽

Van Der Pol Oscillator ◽

Circle Map ◽

Bifurcation Structure ◽

Van Der Pol ◽

One Dimensional ◽

Implicit Equations ◽

Periodic State ◽

Torus Breakdown ◽

Theoretical Results

The bifurcation structure of a constraint Duffing van der Pol oscillator with a diode is analyzed and an objective bifurcation diagram is illustrated in detail in this work. An idealized case, where the diode is assumed to operate as a switch, is considered.In this case, the Poincaré map is constructed as a one-dimensional map: a circle map. The parameter boundary between a torus-generating region where the circle map is a diffeomorphism and a chaos-generating region where the circle map has extrema is derived explicitly, without solving the implicit equations, by adopting some novel ideas. On the bifurcation diagram, intermittency and a saddle-node bifurcation from the periodic state to the quasi-periodic state can be exactly distinguished. Laboratory experiment is also carried out and theoretical results are verified.

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Global Bifurcation Analysis of a Duffing–Van der Pol Oscillator with Parametric Excitation

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500515 ◽

2014 ◽

Vol 24 (04) ◽

pp. 1450051 ◽

Cited By ~ 10

Author(s):

Qun Han ◽

Wei Xu ◽

Xiaole Yue

Keyword(s):

State Space ◽

Invariant Manifolds ◽

Global Analysis ◽

Global Bifurcation ◽

Van Der Pol Oscillator ◽

Fractal Boundary ◽

Van Der Pol ◽

Cell State ◽

Boundary Crisis ◽

Basin Boundaries

In this paper, a composite cell state space is constructed by a multistage division of the continuous phase space. Based on point mapping method, global properties of dynamical systems can be analyzed more accurately and efficiently, and any small regions can be refined for clearly depicting some special basin boundaries in this cell state space. Global bifurcation of a Duffing–Van der Pol oscillator subjected to harmonic parametrical excitation is investigated. Attractors, basins of attraction, basin boundaries, saddles and invariant manifolds have been obtained. As the amplitude of excitation increases, it can be observed that the boundary crisis occurs twice. Then a basin boundary with Wada property appears in the state space and undergoes metamorphosis in the chaotic boundary crisis. At last, two attractors merge into a chaotic one when they simultaneously collide with the chaotic saddle embedded in the fractal boundary. All these results show the effectiveness of the proposed method in global analysis.

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BIFURCATION STRUCTURE OF A VAN DER POL OSCILLATOR SUBJECTED TO NONSINUSOIDAL PERIODIC EXCITATION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412500034 ◽

2012 ◽

Vol 22 (01) ◽

pp. 1250003 ◽

Cited By ~ 5

Author(s):

H. SIMO ◽

P. WOAFO

Keyword(s):

Experimental Investigation ◽

Numerical Simulations ◽

Control Parameter ◽

Van Der Pol Oscillator ◽

Periodic Excitation ◽

Electronic Components ◽

Bifurcation Structure ◽

Van Der Pol ◽

Bifurcation Points ◽

Bifurcation Structures

Bifurcation structures of a Van der Pol oscillator subjected to the effects of nonsinusoidal excitations are obtained both numerically and experimentally. It is found that the bifurcation sequences are similar, but the ranges of a particular behavior and the bifurcation points of the control parameter are different. The experimental investigation using electronic components shows that results are similar to those observed from numerical simulations.

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BIFURCATION STRUCTURE OF A DRIVEN MULTI-LIMIT-CYCLE VAN DER POL OSCILLATOR (II): SYMMETRY-BREAKING CRISIS AND INTERMITTENCY

International Journal of Bifurcation and Chaos ◽

10.1142/s021812749100052x ◽

1991 ◽

Vol 01 (03) ◽

pp. 711-715 ◽

Cited By ~ 19

Author(s):

C. EICHWALD ◽

F. KAISER

Keyword(s):

Symmetry Breaking ◽

Limit Cycles ◽

Critical Exponents ◽

The Self ◽

Van Der Pol Oscillator ◽

Chaotic Attractors ◽

Bifurcation Structure ◽

Van Der Pol ◽

Sustained Oscillations ◽

Symmetry Properties

Bifurcations in the superharmonic region of a generalized version of the van der Pol oscillator which exhibits three limit cycles are investigated. An external force causes the subsequent breakdown of the self-sustained oscillations. Beyond these series of bifurcations chaotic solutions also exist. They display a symmetry-breaking crisis followed by a type III intermittency transition. The bifurcations are discussed with respect to the symmetry properties of chaotic attractors. The critical exponents connected with the bifurcations offer a scaling which partially contradicts that known from literature. An explanation for this behavior is given.

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BIFURCATION STRUCTURE OF A DRIVEN, MULTI-LIMIT-CYCLE VAN DER POL OSCILLATOR (I): THE SUPERHARMONIC RESONANCE STRUCTURE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127491000385 ◽

1991 ◽

Vol 01 (02) ◽

pp. 485-491 ◽

Cited By ~ 20

Author(s):

F. KAISER ◽

C. EICHWALD

Keyword(s):

Limit Cycle ◽

External Force ◽

Limit Cycles ◽

The Self ◽

Van Der Pol Oscillator ◽

Resonance Structure ◽

Bifurcation Structure ◽

Van Der Pol ◽

Superharmonic Resonance ◽

Sustained Oscillations

Bifurcations in the superharmonic region of a generalized version of the van der Pol oscillator which exhibits three limit cycles are investigated. An external force causes the subsequent breakdown of the self-sustained oscillations. Beyond these series of bifurcations chaotic solutions also exist. In this first part we concentrate on a discussion of the bifurcation structure of the system.

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