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

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.

Download Full-text

BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000283 ◽

1993 ◽

Vol 03 (02) ◽

pp. 399-404 ◽

Cited By ~ 5

Author(s):

T. SÜNNER ◽

H. SAUERMANN

Keyword(s):

Bifurcation Diagram ◽

Bifurcation Analysis ◽

Chaotic Behavior ◽

Three Dimensional ◽

Global Bifurcation ◽

Dynamical Behaviour ◽

Van Der Pol Oscillator ◽

Two Dimensional ◽

Van Der Pol ◽

Dimensional Models

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.

Download Full-text

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

Biological Cybernetics ◽

10.1007/bf00206238 ◽

1994 ◽

Vol 72 (1) ◽

pp. 55-67 ◽

Cited By ~ 31

Author(s):

Taishin Nomura ◽

Shunsuke Sato ◽

Shinji Doi ◽

Jose P. Segundo ◽

Michael D. Stiber

Keyword(s):

Global Bifurcation ◽

Van Der Pol Oscillator ◽

Bifurcation Structure ◽

Van Der Pol ◽

Pulse Trains ◽

Periodic Pulse

Download Full-text

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.

Download Full-text

Normal form of double-Hopf singularity with 1:1 resonance for delayed differential equations

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2019.2.6 ◽

2019 ◽

Vol 24 (2) ◽

pp. 241-260

Author(s):

Xiaoqin P. Wu ◽

Liancheng Wang

Keyword(s):

Normal Form ◽

Differential Equations ◽

Delayed Feedback ◽

Van Der Pol Oscillator ◽

Third Order ◽

Van Der Pol ◽

The Third ◽

Delayed Differential Equations ◽

Theoretical Results

In this manuscript, we provide a framework for the double-Hopf singularity with 1:1 resonance for general delayed differential equations (DDEs). The corresponding normal form up to the third-order terms is derived. As an application of our framework, a double-Hopf singularity with 1:1 resonance for a van der Pol oscillator with delayed feedback is investigated to illustrate the theoretical results.

Download Full-text

Chaotic Dynamical Behavior of Coupled One-Dimensional Wave Equations

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127421501157 ◽

2021 ◽

Vol 31 (06) ◽

pp. 2150115

Author(s):

Fei Wang ◽

Junmin Wang ◽

Zhaosheng Feng

Keyword(s):

Numerical Simulations ◽

Wave Equations ◽

Dynamical Behavior ◽

Chaotic Oscillation ◽

Van Der Pol ◽

Velocity Feedback ◽

One Dimensional ◽

Theoretical Results ◽

Dimensional Wave

In this paper, we consider the chaotic oscillation of coupled one-dimensional wave equations. The symmetric nonlinearities of van der Pol type are proposed at the two boundary endpoints, which can cause the energy of the system to rise and fall within certain bounds. At the interconnected point of the wave equations, the energy is injected into the system through an anti-damping velocity feedback. We prove the existence of the snapback repeller when the parameters enter a certain regime, which causes the system to be chaotic. Numerical simulations are presented to illustrate our theoretical results.

Download Full-text

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.

Download Full-text

LOCAL ACTIVITY OF THE VAN DER POL CNN

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127404010552 ◽

2004 ◽

Vol 14 (07) ◽

pp. 2211-2222 ◽

Cited By ~ 2

Author(s):

LEQUAN MIN ◽

GUANRONG CHEN

Keyword(s):

Neural Networks ◽

Activity Theory ◽

Bifurcation Diagram ◽

Fourth Order ◽

Van Der Pol ◽

One Dimensional ◽

Local Activity ◽

Edge Of Chaos ◽

Dynamical Behaviors ◽

Activity Domain

This paper studies a class of coupled Van der Pol (CVDP) cellular neural networks (CNNs) that can be realized via a coupled fourth-order circuit with two synaptic currents. The local activity theory, developed by Chua in 1997, is applied to study the CVDP CNN, thereby revealing that the bifurcation diagram of the CVDP CNN has a local activity domain with an edge of chaos, as well as a one-dimensional locally passive domain. Although no chaotic phenomena have been identified in simulations, many complex dynamical behaviors have been observed, such as the co-existence of one-periodic, divergent, and convergent orbits, at the edge of chaos.

Download Full-text

Parallel Numerical Creation of 2-parametric Bifurcation Diagram of Nonlinear Oscillators

Acta Technica Jaurinensis ◽

10.14513/actatechjaur.v11.n2.453 ◽

2018 ◽

Vol 11 (2) ◽

pp. 61-83

Author(s):

Flóra Hajdu

Keyword(s):

Parallel Algorithm ◽

Bifurcation Diagram ◽

Nonlinear Oscillators ◽

Van Der Pol Oscillator ◽

Test Results ◽

Bifurcation Diagrams ◽

Van Der Pol ◽

Good Efficiency ◽

Speed Up ◽

Number Of Iterations

This paper presents the numerical creation of 2-parametric bifurcation diagrams of nonlinear oscillators with a simple iterative algorithm, which can be easily parallelized. The parallel algorithm was tested with two simple well-known nonlinear oscillators, the Van der Pol oscillator and the Duffing-Holmes oscillator. It was examined how the resolution (number of iterations) affects the speedup and the efficiency. The test results show that a relative good speed up with a good efficiency could be achieved even using a simple desktop.

Download Full-text