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

1991 ◽  
Vol 01 (03) ◽  
pp. 711-715 ◽  
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.

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

1991 ◽  
Vol 01 (02) ◽  
pp. 485-491 ◽  
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.

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

1994 ◽  
Vol 72 (1) ◽  
pp. 55-67 ◽  
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

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

2008 ◽  
Vol 18 (04) ◽  
pp. 1051-1068 ◽  
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.

scholarly journals Self- excited oscillation under the influence of non - linear frictions

1982 ◽  
Vol 4 (3) ◽  
pp. 7-10
Author(s):  
Nguyen Van Dao
Keyword(s):  
Equilibrium Position ◽  
The Self ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Excited System ◽  
Non Linear ◽  
Excited Oscillation

In this article the influence of friction R1, R2 on Van der Pol oscillator is considered. It turned out that the mentioned frictions decrease the amplitude of self – excited oscillations and they stabilize the equilibrium position of the self – excited system.

scholarly journals Algorithms for Controlling chaos: Application to BVP Oscillator

1993 ◽  
Vol 132 ◽  
pp. 45-45
Author(s):  
S. Rajeskar
Keyword(s):  
Computer Simulations ◽  
Limit Cycles ◽  
Digital Computer ◽  
Chaotic Dynamics ◽  
Van Der Pol Oscillator ◽  
Time Dependent ◽  
Van Der Pol ◽  
Small Magnitude ◽  
Controlling Chaos ◽  
Regular Motion

AbstractWe discuss how chaotic dynamics can be converted into regular motion in Bonhoeffer-van der Pol oscillator. Using a control signal proportional to the actual and desired outputs we study the control of fixed points and limit cycles by making time-dependent perturbations of amplitude of external force. We show the round-off induced periodicity in the digital computer simulations of orbits on chaotic attractor. We illustrate the stabilization of unstable periodic or bits by adding periodic pulses of small magnitude.

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

2012 ◽  
Vol 22 (01) ◽  
pp. 1250003 ◽  
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.

On the Existence of Limit Cycles and Relaxation Oscillations in a 3D van der Pol-like Memristor Oscillator

2017 ◽  
Vol 27 (07) ◽  
pp. 1750102
Author(s):  
Marcelo Messias ◽  
Anderson L. Maciel
Keyword(s):  
Phase Space ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Relaxation Oscillator ◽  
Van Der Pol Oscillator ◽  
Relaxation Oscillations ◽  
Van Der Pol ◽  
Piecewise Linear System ◽  
Parameter Values

We study a van der Pol-like memristor oscillator, obtained by substituting a Chua’s diode with an active controlled memristor in a van der Pol oscillator with Chua’s diode. The mathematical model for the studied circuit is given by a three-dimensional piecewise linear system of ordinary differential equations, depending on five parameters. We show that this system has a line of equilibria given by the [Formula: see text]-axis and the phase space [Formula: see text] is foliated by invariant planes transverse to this line, which implies that the dynamics is essentially two-dimensional. We also show that in each of these invariant planes may occur limit cycles and relaxation oscillations (that is, nonsinusoidal repetitive (periodic) solutions), depending on the parameter values. Hence, the oscillator studied here, constructed with a memristor, is also a relaxation oscillator, as the original van der Pol oscillator, although with a main difference: in the case of the memristor oscillator, an infinity of oscillations are produced, one in each invariant plane, depending on the initial condition considered. We also give conditions for the nonexistence of oscillations, depending on the position of the invariant planes in the phase space.

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