A generalization of the S-function method applied to a Duffing–Van der Pol forced oscillator

Homoclinic motions and chaos in the quasiperiodically forced van der Pol-Duffing oscillator with single well potential

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1994.0080 ◽

1994 ◽

Vol 445 (1925) ◽

pp. 597-617 ◽

Cited By ~ 10

Keyword(s):

Periodic Orbits ◽

Invariant Tori ◽

Duffing Oscillator ◽

Double Resonance ◽

Unperturbed System ◽

Van Der Pol ◽

Stable And Unstable Manifolds ◽

Special Cases ◽

Forced Oscillator ◽

Single Well

We study a two-frequency quasiperiodically forced oscillator with single well potential which reduces to the van der Pol and Duffing oscillators in certain special cases. The unperturbed system without damping and forcing terms has a one-parameter family of periodic orbits. We concentrate on the dynamics near the unperturbed resonant periodic orbits. Using the second-order averaging method and a version of Melnikov’s method, we show that when double resonance occurs, the stable and unstable manifolds of normally hyperbolic invariant tori intersect transversely, i. e. transverse homoclinic motions exist, near the unperturbed resonant periodic orbits in certain parameter regions. Such homoclinic motions yield chaotic dynamics characterized by a generalization of the Bernoulli shift. Numerical simulation results are also given to demonstrate the theoretical results.

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Stability analysis for periodic solutions of the Van der Pol–Duffing forced oscillator

Physica Scripta ◽

10.1088/0031-8949/91/1/015201 ◽

2015 ◽

Vol 91 (1) ◽

pp. 015201 ◽

Cited By ~ 7

Author(s):

Jifeng Cui ◽

Jiaming Liang ◽

Zhiliang Lin

Keyword(s):

Stability Analysis ◽

Periodic Solutions ◽

Van Der Pol ◽

Forced Oscillator

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Analysis of a fractional order Van der Pol-like oscillator via describing function method

Nonlinear Dynamics ◽

10.1007/s11071-009-9647-0 ◽

2010 ◽

Vol 61 (1-2) ◽

pp. 265-274 ◽

Cited By ~ 28

Author(s):

Mina Attari ◽

Mohammad Haeri ◽

Mohammad Saleh Tavazoei

Keyword(s):

Fractional Order ◽

Function Method ◽

Describing Function ◽

Van Der Pol ◽

Describing Function Method

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Noise Induced Jumping Dynamics Between Synchronized Modes

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415300347 ◽

2015 ◽

Vol 25 (12) ◽

pp. 1530034 ◽

Cited By ~ 1

Author(s):

Shannon D. Algar ◽

Thomas Stemler ◽

Bernard De Saedeleer

Keyword(s):

Dynamical System ◽

Lyapunov Exponents ◽

Van Der Pol Oscillator ◽

External Forcing ◽

Common Phenomenon ◽

Periodic Forcing ◽

Van Der Pol ◽

Forced Oscillator

Synchronization is a common phenomenon whereby a dynamical system follows the pacemaker provided by an external forcing. Often, such systems have multiple synchronization modes, which are equivalent solutions. We investigate the specific case of two to one synchronization produced by the periodic forcing of a van der Pol oscillator where two possible modes, shifted by one period of the modulation, exist. By studying the flow and the local Lyapunov exponents along the orbit we give an explanation of the noise induced jumps observed in a stochastic forced oscillator. While this investigation gives results that are specific to this system, the framework presented is more general and can be applied to any system showing similar jumping dynamics.

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