Duffing Van Der Pol Oscillator Research Articles

Stochastic Hopf bifurcation: An example

The stochastic Hopf bifurcation behavior of the noisy Duffing-van der Pol oscillator x ̈ = (α + σW)x + β x ̇ − x 2 x ̇ − x 3 , (A) is studied numerically. (α, β are bifurcation parameters, W is white noise, and σ is an intensity parameter.) When the qualitative change of the stationary solution of the Fokker-Planck equation is considered, the stochastic Hopf bifurcation appears as a change from a Dirac measure to a crater-like density. Unfortunately, this behavior is not related to the sample stability of the trivial solution zero. To capture all the stochastic dynamics of the equation, it is necessary to investigate the change of stability of invariant measures and the occurrence of new invariant measures for the generated random dynamical system.

Deterministic and stochastic Duffing-van der Pol oscillators are non-explosive

This paper is concerned with the (non-)explosion behavior of solutions of non-linear random and stochastic differential equations.

Chaos from van der Pol-Duffing Oscillator: Bispectral Analysis

Bispectral analysis techniques provide us with both the theoretical and experimental insight into nonlinear wave process which can not be available from ordinary linear spectral analysis. Bicoherence spectra isolate the nonlinear phase coupling between the triads of Fourier modes. The degree of mode-mode coupling for period doubling sequence route to chaos of van der Pol-Duffing oscillator will be studied using both bispectrum and bicoherence.

Neimark bifurcation, almost-periodicity and chaos in the forced van der Pol-Duffing system in the neighbourhood of the principal resonance

The occurrence of chaotic motion in the forced van der Pol-Duffing oscillator in close neighbourhood of the principal resonance is considered and interpreted in connection with subcritical Neimark bifurcation. Computer simulations and approximate analytical analysis confirm that the chaotic motion region forms a transition zone between two regular motions: the T-periodic and almost-periodic oscillations.

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

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.

On the non-integrability of a family of Duffing-van der Pol oscillators

We investigate the non-integrability of a family of Duffing-van der Pol oscillators x+ alpha x(x2-1)+x+ beta x3= gamma cos omega t by studying the analytic properties of the dynamics in complex time. We find that the solutions of (*) have no worse than algebraic singularities at t*, with only (t-t*)1/2 terms present in their series expansions, unlike, for example, the alpha =0 Duffing case, where, typically, log(t-t*) terms arise. Still, when integrating (*) around long enough contours, a remarkably intricate pattern of square root singularities emerges, on different sheets, which appears to prevent solutions from ever returning to the original sheet. Such evidence of infinitely-sheeted solutions, termed the ISS property, has also been observed in a number of Hamiltonian systems and is illustrated here on a simple example of a single, first-order differential equation. We suggest that the ISS property is a necessary condition for non-integrability, i.e. non-existence of a complete set of analytic, single-valued constants of the motion, which would permit the complete integration of a dynamical system in terms of quadratures.

Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator

This paper investigates the possibility of controlling horseshoe and asymptotic chaos in the Duffing-van der Pol oscillator by both periodic parametric perturbation and addition of second periodic force. Using Melnikov method the effect of weak perturbations on horseshoe chaos is studied. Parametric regimes where suppression of horseshoe occurs are predicted. Analytical predictions are demonstrated through direct numerical simulations. Starting from asymptotic chaos we show the recovery of periodic motion for a range of values of amplitude and frequency of the periodic perturbations. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.

Prediction of horseshoe chaos in BVP and DVP oscillators

An analytical threshold condition for the prediction of horseshoe chaos is obtained for the Bonhoeffer-van der Pol (BVP) oscillator, through the equivalent Duffing-van der Pol (DVP) oscillator, using the Melnikov method. Numerical and analytical predictions of the tangential intersection of stable and unstable orbits of saddle have been compared for both the DVP and BVP oscillators. The analytical threshold of the BVP oscillator, derived by extrapolating that of the DVP oscillator, is found to be in good agreement with the actual numerical analysis of the system.

A route to chaos in a nonlinear oscillator with delay

The paper presents an analysis of the transition from regular to chaotic motion in a Van der Pol-Duffing's oscillator with delay after a Hopf bifurcation. The conditions for the occurrence of the Hopf bifurcation have been determined by means of the approximate method. For the parameters near the bifurcation point a computer simulation of the vibrating system had been performed and the evolution of the system from regular motion to chaos has been analysed at the decrease of the value of the dimensionless damping coefficient.

Observation of chaos in a nonlinear oscillator with delay: A numerical study

The Van der Pol-Duffing's oscillator with time delay is investigated. Some examples of the different behaviour of strange attractors with the changing of delay parameters are given. For certain parameter values the motion of the oscillator is harmonic with only one frequency.

Secondary Bifurcation of Quasi-Periodic Solutions Can Lead to Period Multiplication

Perturbation and asymptotic methods are used to obtain the secondary bifurcation of quasi-periodic solutions from periodic solutions for a model problem. It is a two-cell model consisting of a coupled system of van der Pol-Duffing oscillators. The qualitative features of the secondary bifurcation states depends on the detuning of the oscillators. For special values of the detuning the secondary states are periodic. Then it is possible for period multiplication of solutions to occur at the secondary bifurcation point.

On the frequency response of a Van der Pol-Duffing oscillator

It is shown that the harmonic response curves of a combined Van der Pol-Duffing oscillator without memory represented by x + ε(x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> - 1)x + ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> - αεx <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> /3 = F sin ωt resemble those of an RLC series circuit when the inductance has a wide hysteresis loop.

On the forced response of a class of Van der Pol-Duffing oscillator

A first integral in the form of the well-known Abel's differential equation is obtained for the governing differential equation of a class of Van der Pol-Duffing oscillators with or without external inputs to the system.