Journal ArticleDOI
Influence of Noise on Duffing‐Van der Pol Oscillators
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In this paper, the stochastic motions of oscillators driven by white noise are investigated by using several theoretical methods: analytical and numerical calculations of the stationary probability distribution, analysis of the Hopf bifurcation, computer simulations and study of Lyapunov exponents.Abstract:
The stochastic motions of Duffing-Van der Pol oscillators driven by white noise are investigated by using several theoretical methods: analytical and numerical calculations of the stationary probability distribution, analysis of the Hopf bifurcation, computer simulations and study of Lyapunov exponents. One observes a break of the radial symmetry and noise induced oscillations.
Die stochastischen Bewegungen von Duffing-Van der Pol-Oszillatoren, die durch weises Rauschen angetrieben werden, werden mit Hilfe verschiedener theoretischer Methoden untersucht: Analytische und numerische Berechnungen der stationaren Wahrscheinlichkeitsverteilung, Untersuchung der Hopf-Bifurkation, Computer-Simulationen sowie das Studium der Lyapunov-Exponenten. Man beobachtet einen Bruch der radialen Symmetrie und rauschinduzierte Oszillationen.read more
Citations
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Book
Random Dynamical Systems
TL;DR: This chapter establishes the framework of random dynamical systems and introduces the concept of random attractors to analyze models with stochasticity or randomness.
Journal ArticleDOI
Coherence and stochastic resonance in a two-state system
TL;DR: A two-state dynamics driven by Gaussian white noise and a weak harmonic signal and its output spectra can be calculated for arbitrary noise strength and frequency, allowing characterization of the coherence resonance in the bistable and excitable regimes as well as quantification of nonadiabatic resonances with respect to the external signal in all regimes.
Journal ArticleDOI
Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator.
Anna Zakharova,Tatjana E. Vadivasova,Vadim S. Anishchenko,Aneta Koseska,Juergen Kurths,Juergen Kurths +5 more
TL;DR: It is found that under the influence of noise, Duffing-Van der Pol oscillator with hard excitation and a model of a synthetic genetic oscillator can be well characterized through the concept of stochastic bifurcation, consisting in a qualitative change of the stationary amplitude distribution.
Journal ArticleDOI
Response and stability of strongly non-linear oscillators under wide-band random excitation
TL;DR: In this article, a new stochastic averaging procedure for strongly nonlinear oscillators with lightly linear and non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions.
Journal ArticleDOI
Bifurcation scenarios of the noisy duffing-van der pol oscillator
TL;DR: In this article, the authors present a numerical study of the bifurcation behavior of the noisy Duffing-van der Pol oscillator and propose a new invariant measure for the random dynamical system generated in time one point motion.
References
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Journal ArticleDOI
A nonlinear oscillator with a strange attractor
TL;DR: In this paper, the authors studied the bifurcational behavior of a nonlinear oscillator with a qualitative viewpoint and showed that for a wide range of moderate f extremely complicated nonperiodic motions arise.
Journal ArticleDOI
Georg Duffing, Ingenieur: Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung. Sammlung Vieweg. Heft 41/42, Braunschweig 1918. VI+134 S
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Chaotic behaviour of a parametrically excited damped pendulum
R.W. Leven,B.P. Koch +1 more
TL;DR: In this article, the temporal behavior of a parametrically excited mathematical pendulum is examined by means of the numerical solution of its equation of motion, and it is shown that for certain parameter intervals the pendulum behaves in an apparently chaotic way.
Book ChapterDOI
Statistical Theory of Instabilities in Stationary Nonequilibrium Systems with Applications to Lasers and Nonlinear Optics
TL;DR: The transition of a macroscopic system from a disordered, chaotic state to an ordered more regular state is a very general phenomenon as is testified by the abundance of highly ordered macro-scopic systems in nature as mentioned in this paper.