Non-autonomous Van Der Pol Oscillators Research Articles

Fibonacci stairs and the Afraimovich-Pesin dimension for a stroboscopic section of a nonautonomous van der Pol oscillator

Statistics of Poincaré recurrences is studied in the stroboscopic section of trajectories of a nonautonomous van der Pol oscillator in the framework of the global approach. It is shown that when the oscillator frequency and the frequency of the external force are irrationally related, the set obtained stroboscopically is equivalent to the circle map. For small values of the external amplitude, the Fibonacci stairs is constructed for the golden and silver ratios and its universal properties are confirmed. It is established that the Afraimovich-Pesin dimension for the map in the stroboscopic section is αc = 1 for Diophantine irrational rotation numbers.

Limit cycles and acoustic radiation analysis of a van der Pol oscillator-plate-medium system

The approximate analytical solution of the limit cycle and the acoustic radiation characteristics of a nonlinear vibro-acoustic coupling system consisting of a non-autonomous van der Pol oscillator, a plate, and the surrounding acoustic medium are investigated. The nonlinear coupling equations of the vibro-acoustic coupling system under external excitation are derived using variational theory. The modal expression method and the incoherent superposition assumption are employed to discretize the displacements and the surface acoustic pressure of the plate, and then the incremental harmonic balance (IHB) method is used to drive the approximate analytical expressions for the limit cycles of the coupling system. The acoustic radiation power level and the surface average velocity under multi-frequency excitation are defined to describe the response of the plate. The results are that there are limit cycle motions in the vibration of the vibro-acoustic coupling system due to the van der Pol oscillator’s excitation, and there are more harmonic components in the system when the parameters are increased. The vibro-acoustic characteristics of the plate present nonlinear variation because of the oscillator’s nonlinear damping force. The method combining the IHB method and the modal expression method can be used to investigate the nonlinear vibro-acoustic coupling characteristics of a structure under external and self-excited excitation.

Dynamical inference: where phase synchronization and generalized synchronization meet.

Synchronization is a widespread phenomenon that occurs among interacting oscillatory systems. It facilitates their temporal coordination and can lead to the emergence of spontaneous order. The detection of synchronization from the time series of such systems is of great importance for the understanding and prediction of their dynamics, and several methods for doing so have been introduced. However, the common case where the interacting systems have time-variable characteristic frequencies and coupling parameters, and may also be subject to continuous external perturbation and noise, still presents a major challenge. Here we apply recent developments in dynamical Bayesian inference to tackle these problems. In particular, we discuss how to detect phase slips and the existence of deterministic coupling from measured data, and we unify the concepts of phase synchronization and general synchronization. Starting from phase or state observables, we present methods for the detection of both phase and generalized synchronization. The consistency and equivalence of phase and generalized synchronization are further demonstrated, by the analysis of time series from analog electronic simulations of coupled nonautonomous van der Pol oscillators. We demonstrate that the detection methods work equally well on numerically simulated chaotic systems. In all the cases considered, we show that dynamical Bayesian inference can clearly identify noise-induced phase slips and distinguish coherence from intrinsic coupling-induced synchronization.

Studying the behavior of a nonautonomous Van der Pol oscillator in different time scales with the presence of noise near the synchronization boundary

A new level of organization of the temporal behavior of two coupled complex systems is revealed. We report for the first time the coexistence of two types of intermittent behavior that occurs simultaneously near the boundary of the synchronization regime of coupled chaotic oscillators. This intricate phenomenon was observed both experimentally in a physiological experiment and numerically. The laws for both the distribution and the mean length of the laminar phases versus the control parameter values are analytically deduced. Very good agreement between the theoretical results and the numerically calculated data is shown.

A “saddle-node” bifurcation scenario for birth or destruction of a Smale–Williams solenoid

Formation or destruction of hyperbolic chaotic attractor under parameter variation is considered with an example represented by Smale–Williams solenoid in stroboscopic Poincaré map of two alternately excited non-autonomous van der Pol oscillators. The transition occupies a narrow but finite parameter interval and progresses in such way that periodic orbits constituting a “skeleton” of the attractor undergo saddle-node bifurcation events involving partner orbits from the attractor and from a non-attracting invariant set, which forms together with its stable manifold a basin boundary of the attractor.

Uniformly Hyperbolic Attractor of the Smale–Williams Type for a Poincaré Map in the Kuznetsov System

We propose a general algorithm for computer assisted verification of uniform hyperbolicity for maps which exhibit a robust attractor. The method has been successfully applied to a Poincare map for a system of coupled non-autonomous van der Pol oscillators. The model equation has been proposed by Kuznetsov and the attractor seems to be of the Smale-Williams type.

Hyperbolic attractor in a system of coupled non-autonomous van der Pol oscillators: Numerical test for expanding and contracting cones

We present numerical verification of hyperbolic nature for chaotic attractor in a system of two coupled non-autonomous van der Pol oscillators (Kuznetsov, Phys. Rev. Lett., 95, 144101, 2005). At certain parameter values, in the four-dimensional phase space of the Poincare map a toroidal domain (a direct product of a circle and a three-dimensional ball) is determined, which is mapped into itself and contains the attractor we analyze. In accordance with the computations, in this absorbing domain the conditions of hyperbolicity are valid, which are formulated in terms of contracting and expanding cones in the tangent spaces (the vector spaces of the small state perturbations).

Arnold’s cat map dynamics in a system of coupled nonautonomous van der Pol oscillators

An example of a flow system is presented with an attractor concentrated mostly at a surface of a two-dimensional torus, the dynamics on which is governed by the Arnold cat map. The system is composed of four coupled nonautonomous van der Pol oscillators. Three of them have equal characteristic frequencies, and in the other one the frequency is twice as large. The parameters controlling excitation of the two pairs of oscillators are forced to undergo a slow counterphase periodic modulation in time. At the end of the active stage for one pair of the oscillators, the excitation is passed to another pair, than back, and so on. In terms of a stroboscopic Poincaré section, the respective eight-dimensional (8D) mapping, due to strong phase volume compression, reduces approximately to a 2D map for the phases of one pair of the oscillators that corresponds approximately to the Arnold cat map. The largest two Lyapunov exponents (one positive and one negative) are close to those predicted with the cat map model. Estimates for the fractal dimension of the attractor of the Poincaré map are close to 2.