101 research outputs found
Nonlinear response of a forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two
Non-resonant bifurcations of codimension two may appear in the controlled van der Pol-Duffing oscillator when two critical time delays corresponding to a double Hopf bifurcation have the same value. With the aid of centre manifold theorem and the method of multiple scales, the non-resonant response and two types of primary resonances of the forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. It is shown that the non-resonant response of the forced oscillator may exhibit quasi-periodic motions on a two- or three-dimensional (2D or 3D) torus. The primary resonant responses admit single and mixed solutions and may exhibit periodic motions or quasi-periodic motions on a 2D torus. Illustrative examples are presented to interpret the dynamics of the controlled system in terms of two dummy unfolding parameters and exemplify the periodic and quasi-periodic motions. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation. © 2008 Elsevier Ltd. All rights reserved
Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay
In this paper, the Hopf-pitchfork bifurcation of coupled van der Pol with delay is studied. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, the normal form is gotten by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, bifurcation diagrams and phase portraits are given through analyzing the unfolding structure. Finally, numerical simulations are used to support theoretical analysis
Lag Synchronization in Coupled Multistable van der Pol-Duffing Oscillators
We consider the system of externally excited identical van der Pol-Duffing oscillators unidirectionally coupled in a ring. When the coupling is introduced, each of the oscillator’s trajectories is on different attractor. We study the changes in the dynamics due to the increase in the coupling coefficient. Studying the phase of the oscillators, we calculate the parameter value for which we obtain the antiphase lag synchronization of the system and also the bifurcation values for which we observe qualitative changes in the dynamics of already synchronized system. We give evidence that lag synchronization is typical for coupled multistable systems
Hyperchaos and bifurcations in a driven Van der Pol–Duffing oscillator circuit
We investigate the dynamics of a driven Van der Pol–Duffing oscillator circuit and show the existence of higher-dimensional chaotic orbits (or hyperchaos), transient chaos, strange-nonchaotic attractors, as well as quasiperiodic orbits born from Hopf bifurcating orbits. By computing all the Lyapunov exponent spectra, scanning a wide range of the driving frequency and driving amplitude parameter space, we explore in two-parameter space the regimes of different dynamical behaviours
Normal form of double-Hopf singularity with 1:1 resonance for delayed differential equations
In this manuscript, we provide a framework for the double-Hopf singularity with 1:1 resonance for general delayed differential equations (DDEs). The corresponding normal form up to the third-order terms is derived. As an application of our framework, a double-Hopf singularity with 1:1 resonance for a van der Pol oscillator with delayed feedback is investigated to illustrate the theoretical results
Partial periodic oscillation: an interesting phenomenon for a system of three coupled unbalanced damped Duffing oscillators with delays
In this paper, a system of two coupled damped Duffing resonators driven by a van der Pol oscillator with delays is studied. Some sufficient conditions to ensure the periodic and partial periodic oscillations for the system are established. Computer simulation is given to demonstrate our result
Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation
Complex systems exhibiting critical transitions when one of their governing
parameters varies are ubiquitous in nature and in engineering applications.
Despite a vast literature focusing on this topic, there are few studies dealing
with the effect of the rate of change of the bifurcation parameter on the
tipping points. In this work, we consider a subcritical stochastic Hopf
bifurcation under two scenarios: the bifurcation parameter is first changed in
a quasi-steady manner and then, with a finite ramping rate. In the latter case,
a rate-dependent bifurcation delay is observed and exemplified experimentally
using a thermoacoustic instability in a combustion chamber. This delay
increases with the rate of change. This leads to a state transition of larger
amplitude compared to the one that would be experienced by the system with a
quasi-steady change of the parameter. We also bring experimental evidence of a
dynamic hysteresis caused by the bifurcation delay when the parameter is ramped
back. A surrogate model is derived in order to predict the statistic of these
delays and to scrutinise the underlying stochastic dynamics. Our study
highlights the dramatic influence of a finite rate of change of bifurcation
parameters upon tipping points and it pinpoints the crucial need of considering
this effect when investigating critical transitions
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