768 research outputs found
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
Weakly nonlinear hyperbolic differential equation in Hilbert space
We consider nonlinear perturbations of the hyperbolic equation in the Hilbert
space. Necessary and sufficient conditions for the existence of solutions of
boundary-value problem for the corresponding equation and iterative procedures
for their finding are obtained in the case when the operator in linear part of
the problem hasn't inverse and can have nonclosed set of values. As an
application we consider boundary-value problems for countable systems of
differential equations and van der Pol equation in a separable Hilbert space
Forced large amplitude periodic vibrations of non-linear Mathieu resonators for microgyroscope applications
International audienceThis paper describes a comprehensive non-linear multiphysics model based on the Euler-Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parametric analyses are presented in order to investigate the effect of the proof mass frequency on the bifurcation topology. The model shows that the optimal sensitivity is reached for resonant microgyroscopes designed with sensing frequency four times faster than the actuation one
Dynamics of a Limit Cycle Oscillator under Time Delayed Linear and Nonlinear Feedbacks
We study the effects of time delayed linear and nonlinear feedbacks on the
dynamics of a single Hopf bifurcation oscillator. Our numerical and analytic
investigations reveal a host of complex temporal phenomena such as phase slips,
frequency suppression, multiple periodic states and chaos. Such phenomena are
frequently observed in the collective behavior of a large number of coupled
limit cycle oscillators. Our time delayed feedback model offers a simple
paradigm for obtaining and investigating these temporal states in a single
oscillator.We construct a detailed bifurcation diagram of the oscillator as a
function of the time delay parameter and the driving strengths of the feedback
terms. We find some new states in the presence of the quadratic nonlinear
feedback term with interesting characteristics like birhythmicity, phase
reversals, radial trapping, phase jumps and spiraling patterns in the amplitude
space. Our results may find useful applications in physical, chemical or
biological systems.Comment: VERSION 4: Fig. 10(d) added, an uncited reference removed; (To appear
in Physica D) (17 pages, 21 figures, two column, aps RevTeX); VERSION 3:
Revised. In Section 2, small tau approximation added; Section 3 is divided
into subsections; periodic solution discussed in detail; Figs. 7 and 11
discarded; Figs. 12 and 14 altered; three new figures (now Figs. 10, 11 and
21) added. VERSION 2: Figs. 1 and 2 replace
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