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Register a new userUnstable delayed feedback control to change sign of coupling strength for weakly coupled limit cycle oscillators
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Added by
Viktor Novi\\v{c}enko
Publication date
2020
fields
Physics
and research's language is
English
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Weakly coupled limit cycle oscillators can be reduced into a phase model using phase reduction approach, and the phase model itself is helpful to analyze a synchronization. For example, phase model of two oscillators is one-dimensional differential equation for the evolution of a phase difference, and an existence of fixed points determines frequency-locking solutions. By treating each oscillator as a black-box possessing a single-input single-output one can investigate various control algorithms to change the synchronization of the oscillators. In particular, we are interested in a delayed feedback control algorithm, which applied to oscillator after the phase reduction gives the same phase model as of the control-free case, yet a coupling strength is rescaled. The conventional delayed feedback control is limited to change a magnitude but not a sign of the coupling strength. In this work we present modification of the delayed feedback algorithm supplemented by an additional unstable degree of freedom, which is able to change the sign of the coupling strength. Various numerical calculations performed with Landau-Stuart and FitzHugh-Nagumo oscillators show successful switching between an in-phase and an anti-phase synchronization using provided control algorithm. Additionally we show that the control force becomes non-invasive if our objective is a stabilization of an unstable phase difference for two coupled oscillators.
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We present a linear model, which mimics the response of a spatially extended dissipative medium to a distant perturbation, and investigate its dynamics under delayed feedback control. The time a perturbation needs to propagate to a measurement point is captured by an inherent delay time (or latency). A detailed linear stability analysis demonstrates that a non-zero system delay acts destabilizing on the otherwise stable fixed point for sufficiently large feedback strengths. The imaginary part of the dominant eigenvalue is bounded by twice the feedback strength. In the relevant parameter space it changes discontinuously along specific lines when switching between branches of eigenvalues. When the feedback control force is bounded by a sigmoid function, a supercritical Hopf bifurcation occurs at the stability-instability transition. The frequency and amplitude of the resulting limit cycles respond to parameter changes like the dominant eigenvalue. In particular, they show similar discontinuities along specific lines. These results are largely independent of the exact shape of the sigmoid function. Our findings match well with previously reported results on a feedback-induced instability of vortex diffusion in a rotationally driven Newtonian fluid [M. Zeitz, P. Gurevich, and H. Stark, Eur. Phys. J. E 38, 22 (2015)]. Thus, our model captures the essential features of nonlocal delayed feedback control in spatially extended dissipative systems.
The mechanism of phase synchronization between uncoupled limit-cycle oscillators induced by common external impulsive forcing is analyzed. By reducing the dynamics of the oscillator to a random phase map, it is shown that phase synchronization generally occurs when the oscillator is driven by weak external impulses in the limit of large inter-impulse intervals. The case where the inter-impulse intervals are finite is also analyzed perturbatively for small impulse intensity. For weak Poissonian impulses, it is shown that the phase synchronization persists up to the first order approximation.
We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered. The derivation is performed within the framework of the phase reduction, which fully accounts for the amplitude degrees of freedom. Firstly, we consider the case of identical oscillators subject to intrinsic noise, obtain the synchronization condition, and find that the distribution of phase deviations always possesses lower-law heavy tails. Secondly, we consider the case of nonidentical oscillators. For the average oscillator frequency as a function of the natural frequency mismatch, limiting scaling laws are derived; these laws exhibit the nontrivial phenomenon of frequency repulsion accompanying synchronization under negative coupling. The analytical theory is illustrated with examples of Van der Pol and Van der Pol--Duffing oscillators and the neuron-like FitzHugh--Nagumo system; the results are also underpinned by the direct numerical simulation for ensembles of these oscillators.
Model reduction techniques have been widely used to study the collective behavior of globally coupled oscillators. However, most approaches assume that there are infinitely many oscillators. Here we propose a new ansatz, based on the collective coordinate approach, that reproduces the collective dynamics of the Kuramoto model for finite networks to high accuracy, yields the same bifurcation structure in the thermodynamic limit of infinitely many oscillators as previous approaches, and additionally captures the dynamics of the order parameter in the thermodynamic limit, including critical slowing down that results from a cascade of saddle-node bifurcations.
Collective behavior of pulse-coupled oscillators has been investigated widely. As an example of pulse-coupled networks, fireflies display many kinds of flashing patterns. Mirollo and Strogatz (1990) proposed a pulse-coupled oscillator model to explain the synchronization of South East Asian fireflies ({itshape Pteroptyx malaccae}). However, transmission delays were not considered in their model. In fact, the presence of transmission delays can lead to desychronization. In this paper, pulse-coupled oscillator networks with delayed excitatory coupling are studied. Our main result is that under reasonable assumptions, pulse-coupled oscillator networks with delayed excitatory coupling can not achieve complete synchronization, which can explain why another species of fireflies ({itshape Photinus pyralis}) rarely synchronizes flashing. Finally, two numerical simulations are given. In the first simulation, we illustrate that even if all the initial phases are very close to each other, there could still be big variations in the times to process the pulses in the pipeline. It implies that asymptotical synchronization typically also cannot be achieved. In the second simulation, we exhibit a phenomenon of clustering synchronization.
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