Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays

We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis

The Kuramoto model: A simple paradigm for synchronization phenomena

Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included

Synchronous dynamics of a delayed two-coupled oscillator

This paper presents a detailed analysis on the dynamics of a delayed two-coupled oscillator. Linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. By means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of time delay on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. Moreover, we illustrate our results by numerical simulations

The mathematics behind chimera states

Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott--Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed