Based on Rabinovich system, a 4D Rabinovich system is generalized to study hidden attractors, multiple limit cycles and boundedness of motion. In the sense of coexisting attractors, the remarkable finding is that the proposed system has hidden hyperchaotic attractors around a unique stable equilibrium. To understand the complex dynamics of the system, some basic properties, such as Lyapunov exp...

It has been known that a diffusive coupling between two limit cycle oscillations typically leads to the in-phase synchronization and also that it is the only stable state in the weak-coupling limit. Recently, however, it has been shown that the coupling of the same nature can result in the distinctive dephased synchronization when the limit cycles are close to the homoclinic bifurcation, which ...

Within the context of Liénard equations, we present the FitzHugh–Nagumo model with an idealized nonlinearity. We give an analytical expression (i) for the transient regime corresponding to the emission of a finite number of action potentials (or spikes), and (ii) for the asymptotic regime corresponding to the existence of a limit cycle. We carry out a global analysis to study periodic solutions...

In this paper we study the appearance of bifurcations limit cycles in an epidemic model with two types aware individuals. All transition rates are constant except for alerting decay rate most individuals and creation less individuals, which depend on disease prevalence a non-linear way. For ODE model, numerical computation their stability made by means Poincaré map. Moreover, sufficient conditi...

We develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations. Collective phase equations describing macroscopic rhythms of the system are derived from Langevin-type equations of globally coupled active rotators via a nonlinear Fokker-Planck equation. The theory is an extension of the conventional phase reduction method for...

Motivated by real-world excitable systems such as neuron models and lasers, we consider a paradigmatic model for excitability with a global bifurcation, namely a saddle-node bifurcation on a limit cycle. We study the effect of a time-delayed feedback force in the form of the difference between a system variable at a certain time and at a delayed time. In the absence of delay the only attractor ...

The chapter addresses bifurcations of limit cycles for a general class of nonlinear control systems depending on parameters. A set of simple approximate analytical conditions characterizing all generic limit cycle bifurcations is determined via a first order harmonic balance analysis in a suitable frequency band. Moreover, due to the existing connection between limit cycle bifurcations and rout...

In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and ...