In this paper, we consider a simple discrete-time single-directional network of four neurons. The characteristics equation of the linearized system at the zero solution is a polynomial equation involving very high-order terms. We first derive some sufficient and necessary conditions ensuring that all the characteristic roots have modulus less than 1. Hence, the zero solution of the model is asy...

In this manuscript, we study a Leslie–Gower predator-prey model with hyperbolic functional response and weak Allee effect. The results reveal that the supports coexistence oscillation of both predator prey populations. We also identify regions in parameter space which different kinds bifurcations, such as saddle-node Hopf bifurcations Bogdanov–Takens bifurcations.

We present an unfolding of the codimension-two scenario of the simultaneous occurrence of a discontinuous bifurcation and an Andronov-Hopf bifurcation in a piecewise-smooth, continuous system of autonomous ordinary differential equations in the plane. We find that the Hopf cycle undergoes a grazing bifurcation that may be very shortly followed by a saddle-node bifurcation of the orbit. We deriv...

Due to the encouraging increase in their applications, control theory [Lefschetz, 1965] and [Narendra et al., 1973], design of electric circuits [Chua et al., 1990], neurobiology [FitzHugh, 1961] and [Nagumo et al., 1962] piecewise linear differential systems were studied early from the point of view of qualitative theory of ordinary differential equations [Andronov et al., 1966]. Nowadays, a l...

In this paper, we study the dynamics and stability of a fundamental power system model when a time delay is imposed on the excitation of the generator. It is observed that sustained oscillations can arise in an otherwise stable power system through a delay induced Andronov-Hopf bifurcation. Numerical simulations are conducted to explore the dynamics of the time delayed system after the bifurcat...

Bifurcation theory studies the qualitative changes in the phase portrait when we vary the parameters of the system. In this book chapter we adapt and extend a mathematical model accounting for the subcellular localisation of 14-3-3σ, a protein involved in cell cycle arrest and the regulation of apoptosis. The model is analysed with analytical tools coming from Lyapunov-Andronov theory, and our ...

The bifurcation theory of snap-back repellers in hybrid dynamical systems is developed. Infinite sequences of bifurcations are shown to arise due to the creation of snap-back repellers in non-invertible maps. These are analogous to the cascades of bifurcations known to occur close to homoclinic tangencies for diffeomorphisms. The theoretical results are illustrated with reference to bifurcation...

Abstract. We show that maps describing border collision bifurcations (continuous but non-differentiable discrete time maps) are subject to a curse of dimensionality: it is impossible to reduce the study of the general case to low dimensions, since in every dimension the bifurcation can produce fundamentally different attractors (contrary to the case of local bifurcations in smooth systems). In ...