Delay-Induced Triple-Zero Bifurcation in a Delayed Leslie-Type Predator–Prey Model with Additive Allee Effect

In this paper, a Leslie-type predator–prey model with ratio-dependent functional response and Allee effect on prey is considered. We first study the existence of the multiple positive equilibria and their stability. Then we investigate the effect of delay on the distribution of the roots of characteristic equation and obtain the conditions for the occurrence of simple-zero, double-zero and triple-zero singularities. The formulations for calculating the normal form of the triple-zero bifurcation of the delay differential equations are derived. We show that, under certain conditions on the parameters, the system exhibits homoclinic orbit, heteroclinic orbit and periodic orbit.

HOMOCLINIC INTERACTIONS NEAR A TRIPLE-ZERO DEGENERACY IN CHUA'S EQUATION

In this work, we consider some degeneracies of homoclinic and heteroclinic connections organized by the triple-zero degeneracy, in Chua's equation. This allows us to numerically study the homoclinic-heteroclinic transition exhibited by the curve of Takens–Bogdanov bifurcations as it passes through the triple-zero degeneracy. Several codimension-two degenerate homoclinic and heteroclinic connections organized by the triple-zero bifurcation are involved in this transitional homoclinic-heteroclinic mechanism. In particular, we point out that the existence of a curve of T-points and a curve of Belyakov points (its equilibrium passes from saddle to saddle-focus) is necessary for this process to occur. Closed curves of homoclinic connections with different pulses are also found.

RESONANCES OF PERIODIC ORBITS IN RÖSSLER SYSTEM IN PRESENCE OF A TRIPLE-ZERO BIFURCATION

This paper focuses on resonance phenomena that occur in a vicinity of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point in an autonomous tridimensional system. Namely, by means of blow-up techniques that relate the triple-zero bifurcation to the Kuramoto–Sivashinsky system, we characterize the resonances that appear near the triple-zero bifurcation. Using numerical tools, the results are applied to the Rössler equation, showing a number of interesting bifurcation behaviors associated to these resonance phenomena. In particular, the merging of the periodic orbits appeared in resonances, the existence of two types of Takens–Bogdanov bifurcations of periodic orbits and the presence of Feigenbaum cascades of these bifurcations, joined by invariant tori curves, are pointed out.