Iván Szántó

Limit cycles of a cubic Kolmogorov system

Abstract We show that for certain cubic Kolmogorov systems, four, and no more than four, limit cycles can bifurcate from a critical point in the first quadrant; moreover, three fine foci of order one can coexist and a limit cycle can bifurcate from each.

A cubic Kolmogorov system with six limit cycles

Abstract We consider a class of cubic Kolmogorov systems. We show in particular that a maximum of six small amplitude limit cycles can bifurcate from a critical point in the first quadrant, and we discuss the number of invariant lines.

Coexistence of algebraic and nonalgebraic limit cycles in Kukles systems

A class of Kukles differential systems of degree five having an invariant conic is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, under perturbations of the coefficients of the systems.

Small amplitude limit cycles for cubic systems

In this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systems x = y(ax + y 1 ) ley + sxy(—4 + y) y = ~x(x + by 1 ) + 2dx + exy(-4 + x). With a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.

Chaotic dynamics and coexistence in a three species interaction model

This work deals with a three-dimensional system, which describes a food web model consisting of a prey, a specialist predator and a top predator which is generalist as it consumes the other two species. Using tools of dynamical systems we prove that the trajectories of system are bounded and that open subsets of parameters exist, such that the system in the first octant has at most two singularities. For an open subset of the parameters space, the system is shown to have an invariant compact set and this is a topologically transitive attractor set. Finally, we find another open set in the parameters space, such that the system has two limit cycles each contained in different invariant planes. The work is completed with a numeric simulation showing the attractor is a strange attractor.

Bifurcations of limit cycles in Kukles systems of arbitrary degree with invariant ellipse

Abstract In this paper, for a certain class of Kukles polynomial systems of arbitrary degree n with an invariant ellipse, we show that for certain values of the parameters, the system has an upper bound of limit cycles, where one of the limit cycle is given by an invariant ellipse as an algebraic limit cycle. Writing the system as a perturbation of a Hamiltonian system, we show that the first Poincare–Melnikov integral of the system is a polynomial whose coefficients are the Lyapunov quantities. The maximum number of simple zeros of this polynomial, gives the maximum number of the global limit cycles and the multiplicity of the origin as a root of the polynomial, minus one, gives the maximum weakness that may have the weak focus at the origin.

A predator-prey system involving five limit cycles

Abstract. In this paper we consider the multiparameter system introduced in [M. Scheffer et al. Can.J. Fish. Aquat. Sci. 57(6);1208-1219 (2000)] which corresponds to an extension of the classic minimal Daphnia-algae model. It is shown that there is a neighborhood in the parameter space where the system in the realistic quadrant has a unique equilibrium point which is a repelling weak focus of order four enclosed by a global attractor hyperbolic limit cycle. For a small enough change of the parameters in this neighborhood, bifurcate from the weak focus four infinitesimal Hopf limit cycles (alternating the type of stability) such that the last bifurcated limit cycle is an attractor. Moreover, for certain values of parameters we concluded that this applied model has five concentric limit cycles , three of them being stable hyperbolic limit cycles. This gives a positive answer to a question raised in [C.S. Coleman, Differential Equations Models, V 1. 279-297, Springer-Verlag (1978)] and [N.G. Lloyd et al. Appl. Math. Lett. Vol 9, No. 1, 15-18 (1996)].