This article concerns a symmetrical Lotka-Volterra predator-prey system with delays. By analyzing the associated characteristic equation of the original system at the positive equilibrium and choosing the delay as the bifurcation parameter, the local stability and Hopf bifurcation of the system are investigated. Using the normal form theory, we also establish the direction and stability of the ...

The main modes of behavior of a food chain model, composed of logistic prey and Holling type II predator and superpredator, are discussed in this paper. The study is carried out through bifurcation analysis, alternating between a normal form approach and numerical continuation. The two-parameter bifurcation diagram of the model contains Hopf, fold and transcritical bifurcation curves of equilib...

We examine a heteroclinic bifurcation occurring in families of equivariant vector elds. Within these families, the ows contain structurally stable heteroclinic cycles. The ow can twist around the cycle to produce what is the equivalent of an \inclination-ip" homoclinic orbit. An unfolding of this bifurcation in a generic one-parameter equivariant family shows that Smale horseshoes are embedded ...

This paper is primarily a numerical study of the xed-point bifurcation loci { saddle-node, period-doubling and Hopf bifurcations { present in the family: z ! f (C;A) (z; z) z + z 2 + C + Az where z is a complex dynamic (phase) variable, z its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a secondary parameter, asking how the bifu...

We review some properties of dynamical systems with slowly varying parameters, when a parameter is moved through a bifurcation point of the static system. Bifurcations with a single zero eigenvalue may create hysteresis cycles, whose area scales in a nontrivial way with the adiabatic parameter. Hopf bifurcations lead to the delayed appearance of oscillations. Feedback control theory motivates t...

We study the fine geometric structure of bifurcation currents in the parameter space of cubic polynomials viewed as dynamical systems. In particular we prove that these currents have some laminar structure in a large region of parameter space, reflecting the possibility of quasiconformal deformations. On the other hand, there is a natural bifurcation measure, supported on the closure of rigid p...

One of the major routes to chaotic scattering is through an abrupt bifurcation by which a nonattracting chaotic saddle is created as a system parameter changes through a critical value. In a previously investigated case, however, the fractal dimension of the set of singularities in the scattering function changes continuously through the bifurcation. We describe a type of abrupt bifurcation to ...

The quenching of alternans is considered using a nonlinear cardiac conduction model. The model consists of a nonlinear discrete-time piecewise smooth system. Several authors have hypothesized that alternans arise in the model through a period doubling bifurcation. In this work, it is first shown that the alternans exhibited by the model actually arise through a period doubling border collision ...

In this paper we demonstrate the power of the computer algebra package conley, which enables one to compute connection and transition matrices, two of the main algebraic tools of the Conley index theory. In particular, we study the Cahn-Hilliard equation on the unit square and extend the results obtained in [Maier-Paape et al., 2007] to a bigger range of the bifurcation parameter. Besides provi...

In this paper, we study bifurcation structures of period-adding phenomena in an internal wave model that is a mathematical model for ocean internal waves. It has been suggested that chaotic solutions observed in the internal wave model may be related to the universal property of the energy spectra of ocean internal waves. In numerical bifurcation analyses of the internal wave model, we illustra...