This chapter and Chapter 13 use the theory of normal forms developed in Chapter 9. They contain an introduction to generic bifurcation theory and its applications. Bifurcation theory has grown into a vast subject with a large literature; so, this chapter can only present the basics of the theory. The primary focus of this chapter is the study of periodic solutions, their existence and evolution...

The Hopf bifurcation in slow-fast systems with two slow variables and one fast variable has been studied recently, mainly from a numerical point of view. Our goal is to provide an analytic proof of the existence of the zero Hopf bifurcation exhibited for such systems, and to characterize the stability or instability of the periodic orbit which borns in such zero Hopf bifurcation. Our proofs use...

In this paper the dynamics of a discrete-time prey-predator system is investigated in the closed first quadrant . The existence and stability of fixed points are analyzed algebraically .The conditions of existence for flip bifurcation is derived by busing center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results but also exhibit complex dynamical beha...

A class of Beddington-DeAngelis functional response predator-prey model is considered. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the nor...

Symmetric bifurcation theory is the study of how the trajectories of symmetric vector fields behave as parameters are varied. We introduce some of the basic ideas of this theory in the context of dynamical system models of speciation in evolution. Abstractly, these models are dynamical systems that are equivariant under the natural permutation action of the symmetric group SN on R kN for some i...

In two-parameter systems with symmetry two steady state bifurcation points of diierent symmetry types coalesce generically within one point. Under certain group theoretic conditions involving the action of the symmetry group on the kernels, we show that secondary Hopf bifurcation is borne by the mode interaction. We explain this phenomenon by using linear representation theory. For motivation a...

By applying the bifurcation theory of dynamical systems to the Novikov equation, a new feature of non-smooth traveling wave solutions, two peakons or two cuspons that coexist for the same wave speed, is put forward. It is shown that 0 = g is the peakon bifurcation value in the process of obtaining the bifurcation of phase portraits, where g is a certain integration constant. In particular, we o...

In this work, an analytical and numerical analysis of the transition to chaos in five nonlinear systems ordinary partial differential equations, which are models autocatalytic chemical processes interacting populations, is carried out. It shown analytically numerically that all considered further complication dynamics solutions biological turbulence out full accordance with universal Feigenbaum...