In second order Lagrangian systems bifurcation branches of periodic solutions preserve certain topological invariants. These invariants are based on the observation that periodic orbits of a second order Lagrangian lie on 3-dimensional (non-compact) energy manifolds and the periodic orbits may have various linking and knotting properties. The main ingredients to define the topological invariant...

We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic xed point in a ow onR 3 originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or \pleat," we may apply the theory of templates, or branched two-manifolds, to capture the topology of the ow. This analysis yields a class of ows whic...

This paper discusses a problem of the Hopf bifurcation control for a mathematical model of intracellular calcium oscillations by calculating the curvature coefficient of limit cycle and the bifurcation control theory. We find that the appearance and disappearance of calcium oscillations in this system are due to the supercritical and subcritical Hopf bifurcation of equilibrium points, respectiv...

We examine some properties of attractors for symmetric dynamical systems that show what we refer to as`chaotic intermittency'. These are attractors that contain points with several diierent symmetry types, with the consequence that attracted trajectories come arbitrarily close to possessing a variety of diierent symmetries. Such attractors include heteroclinic attractors, on-oo and in-out inter...

In this paper, we show numerically that the so-called HIV therapy system is regular and converges to a stable equilibrium point for most realistic values (in the medical sense) of its bifurcation parameters. Although there is no rigorous proof of this convergence property, it is conjectured that the system is not chaotic for all positive bifurcation parameters.