North Park Road, Exeter EX4 4QF, UK
College of Engineering, Mathematics and Physical Sciences, University of Exeter
Vitolo R., Broer H. W., Simó C.
Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems
2011, vol. 16, no. 1-2, pp. 154-184
This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.
Broer H. W., Holtman S. J., Vegter G., Vitolo R.
Dynamics and geometry near resonant bifurcations
2011, vol. 16, no. 1-2, pp. 39-50
This paper provides an overview of the universal study of families of dynamical systems undergoing a Hopf–Neimarck–Sacker bifurcation as developed in [1–4]. The focus is on the local resonance set, i.e., regions in parameter space for which periodic dynamics occurs. A classification of the corresponding geometry is obtained by applying Poincaré–Takens reduction, Lyapunov–Schmidt reduction and contact-equivalence singularity theory, equivariant under an appropriate cyclic group. It is a classical result that the local geometry of these sets in the nondegenerate case is given by an Arnol’d resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our approach also provides a skeleton for the local resonant Hopf–Neimarck–Sacker dynamics in the form of planar Poincaré–Takens vector fields. To illustrate our methods a leading example is used: A periodically forced generalized Duffing–Van der Pol oscillator.