Incomplete approach to homoclinicity in a model with bent-slow manifold geometry

The dynamics of a model, originally proposed for a type of instability in p lastic flow, has been investigated in detail. The bifurcation portrait of t he system in two physically relevant parameters exhibits a rich variety of dynamical behavior, including period bubbling and period adding or Farey se quences. The complex bifurcation sequences, characterized by mixed mode osc illations, exhibit partial features of Shilnikov and Gavrilov-Shilnikov sce nario. Utilizing the fact that the model has disparate time scales of dynam ics, we explain the origin of the relaxation oscillations using the geometr ical structure of the bent-slow manifold. Based on a local analysis, we cal culate the maximum number of small amplitude oscillations, s, in the period ic orbit of L-s type, for a given value of the control parameter. This furt her leads to a scaling relation for the small amplitude oscillations. The i ncomplete approach to homoclinicity is shown to be a result of the finite r ate of 'softening' of the eigenvalues of the saddle focus fixed point. The latter is a consequence of the physically relevant constraint of the system which translates into the occurrence of back-to-back Hopf bifurcation. (C) 2000 Elsevier Science B.V. All rights reserved.