2000/01/14 by S. Rajesh, G. Ananthakrishna · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Amplitude #Bent molecular geometry #Bifurcation #Center manifold #Chaos control and synchronization #Farey sequence #Fixed point #Geometry #Homoclinic bifurcation #Hopf bifurcation #Infinite-period bifurcation #Instability #Mathematical analysis #Mathematical optimization #Mathematics #Mechanics #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Phase portrait #Physics #Saddle #Saddle-node bifurcation #Scaling #Slow manifold #cond-mat.mtrl-sci #nlin.CD #stochastic dynamics and bifurcation
paper · pdf · doi:10.1016/s0167-2789(99)00241-9
14 Figures(Postscript); To Appear in Physica D : Nonlinear Phenomena
arxiv created 2000/01/14 · openalex publication_date 2000/06/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The dynamics of a model, originally proposed for a type of instability in plastic flow, has been investigated in detail. The bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical behaviour, including period bubbling and period adding or Farey sequences. The complex bifurcation sequences, characterized by Mixed Mode Oscillations, exhibit partial features of Shilnikov and Gavrilov-Shilnikov scenario. Utilizing the fact that the model has disparate time scales of dynamics, we explain the origin of the relaxation oscillations using the geometrical structure of the bent-slow manifold. Based on a local analysis, we calculate the maximum number of small amplitude oscillations, s, in the periodic orbit of Ls type, for a given value of the control parameter. This further leads to a scaling relation for the small amplitude oscillations. The incomplete approach to homoclinicity is shown to be a result of the finite rate of `softening' of the eigen values 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.