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Incomplete approach to homoclinicity in a model with bent-slow manifold geometry | | S. Rajesh
; G. Ananthakrishna
; | | Date: |
14 Dec 1999 | | Subject: | Chaotic Dynamics; Materials Science | nlin.CD cond-mat.mtrl-sci | | Affiliation: | Materials Research Centre, Indian Institute of Science. India | | Abstract: | 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 $L^s$ 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. | | Source: | arXiv, nlin.CD/0001030 | | Services: | Forum | Review | PDF | Favorites | |
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