Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms
- 1 April 1996
- journal article
- research article
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 16 (6), 1147-1172
- https://doi.org/10.1017/s0143385700009950
Abstract
We study a generic, real analytic unfolding of a planar diffeomorphism having a fixed point with unipotent linear part. In the analogue for vector fields an open parameter domain is known to exist, with a unique limit cycle. This domain is bounded by curves corresponding to a Hopf bifurcation and to a homoclinic connection. In the present case of analytic diffeomorphisms, a similar domain is shown to exist, with a normally hyperbolic invariant circle. It follows that all the ‘interesting’ dynamics, concerning the destruction of the invariant circle and the transition to trivial dynamics by the creation and death of homoclinic points, takes place in an exponentially small part of the parameter-plane. Partial results were stated in [5]. Related numerical results appeared in [16].Keywords
This publication has 1 reference indexed in Scilit:
- Averaging under Fast Quasiperiodic ForcingPublished by Springer Science and Business Media LLC ,1994