Shil’nikov Chaos: How to Characterize Homoclinic and Heteroclinic Behaviour
We introduce the concepts of Shil’nikov chaos and competing instabilities in a nonlinear dynamics including at least a saddle focus and a saddle point, a parameter change induces a smooth transition from a homoclinic to a heteroclinic trajectory.
In terms of the return map to a given Poincaré section, the two trajectories have the following characterization. In the homoclinic case, the global behavior is recovered from the local linear dynamics within a unit box around the saddle focus. The heteroclinic case requires the composition of two linearized maps around the two unstable points.
By an exponential transformation the geometrical map yields the return map of the orbital times. This new map represents the most appropriate indicator for experimental situations whenever a symbolic dynamics built on geometric position does not offer a sensitive test. Furthermore the time maps display a large sensitivity to noise. This offers a criterion to discriminate between a simulation (either analog or digital) with a few variables and experiment dealing with the physical variables embedded in the real world and thus acted upon by noise.
KeywordsHomoclinic Orbit Saturable Absorber Population Inversion Symbolic Dynamic Unstable Point
Unable to display preview. Download preview PDF.
- L.P. Shil’nikov, Mat. Sb. 81, 92, 123 (1970).Google Scholar
- 6.F.T. Arecchi, A. Lapucci, R. Meucci, J.A. Roversi, and P. Coullet, i) ELITE Meeting, Torino, Italy, march 1987.Google Scholar
- F.T. Arecchi, A. Lapucci, R. Meucci, J.A. Roversi, and P. Coullet, ii) International Workshop on Instabilities and Chaos in Nonlinear Optical systems, Il Ciocco, Italy, July 1987 paper WC 13–2.Google Scholar
- 8.J.P. Crutchfield, D. Farmer, and B.A. Hubermann, Phys. Rev. 92, 45 (1982).Google Scholar
- F.T. Arecchi, and A. Politi, and L. Ulivi, Nuovo Cimento 71B, 119, (1982).Google Scholar
- 15.F. Papoff, A. Fioretti, E. Arimondo, and N.B. Abraham, “Return time and distribution in the laser with saturable absorber”, this same ConferenceGoogle Scholar
- 16.F.T. Arecchi, A. Lapucci, and R. Meucci, to be published.Google Scholar