N-dimensional dynamical systems exploiting instabilities in full

We report experimental and numerical results showing how certain N-dimensio nal dynamical systems are able to exhibit complex time evolutions based on the nonlinear combination of N-1 oscillation modes. The experiments have be en done with a family of thermo-optical systems of effective dynamical dime nsion varying from 1 to 6. The corresponding mathematical model is an N-dim ensional vector field based on a scalar-valued nonlinear function of a sing le variable that is a linear combination of all the dynamic variables. We s how how the complex evolutions appear associated with the occurrence of suc cessive Hopf bifurcations in a saddle-node pair of fixed points up to exhau st their instability capabilities in N dimensions. For this reason the obse rved phenomenon is denoted as the full instability behavior of the dynamica l system. The process through which the attractor responsible for the obser ved time evolution is formed may be rather complex and difficult to charact erize. Nevertheless, the well-organized structure of the time signals sugge sts some generic mechanism of nonlinear mode mixing that we associate with the cluster of invariant sets emerging from the pair of fixed points and wi th the influence of the neighboring saddle sets on the flow nearby the attr actor. The generation of invariant tori is likely during the full instabili ty development and the global process may be considered as a generalized La ndau scenario for the emergence of irregular and complex behavior through t he nonlinear superposition of oscillatory motions. (C) 2000 American Instit ute of Physics. [S1054-1500(00)01004-1].