Abstract

We will present through a toy example given by three coupled first order ODEs a mechanism of transition from simple periodic orbits to motions of increasing complexity. The mechanism is based on the presence of many separatrices (eventually a dense set) and can be understood by regarding the evolution in real time as a curve on a Riemann surface, the one associated to the solution (as a function of some suitably defined "complex time") of the equations of motion. The presence of a high amount of branching produces complicated orbits. We will also discuss in which sense this mechanism also features sensitive dependence on the initial data, albeit with no exponential divergence of nearby trajectories. The explanation will be complemented by the projection of computer simulations of the different motions obtained through a numerical integration of the equations. All these concepts will be explained for the toy model in question, although the mechanism is believed to be present in many systems governed by differential equations.

This is joint work with Francesco Calogero (Roma La Sapienza), Paolo Santini (Roma La Sapienza) and Matteo Sommacal (SISSA Trieste).