Globally coupled chaotic systems with bistable thermodynamic limit
- When?
- Friday 5 November 2010, 16:00 to 17:00
- Where?
- 22AA04
- Open to:
- Staff, Students
- Speaker:
- Roland Zweimueller (Surrey)
Abstract: I will report on joint work with Gerhard Keller (Erlangen) and Jean-Baptiste Bardet (Rouen). We study systems consisting of a large number of identical chaotic components which interact via a mean-field coupling rule, meaning that the evolution of each component depends on the global average of all states.
Specifically, we investigate the asymptotic behaviour of the distribution of this global average in a particular model. In the "thermodynamic limit" (number of components to infinity) the evolution of these distributions is given by a nonlinear self-consistent Perron-Frobenius operator (SCPFO). For the range of coupling (strength) parameters we consider, our finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. However, within this range of parameters, the SCPFO undergoes a bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium. This gives the first rigorous explanation of a numerically observed phenomenon called the "violation of the law of large numbers" in mean-field coupled maps.
