- Series
- CDSNS Colloquium
- Time
- Monday, October 21, 2013 - 4:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Bastien Fernandez – CPT Luminy
- Organizer
- Leonid Bunimovich
To identify and to explain coupling-induced phase transitions in
Coupled Map Lattices (CML) has been a lingering enigma for about two
decades. In numerical simulations, this phenomenon has always been observed
preceded by a lowering of the Lyapunov dimension, suggesting that the
transition might require changes of linear stability. Yet, recent proofs of
co-existence of several phases in specially designed models work in the
expanding regime where all Lyapunov exponents remain positive.
In this talk, I will consider a family of CML composed by piecewise
expanding individual map, global interaction and finite number N of sites,
in the weak coupling regime where the CML is uniformly expanding.
I will show, mathematically for N=3 and numerically for N>3, that a
transition in the asymptotic dynamics occurs as the coupling strength
increases. The transition breaks the (Milnor) attractor into several
chaotic pieces of positive Lebesgue measure, with distinct empiric
averages. It goes along with various symmetry breaking, quantified by means
of magnetization-type characteristics.
Despite that it only addresses finite-dimensional systems, to some extend,
this result reconciles the previous ones as it shows that loss of
ergodicity/symmetry breaking can occur in basic CML, independently of any
decay in the Lyapunov dimension.