Abstract:

This paper investigates the mixing property of a class of piecewise linear Lorenz maps and the existence of absolutely continuous invariant measures for their stochastically coupled counterparts. Applying renormalization theory, it is rigorously proved that two deterministic maps within this Lorenz framework exhibit mixing. Based on this foundation, numerical experiments are designed, revealing a peculiar phenomenon of synchronized chaos: while both deterministic maps have positive Lyapunov exponents, the corresponding stochastic map possesses a negative Lyapunov exponent while still maintaining an absolutely continuous invariant measure. Further research demonstrates that this stochastic map not only possesses global ergodicity but can also induce the emergence of synchronization phenomena.

Key words: Lorenz map, trivial renormalization, random map, absolutely continuous invariant measure.