Lyapunov exponents and extensivity of strongly coupled chaotic maps in regular graphs

In Thermodynamics and Statistical Physics, a system’s property is extensive when it grows with the system size. When it happens, the system can be decomposed into separate components, which has been done in many systems with weakly interacting components, such as for various gas models. Similarly, Ruelle conjectured 40 years ago that the Lyapunov exponents (LEs) of some sufficiently large chaotic systems are extensive, which led to study the extensivity properties of chaotic systems with strong interactions. Because of the complexities in these systems, most results achieved so far are restricted to numerical simulations. Here, we derive closed-form expressions for the LEs and entropy rate of coupled maps in finite- and infinite-sized regular graphs, according to the coupling strength, map’s chaoticity, and graph’s spectral properties. We show that this type of system has either 4 or 5 cases for the LEs, depending on the graph’s extreme Laplacian eigenvalues. These cases represent qualitatively different collective behaviours emerging in parameter space, including chaotic synchronisation (N−1 negative LEs) and incoherent chaos (N positive LEs). From the entropy rate, we show that the ring and complete graphs (nearest-neighbour and all-to-all couplings, respectively) are extensive in all parameter regions outside the chaotic synchronisation region. Although our derivations are restricted to one-dimensional maps with constant positive derivative (i.e., chaotic), our approach can be used to find LE and entropy rates for other regular graphs (such as for cyclic graphs) or be the basis for tackling small world graphs via perturbative methods.