Collective Lyapunov modes

Abstract

Abstract. We show, using covariant Lyapunov vectors in addition to standard Lyapunov analysis, that there exists a set of collective Lyapunov modes in large chaotic systems exhibiting collective dynamics. Associated with delocalized Lyapunov vectors, they act collectively on the trajectory and hence characterize the instability of its collective dynamics. We further develop, for globally-coupled systems, a connection between these collective modes and the Lyapunov modes in the corresponding PerronFrobenius equation. We thereby address the fundamental question of the effective dimension of collective dynamics and discuss the extensivity of chaos in presence of collective dynamics.

Chaotic macroscopic phases in one-dimensional oscillators

“…This localized structure itself is, however, unexpected. In fact Λ max is rather close to λ min and, on the basis of the observations reported in reference [12], one might expect to find evidence of the macroscopic instability in the structure of the usual microscopic Lyapunov exponents. This is an additional evidence that the relationship between macroscopic and microscopic instabilities is subtler than believed so far.…”

“…However, one cannot dismiss the results contained in reference [12], based on the structure of the microscopic covariant Lyapunov vectors [19][20][21].…”

“…Extensivity is, therefore, a requisite for a covariant vector to have a macroscopic interpretation. In fact, extensive vectors have been found in some mean-field models [12], especially in an ensemble of noisy logistic maps, where one could make a comparison with the Frobenius-Perron operator. So it is legitimate to ask how it is possible that in some cases the saturation induced by the reordering of the microscopic variables (see Fig.…”

“…On the one hand, it has been found that in some model, the most relevant exponents which control the stability of the collective motion can be singled out within the microscopic Lyapunov spectra [12]. In other cases, it has been found that the macroscopic stability is controlled by the evolution of finite perturbations [13].…”

“…This is not, however, always the case. Coupled logistic maps appear to be characterized by an infinite dimensional collective dynamics: this has been conjectured by adding noise to the microscopic dynamics (in order to smooth out the singularities of the density) and thereby decreasing the noise amplitude [12,18]. As a result, it has been found that the dimension increases logarithmically upon decreasing the noise amplitude.…”