Convergent Chaos

Chaos is widely understood as being a consequence of sensitive dependence upon initial conditions. This is the result of an instability in phase space, which separates trajectories exponentially. Here, we investigate the dynamics of a chaotic one-dimensional model for inertial particles in a random velocity field and demonstrate that despite their overall intrinsic instability, trajectories may be very strongly convergent in phase space over extremely long periods. We establish that this strong convergence is a multi-facetted phenomenon, in which the clustering is intense, widespread and balanced by lacunarity of other regions [1]. Power laws, indicative of scale-free features, characterize the distribution of particles in the system. We use large-deviation theory and extreme-value statistics to explain this effect, and in particular, we develop the large-deviation theory for fluctuations of the finite-time Lyapunov exponent of this system. We show that the determination of the entropy function for the distribution reduces to the analysis of a Schrödinger equation, which is tackled by semi-classical methods [2]. The system has ‘generic’ instability properties, and we consider the broader implications of our observation of long-term stability in chaotic systems. Our results show that the interpretation of the ‘butterfly effect’ needs to be carefully qualified. [1] M. Pradas, A. Pumir, G. Huber, M. Wilkinson, J. Phys A 50, 275101 (2017); [2] G. Huber, M. Pradas, A. Pumir, M. Wilkinson, Physica A (in press, 2017).