Neural networks are dense in the space of dynamical systems. We present a Monte Carlo study of the dynamic properties along the route to chaos over random dynamical system function space by randomly sampling the neural network function space. Our results show that as the dimension of the system (the number of dynamical variables) is increased, the probability of chaos approaches unity. We prese...

Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann–Gibbs (BG) statistical mechanics has suggested the potential verification of these generalizations at the onset of chaos, when the only Lyapunov expon...

The natural order in the space of binary sequences permits to recover the U -sequence. Also the scaling laws of the period-doubling cascade and the intermittency route to chaos defined in that ordered set are explained. These arise as intrinsic properties of this ordered set, and independent from any consideration about dynamical systems.

With a view to understanding the "rheochaos" observed in recent experiments in a variety of orientable fluids, we study numerically the equations of motion of the spatiotemporal evolution of the traceless symmetric order parameter of a sheared nematogenic fluid. In particular we establish, by decisive numerical tests, that the irregular oscillatory behavior seen in a region of parameter space w...

In this paper, the analytical methods for approximate solutions of periodic motions to chaos in nonlinear dynamical systems are reviewed. Briefly discussed are the traditional analytical methods including the Lagrange stand form, perturbation methods, and method of averaging. A brief literature survey of approximate methods in application is completed, and the weakness of current existing appro...

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos, and the persistence o...

One of the predominant themes of nonlinear dynamics during the past twenty years has been the characterization of the \routes to chaos". Within generic families of vector elds, there are ubiquitous patterns that describe how qualitative properties of vector elds can change with a varying parameter. Bifurcation theory classiies these patterns, and they form the substrate for determining the rout...