This paper presents a comprehensive introduction and systematic derivation of the evolutionary equations for absolute entropy H and relative entropy D, some of which exist sporadically in the literature in different forms under different subjects, within the framework of dynamical systems. In general, both H and D are dissipated, and the dissipation bears a form reminiscent of the Fisher inform...

It is traditionally believed that the macroscopic randomness has nothing to do with the micro-level uncertainty. Besides, the sensitive dependence on initial condition (SDIC) of Lorenz chaos has never been considered togetherwith the so-called continuum-assumption of fluid (on which Lorenz equations are based), from physical and statistic viewpoints. A very fine numerical technique [6] with neg...

We describe a MIMD parallel code to solve a general class of ordinary di erential equations, with particular emphasis on the large, sparse systems arising from space discretization of systems of parabolic partial di erential equations. The main goals of this work are sharp bounds on the accuracy of the computed solution and exibility of the software. We discuss the sources of error in solving d...

In this paper, the slow manifolds of Lorenz-Haken system are discussed. By using three different methods, associated with the slow manifold equations of the L-H system are obtained. Firstly, the slow manifold equation of the nonlinear chaotic dynamic system is obtained by considering that the slow manifold is locally defined by a plane orthogonal to the tangent system’s left fast eigenvector. O...

We describe a MIMD parallel code to solve a general class of ordinary diierential equations, with particular emphasis on the large, sparse systems arising from space discretization of systems of parabolic partial diierential equations. The main goals of this work are sharp bounds on the accuracy of the computed solution and exibility of the software. We discuss the sources of error in solving d...

It is formally proved that the general smoother for nonlinear dynamics can be formulated as a sequential method, that is, observations can be assimilated sequentially during a forward integration. The general filter can be derived from the smoother and it is shown that the general smoother and filter solutions at the final time become identical, as is expected from linear theory. Then, a new sm...

is considered in which complex behavior of trajectories has been discovered [1] by means of computer simulation. These equations were put forward in [1] as a model for studying the dynamics of the Lorenz system for large Rayleigh number. Some physical applications of the model were pointed out in [2]. It was shown in papers [3,4] that there are two types of Lorenz-like attractors in this model....

In this paper we first analyze the structure of Maxwell equations in a Lorentzian spacetime when the potential is A = eK, i.e., proportional to a 1-form K physically equivalent Killing vector field. We show that A necessarily obeys the Lorenz gauge δA = 0. Moreover we determine the form of the current associated with this potential showing that it is of a superconducting type, i.e., proportiona...

In this talk we present a few recent results on regularity properties and numerical approximations for stochastic ordinary and partial differential equations with non-globally monotone nonlinearities. In particular, we establish strong convergence rates for Cox-Ingersoll-Ross (CIR) processes, stochastic Duffing-van der Pol oscillators, stochastic Lorenz equations, and Cahn-Hilliard-Cook equatio...

An algorithm is presented for detecting switch dynamics in chaotic time-waveform. By the “switch dynamics,” we mean that the chaotic time-waveform is measured from a dynamical system whose bifurcation parameters are occasionally switched among a set of slightly different parameter values. First, the switched chaotic time-waveform is divided into windows of short-term time-waveforms. From the se...