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A discrete-time version of the replicator equation for two-strategy games is studied. The stationary properties differ from that of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replace…
We describe classes of ergodic dynamical systems for which some statistical properties are known exactly. These systems have integer dimension, are not globally dissipative, and are defined by a probability density and a two-form. This…
We analyze a system of reacting elements harmonically coupled to nearest neighbors in the continuum limit. An analytic solution is found for traveling waves. The procedure is used to find oscillatory as well as solitary waves. A comparison…
Sampling equation method is presented to look for exact solutions of nonlinear differential equations. Application of this approach to one of the extensive chaos model is considered. Exact solutions of this model in travelling wave are…
We review some of the exactly solvable one dimensional continuum fluid models of equilibrium classical statistical mechanics under the unified setting of functional integration in one dimension. We make some further developments and remarks…
Recent work in dynamical systems theory has shown that many properties that are associated with irreversible processes in fluids can be understood in terms of the dynamical properties of reversible, Hamiltonian systems. That is,…
Discrete autonomous dynamical systems in dimension 1 can exhibit chaotic behavior, whereas the corresponding continuous evolution equations rule it out, and cannot even possess a nontrivial periodic solution. Therefore the passage from…
In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and…
Various kinematical quantities associated with the statistical properties of dynamical systems are examined: statistics of the motion, dynamical bases and Lyapunov exponents. Markov partitons for chaotic systems, without any attempt at…
We address the linear-mode analysis performed near an equilibrium configuration in the fluid rest frame with a dynamical magnetic field perturbed on a constant configuration. We develop a simple and general algorithm for an analytic…
In various fields of natural science, the chaotic systems of differential equations are considered more than 50 years. The correct prediction of the behaviour of solutions of dynamical model equations is important in understanding of…
Classical cellular automata represent a class of explicit discrete spacetime lattice models in which complex large-scale phenomena emerge from simple deterministic rules. With the goal to uncover different physically distinct classes of…
We investigate a one dimensional flow described with the non-compressible coupled Euler and non-compressible Navier-Stokes equations in Cartesian coordinate systems. We couple the two fluids through the continuity equation where different…
We discuss recent advances in the regularity problem of a variety of fluid equations and systems. The purpose is to illustrate the advantage of harmonic analysis techniques in obtaining sharper conditional regularity results when compared…
We present an optimization process to estimate parameters in systems of ordinary differential equations from chaotic time series. The optimization technique is based on a variational approach, and numerical studies on noisy time series…
We introduce methods for deriving analytic solutions from differential-algebraic systems of equations (DAEs), as well as methods for deriving governing equations for analytic characterization which is currently limited to very small systems…
In this paper the exact analytical solution of the motion of a rigid body with arbitrary mass distribution is derived in the absence of forces or torques. The resulting expressions are cast into a form where the dependence of the motion on…
In a smooth flow, the leading-order response of trajectories to infinitesimal perturbations in their initial conditions is described by the finite-time Lyapunov exponents and associated characteristic directions of stretching. We give a…
Even if it is nonintegrable, a differential equation may nevertheless admit particular solutions which are globally analytic. On the example of the dynamical system of Kuramoto and Sivashinsky, which is generically chaotic and presents a…
The use of numerical simulation for prediction of characteristics of chaotic dynamical systems inherently involves unpredictable processes. In this work, we develop a model for the expected error in the simulation of ergodic, chaotic ODE…