Invariant Measure of Nonlinear Dynamical Systems and Chaotic Behaviors

LI-YORKE CHAOTIC GENERALIZED SHIFT DYNAMICAL SYSTEMS

‎In this text we prove that in generalized shift dynamical system $(X^Gamma,sigma_varphi)$‎ ‎for finite discrete $X$ with at least two elements‎, ‎infinite countable set $Gamma$ and‎ ‎arbitrary map $varphi:GammatoGamma$‎, ‎the following statements are equivalent‎: ‎ - the dynamical system $(X^Gamma,sigma_varphi)$ is‎ Li-Yorke chaotic;‎ - the dynamical system $(X^Gamma,sigma_varphi)$ has‎ an scr...

Animated Phase Portraits of Nonlinear and Chaotic Dynamical Systems

li-yorke chaotic generalized shift dynamical systems

‎in this text we prove that in generalized shift dynamical system $(x^gamma,sigma_varphi)$‎ ‎for finite discrete $x$ with at least two elements‎, ‎infinite countable set $gamma$ and‎ ‎arbitrary map $varphi:gammatogamma$‎, ‎the following statements are equivalent‎: ‎ - the dynamical system $(x^gamma,sigma_varphi)$ is‎ li-yorke chaotic;‎ - the dynamical system $(x^gamma,sigma_varphi)$ has‎ an scr...

Sporadicity: Between periodic and chaotic dynamical behaviors.

We define the class of sporadic dynamical systems as the systems where the algorithmic complexity of Kolmogorov [Kolmogorov, A. N. (1983) Russ. Math. Surv. 38, 29-40] and Chaitin [Chaitin, G. J. (1987) Algorithmic Information Theory (Cambridge Univ. Press, Cambridge, U.K.)] as well as the logarithm of separation of initially nearby trajectories grow as n(v(0) )(log n)(v(1) ) with 0 < v(0) < 1 o...

Numerical Modeling of Toroidal Dynamical Systems with Invariant Lebesgue Measure

Computer simulat ions of dynamical systems can cont ain bot h discretizations, in which finite machine arithmet ic replaces cont inuum state spaces , and realizations , in which a continuous syste m is replaced by some approximation such as a computational method. In some circumstances, complicated theoret ical behavior collapses to t rivial and degenerate behavior. In others, the computation m...