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We discuss how the concept of the quantum action can be used to characterize quantum chaos. As an example we study quantum mechanics of the inverse square potential in order to test some questions related to quantum action. Quantum chaos is…
In problems where the temporal evolution of a nonlinear system cannot be followed, a method for studying the fluctuations of spatial patterns has been developed. That method is applied to well-known problems in deterministic chaos (the…
We study the exponential stability of evolutionary equations. The focus is laid on second order problems and we provide a way to rewrite them as a suitable first order evolutionary equation, for which the stability can be proved by using…
In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set it is not possible to reconstruct the invariant measure up to arbitrary fine resolution and arbitrary high…
We give a distribution-dependent concentration inequality for functions of independent variables. The result extends Bernstein's inequality from sums to more general functions, whose variation in any argument does not depend too much on the…
Previous research on nonlinear oscillator networks has shown that chaos synchronization is attainable for identical oscillators but deteriorates in the presence of parameter mismatches. Here, we identify regimes for which the opposite…
Power spectrum of the distributed chaos can be represented by a weighted superposition of the exponential functions which is converged to a stretched exponential $\propto \exp-(k/k_{\beta})^{\beta }$. An asymptotic theory has been developed…
The relationship between chaos and quantum mechanics has been somewhat uneasy -- even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our…
Exponential distribution is ubiquitous in the framework of multi-agent systems. Usually, it appears as an equilibrium state in the asymptotic time evolution of statistical systems. It has been explained from very different perspectives. In…
The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov…
We characterize the exponential distribution as the only one which satisfies a regression condition. This condition involves the regression function of a fixed record value given two other record values, one of them being previous and the…
We prove stochastic stability of chaotic maps for a general class of Markov random perturbations (including singular ones) satisfying some kind of mixing conditions. One of the consequences of this statement is the proof of Ulam's…
The missing mass refers to the probability of elements not observed in a sample, and since the work of Good and Turing during WWII, has been studied extensively in many areas including ecology, linguistic, networks and information theory.…
Classical chaos refers to the property of trajectories to diverge exponentially as time tends to infinity. It is characterized by a positive Lyapunov exponent. There are many different descriptions of quantum chaos. The one related to the…
Chaos is usually referred to the sensitivity to initial conditions in which the nonlinearity plays a crucial role. Beyond such a mathematical description, the understanding of the underlying physical origin of the chaos is still not very…
By Girsanov's thoerem and using the existing log-Harnack inequality for distribution independent SDEs, the log-Harnack inequality is derived for path-distribution dependent stochastic Hamiltonian systems. As an application, the exponential…
Chaos is a fundamental phenomenon in nonlinear dynamics, manifesting as irregular and unpredictable behavior across various physical systems. Among the diverse routes to chaos, intermittent chaos is a distinct transition pathway,…
The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a…
A quantum characteristic exponent may be defined, with the same operational meaning as the classical Lyapunov exponent when the latter is expressed as a functional of densities. Existence conditions and supporting measure properties are…
We study monotone skew-product semiflows generated by families of nonautonomous neutral functional differential equations with infinite delay and stable D-operator, when the exponential ordering is considered. Under adequate hypotheses of…