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A quantum system is described, whose wave function has a complexity which increases exponentially with time. Namely, for any fixed orthonormal basis, the number of components required for an accurate representation of the wave function…
This paper introduces a new notion of chaotic algorithms. These algorithms are iterative and are based on so-called chaotic iterations. Contrary to all existing studies on chaotic iterations, we are not interested in stable states of such…
This study redefines the analysis of Devaney chaos in multiple mappings from a set-valued perspective and introduces new conditions to characterize their chaotic behavior. As an innovative advancement, we develop computational algorithms to…
A method to estimate Lyapunov spectra from spatio-temporal data is presented, which is well-suited to be applied to experimental situations. It allows to characterize the high-dimensional chaotic states, with possibly a large number of…
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,…
We show, using covariant Lyapunov vectors, that the chaotic solutions of spatially extended dissipative systems evolve within a manifold spanned by a finite number of physical modes hyperbolically isolated from a set of residual degrees of…
This study addresses a longstanding question regarding the mathematical proof of chaotic behavior in kinetic differential equations. Following the numerous numerical and experimental results in the past 50 years, we introduce two formal…
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…
First order optimization algorithms play a major role in large scale machine learning. A new class of methods, called adaptive algorithms, were recently introduced to adjust iteratively the learning rate for each coordinate. Despite great…
Chaos is omnipresent in nature, and its understanding provides enormous social and economic benefits. However, the unpredictability of chaotic systems is a textbook concept due to their sensitivity to initial conditions, aperiodic behavior,…
In this study linear and nonlinear higher order singularly perturbed problems are examined by a numerical approach, the differential quadrature method. Here, the main idea is using Chebyshev polynomials to acquire the weighting coefficient…
We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite- time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered…
Many research works deal with chaotic neural networks for various fields of application. Unfortunately, up to now these networks are usually claimed to be chaotic without any mathematical proof. The purpose of this paper is to establish,…
We analyze the consequences of iterative measurement-induced nonlinearity on the dynamical behavior of qubits. We present a one-qubit scheme where the equation governing the time evolution is a complex-valued nonlinear map with one complex…
Many systems in biology, physics and engineering can be described by systems of ordinary differential equation containing many parameters. When studying the dynamic behavior of these large, nonlinear systems, it is useful to identify and…
Stochastic differential equations (sdes) play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability. A general strategy for developing accurate and efficient schemes…
Chaotic systems have been investigated in several areas of engineering. In control theory, such systems have instigated the emergence of new techniques as well, have been used as a source of noise generation. The application of chaotic…
Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this…
Fractal basin boundaries provide an important means of characterizing chaotic systems. We apply these ideas to general relativity, where other properties such as Lyapunov exponents are difficult to define in an observer independent manner.…
We discuss various numerical approaches for studying the chaotic dynamics of multidimensional Hamiltonian systems, focusing our analysis on the chaotic evolution of initially localized energy excitations in the disordered Klein-Gordon…