Related papers: Defining Chaos
We propose a new way of defining entropy of a system, which gives a general form which may be nonextensive as Tsallis entropy, but is linearly dependent on component entropies, like Renyi entropy, which is extensive. This entropy has a…
Catastrophes of all kinds can be roughly defined as short duration-large amplitude events following and followed by long periods of "ripening". Major earthquakes surely belong to the class of 'catastrophic' events. Because of the space-time…
Chaotic cryptography is based on the properties of chaos as source of entropy. Many different schemes have been proposed to take advantage of those properties and to design new strategies to encrypt information. However, the right and…
Complexity is an interdisciplinary concept which, first of all, addresses the question of how order emerges out of randomness. For many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related…
Chaotic iterations have been introduced on the one hand by Chazan, Mi- ranker [6] and Miellou [10] in a numerical analysis context, and on the other hand by Robert [12] and Pellegrin [11] in the discrete dynamical systems frame- work. In…
In discrete dynamical system $(X, f)$ where $X$ is a topological space and $f \in C(X,X)$, three notions of distributional chaos were defined. They were denoted by $DC1, DC2$ and $DC3$. For interval systems such three notions coincide and…
The definition of nonequilibrium entropy is provided for the general nonequilibrium processes by connecting thermodynamics with statistical physics, and the principle of entropy increment in the nonequilibrium processes is also proved in…
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…
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…
It is a fundamental problem how the universal concept of classical chaos emerges from the microscopic description of quantum mechanics. We here study standard classical chaos in a framework of quantum mechanics. In particular, we design a…
In nonequilibrium thermodynamics macroscopic entropy creation plays an important role. Here we study, from various viewpoints, its relation with the phase space contraction, which has been recently proposed as an apparently alternative…
Complexity is an important metric for appropriate characterization of different classes of irregular signals, observed in the laboratory or in nature. The literature is already rich in the description of such measures using a variety of…
We derive quantitative sufficient conditions for rotational chaos and diffusion in annular homeomorphisms, building on the topological criteria established in [31]. These conditions depend only on basic properties of the maps, making their…
The concept of emergence, or synergy in its simplest form, is widely used but lacks a rigorous definition. Our work connects information and set theory to uncover the mathematical nature of synergy as the failure of distributivity. It…
Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we…
Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize…
Morphogenesis, as it is understood in a wide sense by Ren\'e Thom, is considered for various types of chaos. That is, those, obtained by period-doubling cascade, Devaney's and Li-Yorke chaos. Moreover, in discussion form we consider…
The problem of characterizing complexity of quantum dynamics - in particular of locally interacting chains of quantum particles - will be reviewed and discussed from several different perspectives: (i) stability of motion against external…
Here we define natural chaotic systems, like the earths weather and climate system, as chaotic systems which are open to the world so have constantly changing boundary conditions, and measurements of their states are subject to errors. In…
We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of…