Related papers: Topological chaos: what may this mean ?
We propose a formalism which makes the chaos to be quantized. Quantum mechanical equation is derived for describing the chaos for a particle moving in an electromagnetic field.
One of the common characteristics of chaotic maps or flows in high dimensions is "unstable dimensional variability", in which there are periodic points whose unstable manifolds have different dimensions. In this paper, in trying to…
The chaotic hypothesis is proposed as a basis for a general theory of nonequilibrium stationary states. Version 2: new comments added after presenting this talk at the Meeting mentioned in the Acknowledgement. One typo corrected.
We study the coherent dynamics of globally coupled maps showing macroscopic chaos. With this term we indicate the hydrodynamical-like irregular behaviour of some global observables, with typical times much longer than the times related to…
The notion of Bohr chaos was introduced in [3, 4]. We answer a question raised in [3] of whether a non uniquely ergodic minimal system of positive topological entropy can be Bohr chaotic. We also prove that all systems with the…
A particular example of chaos can be conceived in the interaction of non-linear oscillator with a harmonic gravitational wave. When we replace the linear potential forces by the therm SIN(x), the type of solution becomes subject to external…
In the sciences in general, the phrase "route to chaos" has come to refer to a metaphor when some physical, biological, economic, or social system transitions from one exhibiting order to one displaying randomness (or chaos). Sometimes the…
Is it possible to define what we could mean by chaos in a space-time metric (even in the simplest toy-model studies)? Is it of importance for phenomena we may search for in Nature?
The role of global topology in the dynamics of the Universe is poorly understood. Along with observational programmes for determining the topology of the Universe, some small theoretical steps have recently been made. Heuristic…
This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multivalued inverses, graphical convergence of a net of…
Synchronization of chaos arises between coupled dynamical systems and is very well understood as a temporal phenomena which leads the coupled systems to converge or develop a dependence with time. In this work, we provide a complementary…
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 this paper, we extend the formal definition of topological surgery by introducing new notions in order to model natural phenomena exhibiting it. On the one hand, the common features of the presented natural processes are captured by our…
Using standard definitions of chaos (as positive Kolmogorov-Sinai entropy) and diffusion (that multiple time distribution functions are Gaussian), we show numerically that both chaotic and nonchaotic systems exhibit diffusion, and hence…
This work redefines the framework of chaos in dynamical systems by extending Devaney's definition to multiple mappings, emphasizing the pivotal role of nonlinearity. We propose a novel theorem demonstrating how nonlinear dynamics within a…
Some intriging connections between the properties of nonlinear noise driven systems and the nonlinear dynamics of a particular set of Hamilton's equation are discussed. A large class of Fokker-Planck Equations, like the Schr\"odinger…
Nowadays, chaos theory and nonlinear dynamics lack research focuses. Here we mention a few major open problems: 1. an effective description of chaos and turbulence, 2. rough dependence on initial data, 3. arrow of time, 4. the paradox of…
We explore the border between regular and chaotic quantum dynamics, characterized by a power law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. This…
A new phenomenon, entrainment of chaos, which is understood as a seizure of an irregular behavior by limit cycles, is discussed. As a result, chaotic cycles appear if the chaos amplitude is small. Otherwise, the chaos is not necessarily…
We consider stable periodic helixes as a generalization of stable periodic orbits. We see that in the studied class of iterated functions Chaos always arise suddenly. Therefore, we shall study the route from chaos to order rather than the…