Statistical properties for compositions of standard maps with increasing coefficient

2020 ◽  
pp. 1-44
Author(s):  
ALEX BLUMENTHAL
Keyword(s):  
Phase Space ◽  
Chaotic Behavior ◽  
Positive Lebesgue Measure ◽  
Large Set ◽  
Rigorous Analysis ◽  
Strong Law ◽  
Standard Map ◽  
Positive Volume ◽  
Open Question ◽  
Volume Preserving

The Chirikov standard map is a prototypical example of a one-parameter family of volume-preserving maps for which one anticipates chaotic behavior on a non-negligible (positive-volume) subset of phase space for a large set of parameters. Rigorous analysis is notoriously difficult and it remains an open question whether this chaotic region, the stochastic sea, has positive Lebesgue measure for any parameter value. Here we study a problem of intermediate difficulty: compositions of standard maps with increasing coefficient. When the coefficients increase to infinity at a sufficiently fast polynomial rate, we obtain a strong law, a central limit theorem, and quantitative mixing estimates for Holder observables. The methods used are not specific to the standard map and apply to a class of compositions of ‘prototypical’ two-dimensional maps with hyperbolicity on ‘most’ of phase space.

The Tangled Tale of Phase Space

2018 ◽  
Author(s):  
David D. Nolte
Keyword(s):  
Phase Space ◽  
Chaotic Behavior ◽  
Three Body Problem ◽  
Henri Poincaré ◽  
History Of ◽  
The Stability ◽  
Three Body ◽  
Space Phase ◽  
Central Paradigm ◽  
System Phase

This chapter presents the history of the development of the concept of phase space. Phase space is the central visualization tool used today to study complex systems. The chapter describes the origins of phase space with the work of Joseph Liouville and Carl Jacobi that was later refined by Ludwig Boltzmann and Rudolf Clausius in their attempts to define and explain the subtle concept of entropy. The turning point in the history of phase space was when Henri Poincaré used phase space to solve the three-body problem, uncovering chaotic behavior in his quest to answer questions on the stability of the solar system. Phase space was established as the central paradigm of statistical mechanics by JW Gibbs and Paul Ehrenfest.

Analysis of the Dynamic Characteristics of the Firefly Algorithm

2020 ◽  
pp. 100-115
Author(s):  
Takuya Shindo
Keyword(s):  
Heuristic Algorithm ◽  
Dynamic Characteristics ◽  
Firefly Algorithm ◽  
Chaotic Behavior ◽  
Fundamental Principle ◽  
Deterministic System ◽  
Rigorous Analysis ◽  
The Individual ◽  
Searching Ability ◽  
The Relationship

The firefly algorithm is a meta-heuristic algorithm, the fundamental principle of which mimics the characteristics associated with the blinking of natural fireflies. This chapter presents a rigorous analysis of the dynamics of the firefly algorithm, which the authors performed by applying a deterministic system that removes the stochastic factors from the state update equation. Depending on its parameters, the individual deterministic firefly algorithm exhibits chaotic behavior. This prompted us to investigate the relationship between the behavior of the algorithm and its parameters as well as the extent to which the chaotic behavior influences the searching ability of the algorithm.

From Dispersion to Laminarity in Dynamical Systems

1994 ◽  
Vol 49 (12) ◽  
pp. 1241-1247 ◽  
Author(s):  
G. Zumofen ◽  
J. Klafter
Keyword(s):  
Random Walk ◽  
Dynamical Systems ◽  
Continuous Time ◽  
Chaotic Behavior ◽  
Continuous Time Random Walk ◽  
Standard Map ◽  
One Dimensional ◽  
Enhanced Diffusion

Abstract We study transport in dynamical systems characterized by intermittent chaotic behavior with coexistence of dispersive motion due to periods of localization, and of enhanced diffusion due to periods of laminar motion. This transport is discussed within the continuous-time random walk approach which applies to both dispersive and enhanced motions. We analyze the coexistence for the standard map and for a one-dimensional map.

scholarly journals When is a dynamical system mean sensitive?

2017 ◽  
Vol 39 (06) ◽  
pp. 1608-1636 ◽  
Author(s):  
FELIPE GARCÍA-RAMOS ◽  
JIE LI ◽  
RUIFENG ZHANG
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
The Other ◽  
Positive Entropy ◽  
Topological Dynamical System ◽  
Other Hand ◽  
Open Question ◽  
Uniquely Ergodic ◽  
Mean Sensitivity ◽  
Transitive System

This article is devoted to studying which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things, we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand, we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.

Stochasticity in the Josephson map

1996 ◽  
Vol 56 (3) ◽  
pp. 493-506 ◽  
Author(s):  
Y. Nomura ◽  
Y. H. Ichikawa ◽  
A. T. Filipov
Keyword(s):  
Nonlinear Dynamics ◽  
Diffusion Coefficient ◽  
Phase Space ◽  
Characteristic Function ◽  
Numerical Investigation ◽  
Stochastic Diffusion ◽  
Standard Map ◽  
External Bias ◽  
Stochastic Parameter ◽  
Phase Space Portrait

The Josephson map describes the nonlinear dynamics of systems characterized by the standard map with a uniform external bias superposed. The intricate structures of the phase-space portrait of the Josephson map are examined here on the basis of the associated tangent map. A numerical investigation of stochastic diffusion in the Josephson map is compared with the renormalized diffusion coefficient calculated using the characteristic function. The global stochasticity of the Josephson map occurs at far smaller values of the stochastic parameter than is the case of the standard map.

THE DYNAMICS OF SYSTEMS OF COMPLEX NONLINEAR OSCILLATORS: A REVIEW

2004 ◽  
Vol 14 (11) ◽  
pp. 3821-3846 ◽  
Author(s):  
GAMAL M. MAHMOUD ◽  
TASSOS BOUNTIS
Keyword(s):  
Phase Space ◽  
Periodic Solutions ◽  
Chaotic Behavior ◽  
Nonlinear Oscillators ◽  
Complex Variables ◽  
Point Of View ◽  
Stability And Control ◽  
Ginzburg Landau Equations ◽  
The Stability ◽  
And Control

Dynamical systems in the real domain are currently one of the most popular areas of scientific study. A wealth of new phenomena of bifurcations and chaos has been discovered concerning the dynamics of nonlinear systems in real phase space. There is, however, a wide variety of physical problems, which, from a mathematical point of view, can be more conveniently studied using complex variables. The main advantage of introducing complex variables is the reduction of phase space dimensions by a half. In this survey, we shall focus on such classes of autonomous, parametrically excited and modulated systems of complex nonlinear oscillators. We first describe appropriate perturbation approaches, which have been specially adapted to study periodic solutions, their stability and control. The stability analysis of these fundamental periodic solutions, though local by itself, can yield considerable information about more global properties of the dynamics, since it is in the vicinity of such solutions that the largest regions of regular or chaotic motion are observed, depending on whether the periodic solution is, respectively, stable or unstable. We then summarize some recent studies on fixed points, periodic solutions, strange attractors, chaotic behavior and the problem of chaos control in systems of complex oscillators. Some important applications in physics, mechanics and engineering are mentioned. The connection with a class of complex partial differential equations, which contains such famous examples, as the nonlinear Schrödinger and Ginzburg–Landau equations is also discussed. These complex equations play an important role in many branches of physics, e.g. fluids, superconductors, plasma physics, geophysical fluids, modulated optical waves and electromagnetic fields.

The phase-space hydrodynamic model for the quantum standard map

1991 ◽  
Vol 63 (1-3) ◽  
pp. 279-305 ◽  
Author(s):  
Alejandro Spina ◽  
Rex.T. Skodje
Keyword(s):  
Phase Space ◽  
Hydrodynamic Model ◽  
Standard Map

scholarly journals The Spectral Underpinning of word2vec

2020 ◽  
Vol 6 ◽  
Author(s):  
Ariel Jaffe ◽  
Yuval Kluger ◽  
Ofir Lindenbaum ◽  
Jonathan Patsenker ◽  
Erez Peterfreund ◽  
...  
Keyword(s):  
Natural Language Processing ◽  
Spectral Method ◽  
Language Processing ◽  
Rigorous Analysis ◽  
Theoretical Justification ◽  
Nonlinear Functional ◽  
Nonlinear Properties ◽  
Frequent Use ◽  
Highly Nonlinear ◽  
Open Question

Word2vec introduced by Mikolov et al. is a word embedding method that is widely used in natural language processing. Despite its success and frequent use, a strong theoretical justification is still lacking. The main contribution of our paper is to propose a rigorous analysis of the highly nonlinear functional of word2vec. Our results suggest that word2vec may be primarily driven by an underlying spectral method. This insight may open the door to obtaining provable guarantees for word2vec. We support these findings by numerical simulations. One fascinating open question is whether the nonlinear properties of word2vec that are not captured by the spectral method are beneficial and, if so, by what mechanism.

scholarly journals Time Recurrence Analysis of a Near Singular Billiard

2019 ◽  
Vol 24 (2) ◽  
pp. 50 ◽  
Author(s):  
Rodrigo Simile Baroni ◽  
Ricardo Egydio de Carvalho ◽  
Bruno Castaldi ◽  
Bruno Furlanetto
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Singular Limit ◽  
Largest Lyapunov Exponent ◽  
Classical Dynamics ◽  
Sharp Drop ◽  
Nonlinear Part ◽  
Return Times

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.

Plankton Models and Its Attractors in a Local Approximation

2019 ◽  
pp. 59-90
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Chaotic Behavior ◽  
Lyapunov Stability Theory ◽  
Local Approximation ◽  
Natural Mortality ◽  
Environmental Characteristic ◽  
Systems Of Differential Equations ◽  
Interacting Populations ◽  
Plankton Models

The previously accepted models of plankton consisting of two interacting populations—phytoplankton and zooplankton—are considered in a local approximation. The analysis of models is carried out with the help of a qualitative study of systems of differential equations as a whole (i.e., in the entire phase space of systems, not limited to a neighborhood of equilibrium positions). Analytical conditions for the occurrence of a Hopf bifurcation are obtained for each model using the Lyapunov stability theory. A comparison of various models is given, and their shortcomings associated with the incompleteness of research are indicated. It has been established that in some cases the loss of stability of the equilibrium position does not lead to the formation of a limit cycle (Hopf bifurcation) but to the formation of a limit continuum with a chaotic behavior of the trajectories in a large part of the phase space. It is shown that the parameters significantly influencing the dynamics of the development of plankton are the natural mortality of populations as an environmental characteristic of the environment.

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