scholarly journals Quasistatic dynamical systems

2016 ◽  
Vol 37 (8) ◽  
pp. 2556-2596 ◽  
Author(s):  
NEIL DOBBS ◽  
MIKKO STENLUND
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Time Evolution ◽  
Chaotic Maps ◽  
Martingale Problem ◽  
Initial Distribution ◽  
Initial State ◽  
Statistical Behavior ◽  
Long Time ◽  
Well Posed

We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. Time evolution of states under a quasistatic dynamical system is entirely deterministic, but choosing the initial state randomly renders the process a stochastic one. In the prototypical setting where the time evolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behavior as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a well-posed martingale problem. We also consider various admissible ways of centering the process, with the curious conclusion that the ‘obvious’ centering suggested by the initial distribution sometimes fails to yield the expected diffusion.

scholarly journals Intermittent quasistatic dynamical systems: weak convergence of fluctuations

2018 ◽  
Vol 5 (1) ◽  
pp. 8-34 ◽  
Author(s):  
Juho Leppänen
Keyword(s):  
Dynamical Systems ◽  
Weak Convergence ◽  
Time Evolution ◽  
Martingale Problem ◽  
Statistical Properties ◽  
Time Dependent ◽  
Stochastic Diffusion ◽  
Interval Maps ◽  
Well Posed ◽  
Over Time

Abstract This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due to external influences. We focus on the case where the time-evolution is described by intermittent interval maps (Pomeau-Manneville maps) with time-dependent parameters. In a suitable range of parameters, we obtain a description of the statistical properties as a stochastic diffusion, by solving a well-posed martingale problem. The results extend those of a related recent study due to Dobbs and Stenlund, which concerned the case of quasistatic (uniformly) expanding systems.

CHAOTIC AND QUASIPERIODIC MOTIONS OF THREE PLANAR CHARGED PARTICLES

2001 ◽  
Vol 11 (07) ◽  
pp. 1937-1951
Author(s):  
SHU-MING CHANG ◽  
WEN-WEI LIN ◽  
TAI-CHIA LIN
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Charged Particles ◽  
Time Dynamics ◽  
Long Time ◽  
Long Time Dynamics

We study two dynamical systems for the motion of three planar charged particles with charges nj ∈ {±1}, j = 1, 2, 3. Both dynamical systems are parametric with a parameter α ∈ [0, 1] and have the same nonlinear terms. As α = 0, 1, the dynamical systems have no chaos. However, one dynamical system may create chaos as α varies from zero to one. This may provide an example to show that the homotopy deformation of dynamical systems cannot preserve the long-time dynamics even though the dynamical systems have the same nonlinear terms.

INTEGRAL TREATMENT FOR TIME EVOLUTION: THE GENERAL INTERACTIVITY

1999 ◽  
Vol 14 (20) ◽  
pp. 3239-3252
Author(s):  
E. CASUSO
Keyword(s):  
Dynamical Systems ◽  
Time Evolution ◽  
Time Distribution ◽  
Chaotic Behavior ◽  
Order Approximation ◽  
Natural Systems ◽  
Structural Discontinuity ◽  
Long Time ◽  
Integral Treatment ◽  
First Order Approximation

Assuming that the unpredictability associated with many dynamical systems is an artefact of the differential treatment of their time evolution, we propose here an integral treatment as an alternative. We make the assumption that time is two-dimensional, and that the time distribution in the past of observables characterizing the dynamical system, is some characteristic "projection" of its time distribution in the future. We show here how this method can be used to predict the time evolution of several dynamically complex systems over long time intervals. The present work can be considered as the natural next step to the assumption of nonderivability for subatomic dynamical systems to explain the connection between Quantum Mechanics and General Relativity. Here we propose that matter and space–time are not only nonderivable but also show structural discontinuity. Starting with this premise we use continuity and derivability, but only as a first order approximation to reality. Extrapolation to very large or very small scales, or to predictions over long time scales for many natural systems on intermediate scales (human scales), may lead to chaotic behavior, or to nondeterministic or probabilistic theories.

scholarly journals Sensitivity Analysis and Stabilization for Two Dynamical Systems

2018 ◽  
pp. 33-40
Author(s):  
Frank Etin-Osa Bazuaye
Keyword(s):  
Dynamical System ◽  
Sensitivity Analysis ◽  
Dynamical Systems ◽  
Political Parties ◽  
Initial Data ◽  
Differential Equations ◽  
Democratic Process ◽  
Initial State ◽  
Mathematical Problems ◽  
The Impact

This paper focuses on the sensitivity analysis for two dominant political parties. In contrast to Misra, Bazuaye and Khan, who developed the model without investigating the impact of varying the initial state of political parties on the solution trajectory of each political parties, we have developed a sound numerical algorithm to analyze the impact of change on the initial data on the behavior of the democratic process which is a rare contribution to knowledge. Two Matlab standard solvers for ordinary differential equations, ode45 and ode23, have been utilized to handle these formidable mathematical problems. Our findings indicate that as the initial data varies, the dynamical system describing the interaction between two political parties is stabilized over a period of eight years. As duration increases, the systems get de-stabilized.

scholarly journals ESTIMATION OF INITIAL CONDITIONS AND SECURE COMMUNICATION

2003 ◽  
Vol 13 (10) ◽  
pp. 3079-3084 ◽  
Author(s):  
ANIL MAYBHATE ◽  
R. E. AMRITKAR ◽  
D. R. KULKARNI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Time Evolution ◽  
Secure Communication ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Important Application ◽  
Secure Communications ◽  
Newton Raphson ◽  
Raphson Method

We estimate the initial conditions of a multivariable dynamical system from a scalar signal, using a modified Newton–Raphson method incorporating the time evolution. We can estimate initial conditions of periodic and chaotic systems and the required length of scalar signal is very small. An important application of the method is in secure communications. The communication procedure has several advantages as compared to others using dynamical systems.

Definitions—Background

2019 ◽  
pp. 43-63
Author(s):  
Pierre-Loïc Garoche
Keyword(s):  
Dynamical System ◽  
Signal Processing ◽  
Dynamical Systems ◽  
Convex Optimization ◽  
Physical System ◽  
Discrete Dynamical Systems ◽  
Target Language ◽  
Initial State ◽  
Typical Object ◽  
Tools And Methods

This chapter presents the formalisms describing discrete dynamical systems and gives an overview on the convex optimization tools and methods used to compute the analyses. A dynamical system is a typical object used in control systems or in signal processing. In some cases, it is eventually implemented in a program to perform the desired feedback control to a cyber-physical system. Language-wise, model-based languages such as LUSTRE, ANSYS SCADE, or MATLAB Simulink provide primitives to build these dynamical systems or controllers relying on simpler constructs. In terms of programs, such dynamical systems can easily be implemented as a “while true loop” initialized by the initial state and performing the update f. The simplest systems are usually directly coded in the target language, while more advanced systems are compiled through autocoders.

scholarly journals Time evolution of the anisotropies of the hydrodynamically expanding SQGP

2016 ◽  
Vol 31 (28n29) ◽  
pp. 1645016
Author(s):  
A. Bagoly ◽  
M. Csanád
Keyword(s):  
Time Evolution ◽  
Heavy Ion ◽  
Gluon Plasma ◽  
High Energy ◽  
Initial Distribution ◽  
Hadronic Matter ◽  
Initial State ◽  
Ion Collisions ◽  
Hydrodynamic Evolution ◽  
Plane Anisotropy

In high energy heavy ion collisions of RHIC and LHC, a strongly interacting quark gluon plasma (sQGP) is created. This medium undergoes a hydrodynamic evolution, before it freezes out to form a hadronic matter. The initial state of the sQGP is determined by the initial distribution of the participating nucleons and their interactions. Due to the finite number of nucleons, the initial distribution fluctuates on an event-by-event basis. The transverse plane anisotropy of the initial state can be translated into a series of anisotropy coefficients or eccentricities: second, third, fourth-order anisotropy etc. These anisotropies then evolve in time, and result in measurable momentum-space anisotropies, to be measured with respect to their respective symmetry planes. In this paper we investigate the time evolution of the anisotropies. With a numerical hydrodynamic code, we analyze how the speed of sound and viscosity influence this evolution.

scholarly journals A Constructive Method for Computing Generalized Manley-Rowe Constants of Motion

2013 ◽  
Vol 14 (4) ◽  
pp. 1094-1102
Author(s):  
Elena Kartashova ◽  
Loredana Tec
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conservation Laws ◽  
Time Evolution ◽  
Numerical Study ◽  
Constructive Method ◽  
Constants Of Motion

AbstractThe Manley-Rowe constants of motion (MRC) are conservation laws written out for a dynamical system describing the time evolution of the amplitudes in resonant triad. In this paper we extend the concept of MRC to resonance clusters of any form yielding generalized Manley-Rowe constants (gMRC) and give a constructive method how to compute them. We also give details of a Mathematica implementation of this method. While MRC provide integrability of the underlying dynamical system, gMRC generally do not but may be used for qualitative and numerical study of dynamical systems describing generic resonance clusters.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

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