Analysis of a Nonlinear System Exhibiting Chaotic, Noisy Chaotic, and Random Behaviors

1996 ◽  
Vol 63 (2) ◽  
pp. 509-516 ◽  
Author(s):  
H. Lin ◽  
S. C. S. Yim
Keyword(s):  
Nonlinear System ◽  
Probability Density ◽  
Planck Equation ◽  
Stochastic Approach ◽  
External Noise ◽  
Solution Procedure ◽  
Mathematical Representation ◽  
Random Dynamics ◽  
Different Response ◽  
Noisy Chaos

This study presents a stochastic approach for the analysis of nonchaotic, chaotic, random and nonchaotic, random and chaotic, and random dynamics of a nonlinear system. The analysis utilizes a Markov process approximation, direct numerical simulations, and a generalized stochastic Melnikov process. The Fokker-Planck equation along with a path integral solution procedure are developed and implemented to illustrate the evolution of probability density functions. Numerical integration is employed to simulate the noise effects on nonlinear responses. In regard to the presence of additive ideal white noise, the generalized stochastic Melnikov process is developed to identify the boundary for noisy chaos. A mathematical representation encompassing all possible dynamical responses is provided. Numerical results indicate that noisy chaos is a possible intermediate state between deterministic and random dynamics. A global picture of the system behavior is demonstrated via the transition of probability density function over its entire evolution. It is observed that the presence of external noise has significant effects over the transition between different response states and between co-existing attractors.

Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

1988 ◽  
Vol 55 (3) ◽  
pp. 702-705 ◽  
Author(s):  
Y. K. Lin ◽  
Guoqiang Cai
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Probability Distribution ◽  
White Noise ◽  
Probability Density ◽  
Detailed Balance ◽  
Stationary Probability ◽  
Probability Flow ◽  
Parametric And External Excitations ◽  
White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

2016 ◽  
Vol 17 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Xu Sun ◽  
Xiaofan Li ◽  
Yayun Zheng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lévy Processes ◽  
Planck Equation ◽  
Levy Processes ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

PROBABILITY DENSITY EVOLUTION EQUATIONS — A HISTORICAL INVESTIGATION

2009 ◽  
Vol 03 (03) ◽  
pp. 209-226 ◽  
Author(s):  
LI JIE ◽  
CHEN JIANBING
Keyword(s):  
Evolution Equation ◽  
Probability Density ◽  
Evolution Equations ◽  
Planck Equation ◽  
Point Of View ◽  
Lagrangian Description ◽  
Density Evolution ◽  
Physical Point ◽  
Probability Density Evolution ◽  
Generalized Probability

The paper aims at clarifying the essential relationship between traditional probability density evolution equations and the generalized probability density evolution equation which is developed by the authors in recent years. Using the principle of preservation of probability as a uniform fundamental, the probability density evolution equations, including the Liouville equation, Fokker–Planck equation and the Dostupov–Pugachev equation, are derived from the physical point of view. It is pointed out that combining with Eulerian or Lagrangian description of the associated dynamical system will lead to different probability density evolution equations. Particularly, when both the principle and dynamical systems are viewed from Lagrangian description, we are led to the generalized probability density evolution equation.

A Stochastic Approach for the Control of a Nonlinear System

1995 ◽  
Vol 28 (8) ◽  
pp. 55-60
Author(s):  
L. Autrique ◽  
J.E. Souza de Cursi
Keyword(s):  
Nonlinear System ◽  
Stochastic Approach

scholarly journals On universal features of the turbulent cascade in terms of non-equilibrium thermodynamics

2018 ◽  
Vol 848 ◽  
pp. 117-153 ◽  
Author(s):  
Nico Reinke ◽  
André Fuchs ◽  
Daniel Nickelsen ◽  
Joachim Peinke
Keyword(s):  
Probability Density ◽  
Entropy Production ◽  
Turbulent Flows ◽  
Planck Equation ◽  
Small Scale ◽  
Equilibrium Thermodynamics ◽  
Fokker Planck Equation ◽  
Velocity Increments ◽  
Turbulent Cascade ◽  
Fokker Planck

Features of the turbulent cascade are investigated for various datasets from three different turbulent flows, namely free jets as well as wake flows of a regular grid and a cylinder. The analysis is focused on the question as to whether fully developed turbulent flows show universal small-scale features. Two approaches are used to answer this question. First, two-point statistics, namely structure functions of longitudinal velocity increments, and, second, joint multiscale statistics of these velocity increments are analysed. The joint multiscale characterisation encompasses the whole cascade in one joint probability density function. On the basis of the datasets, evidence of the Markov property for the turbulent cascade is shown, which corresponds to a three-point closure that reduces the joint multiscale statistics to simple conditional probability density functions (cPDFs). The cPDFs are described by the Fokker–Planck equation in scale and its Kramers–Moyal coefficients (KMCs). The KMCs are obtained by a self-consistent optimisation procedure from the measured data and result in a Fokker–Planck equation for each dataset. Knowledge of these stochastic cascade equations enables one to make use of the concepts of non-equilibrium thermodynamics and thus to determine the entropy production along individual cascade trajectories. In addition to this new concept, it is shown that the local entropy production is nearly perfectly balanced for all datasets by the integral fluctuation theorem (IFT). Thus, the validity of the IFT can be taken as a new law of the turbulent cascade and at the same time independently confirms that the physics of the turbulent cascade is a memoryless Markov process in scale. The IFT is taken as a new tool to prove the optimal functional form of the Fokker–Planck equations and subsequently to investigate the question of universality of small-scale turbulence in the datasets. The results of our analysis show that the turbulent cascade contains universal and non-universal features. We identify small-scale intermittency as a universality breaking feature. We conclude that specific turbulent flows have their own particular multiscale cascades, in other words, their own stochastic fingerprints.

A Practical Technique for Spectral Analysis of Nonlinear Systems Under Stochastic Parametric and External Excitations

1991 ◽  
Vol 113 (4) ◽  
pp. 516-522 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Spectral Response ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Matching Condition ◽  
External Noise ◽  
Equivalent Linearization ◽  
Parametric Noise ◽  
Parametric And External Excitations

A practical approach is developed for analyzing the spectral response of a nonlinear system subjected to both parametric and external Gaussian white noise excitations. The technique is implemented through the combined methods of equivalent external excitation and equivalent linearization to derive an equivalent linear system under equivalent external noise excitation. The spectral response is then obtained through utilizing the input/output spectral relation and covariance matching condition. A parametric noise excited linear system, Duffing oscillator, and nonlinear system with hysteretic nonlinearity are selected for investigation. The validity of the proposed method for analyzing spectral response is further supported by some analytical solutions and FFT technique through Monte Carlo simulations.

Random Vibration of a Nonlinear System With a Set-up Spring

1962 ◽  
Vol 29 (3) ◽  
pp. 477-482 ◽  
Author(s):  
S. H. Crandall
Keyword(s):  
Probability Density ◽  
Random Vibration ◽  
Planck Equation ◽  
Approximate Solutions ◽  
Equivalent Linearization ◽  
Zero Crossings ◽  
Mean Square Response ◽  
Spring System ◽  
Set Up ◽  
The Relationship

Several aspects of the problem of random vibration in nonlinear systems are discussed in terms of a particular set-up spring system. Exact solutions for the mean square response and the expected frequency of zero-crossings are obtained by means of the Fokker-Planck equation. These are compared with approximate solutions obtained from equivalent linearization techniques. The distribution of response peaks is then studied and a probability density is derived for the peaks on a relative frequency basis. The probability density of the response envelope is also obtained and the relationship between the two distributions is discussed.

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