Probabilistic Analysis of Bifurcations in Stochastic Nonlinear Dynamical Systems

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Ehsan Mirzakhalili ◽  
Bogdan I. Epureanu
Keyword(s):  
Dynamical Systems ◽  
Probability Density Function ◽  
Probability Density ◽  
Lorenz System ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical ◽  
Pitchfork Bifurcations ◽  
The Lorenz System

Bifurcation diagrams are limited most often to deterministic dynamical systems. However, stochastic dynamics can substantially affect the interpretation of such diagrams because the deterministic diagram often is not simply the mean of the probabilistic diagram. We present an approach based on the Fokker-Planck equation (FPE) to obtain probabilistic bifurcation diagrams for stochastic nonlinear dynamical systems. We propose a systematic approach to expand the analysis of nonlinear and linear dynamical systems from deterministic to stochastic when the states or the parameters of the system are noisy. We find stationary solutions of the FPE numerically. Then, marginal probability density function (MPDF) is used to track changes in the shape of probability distributions as well as determining the probability of finding the system at each point on the bifurcation diagram. Using MPDFs is necessary for multidimensional dynamical systems and allows direct visual comparison of deterministic bifurcation diagrams with the proposed probabilistic bifurcation diagrams. Hence, we explore how the deterministic bifurcation diagrams of different dynamical systems of different dimensions are affected by noise. For example, we show that additive noise can lead to an earlier bifurcation in one-dimensional (1D) subcritical pitchfork bifurcation. We further show that multiplicative noise can have dramatic changes such as changing 1D subcritical pitchfork bifurcations into supercritical pitchfork bifurcations or annihilating the bifurcation altogether. We demonstrate how the joint probability density function (PDF) can show the presence of limit cycles in the FitzHugh–Nagumo (FHN) neuron model or chaotic behavior in the Lorenz system. Moreover, we reveal that the Lorenz system has chaotic behavior earlier in the presence of noise. We study coupled Brusselators to show how our approach can be used to construct bifurcation diagrams for higher dimensional systems.

scholarly journals Chaos and Complexity Dynamics of Evolutionary Systems

2020 ◽  
Author(s):  
Lal Mohan Saha
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Chaotic Motion ◽  
Correlation Dimension ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Attractors ◽  
Tabular Form ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical

Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.

scholarly journals The Steady-State Response of a Class of Dynamical Systems to Stochastic Excitation

1982 ◽  
Vol 49 (3) ◽  
pp. 629-632 ◽  
Author(s):  
T. K. Caughey ◽  
F. Ma
Keyword(s):  
Steady State ◽  
Dynamical Systems ◽  
Probability Density Function ◽  
Nonlinear Dynamical Systems ◽  
Steady State Response ◽  
Stochastic Excitation ◽  
Steady State Probability ◽  
Nonlinear Dynamical ◽  
State Response ◽  
Classical Oscillator

In this paper a class of coupled nonlinear dynamical systems subjected to stochastic excitation is considered. It is shown how the exact steady-state probability density function for this class of systems can be constructed. The result is then applied to some classical oscillator problems.

Synergetics and Acoustic Emission Approach for Crazing Nonlinear Dynamical Systems

2020 ◽  
Vol 30 (03) ◽  
pp. 2050043
Author(s):  
Guodong Gao ◽  
Yongming Xing
Keyword(s):  
Acoustic Emission ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
The Self ◽  
Self Organization ◽  
Mathematical Explanation ◽  
Theoretical Derivation ◽  
Nonlinear Dynamical ◽  
Loading Mode

This paper reports that synergetics are used to analyze the crazing evolution. On this basis, chaotic effect is explored. The chaos equation is established and verified. The theoretical derivation are consistent with the experimental results. We design a special specimen with a special loading mode, the transient monitoring function of acoustic emission (AE) technology is used to track and detect the crazing inside the PMMA in real time, and the experiments show that synergetics can explain the crazing properties of polymer. Importantly, the mathematical explanation is also given. The AE analysis, synergetics, and craze photo reached a conclusion that the crazing has chaotic behavior. After analyzing the AE events and crazing at different stress levels, the accuracy of synergetic approach for crazing is verified. By studying the course of AE events and crazing, the self-organization effect is proposed. The research results will provide data support for the application of PMMA in ship, aircraft, and precision instruments.

scholarly journals Controlling nonlinear dynamical systems into arbitrary states using machine learning

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Alexander Haluszczynski ◽  
Christoph Räth
Keyword(s):  
Machine Learning ◽  
Dynamical Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Large Data ◽  
Data Sets ◽  
Initial State ◽  
Nonlinear Dynamical ◽  
Phase Space Methods

AbstractControlling nonlinear dynamical systems is a central task in many different areas of science and engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing approaches either require knowledge about the underlying system equations or large data sets as they rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on machine learning (ML), which generalizes control techniques of chaotic systems without requiring a mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities, we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states coming from any initial state. We outline and validate our approach using the examples of the Lorenz and the Rössler system and show how these systems can very accurately be brought not only to periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control scheme with little demands on the amount of required data on hand, we briefly discuss possible applications ranging from engineering to medicine.

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