THE CORRESPONDENCE BETWEEN STOCHASTIC RESONANCE AND BIFURCATION OF MOMENT EQUATIONS OF NOISY NONLINEAR DYNAMICAL SYSTEM

2007 ◽  
Vol 17 (11) ◽  
pp. 4081-4098 ◽  
Author(s):  
GUANG-JUN ZHANG ◽  
JIAN-XUE XU
Keyword(s):  
Fixed Point ◽  
Stochastic Resonance ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical System ◽  
Original System ◽  
Bistable System ◽  
Moment Equations ◽  
Nonlinear Dynamical ◽  
Independent Variable ◽  
The Moment

There is a kind of correspondence between stochastic resonance and bifurcation of the moment equations of a noisy nonlinear system with the same noise intensity as the resonance independent variable and the bifurcation parameter, respectively. In this paper, this correspondence is examined and revealed in the noisy one-dimensional bistable system and the noisy two-dimensional Duffing oscillator. The bifurcation of the moment equations of each noisy system is the bifurcation with double-branch of fixed-point shift. Besides classical stochastic resonance, a kind of complex stochastic resonance corresponds to the bifurcation of moment equations. This complex stochastic resonance is induced by the stochastic transitions of system motion among the three fixed point attractors on both sides of the bifurcation point of the original system, which is predicted semi-analytically. Finally, due to this correspondence being examined, the mechanism of stochastic resonance can be provided through analyzing the change of the energy transfer induced by the bifurcation of the moment equations.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

2020 ◽  
Vol 22 (4) ◽  
pp. 983-990
Author(s):  
Konrad Mnich
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Nonlinear Dynamical System ◽  
Tuned Mass Damper ◽  
Nonlinear Dynamical ◽  
Duffing System ◽  
Mass Damper ◽  
Basin Stability ◽  
Stability Concept

AbstractIn this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

ON THE MATHEMATICAL CLARIFICATION OF THE SNAP-BACK-REPELLER IN HIGH-DIMENSIONAL SYSTEMS AND CHAOS IN A DISCRETE NEURAL NETWORK MODEL

2002 ◽  
Vol 12 (05) ◽  
pp. 1129-1139 ◽  
Author(s):  
WEI LIN ◽  
JIONG RUAN ◽  
WEIRUI ZHAO
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Fixed Point ◽  
Numerical Simulations ◽  
Neural Network Model ◽  
Jacobian Matrix ◽  
Nonlinear Dynamical System ◽  
High Dimensional ◽  
Nonlinear Dynamical

We investigate the differences among several definitions of the snap-back-repeller, which is always regarded as an inducement to produce chaos in nonlinear dynamical system. By analyzing the norms in different senses and the illustrative examples, we clarify why a snap-back-repeller in the neighborhood of the fixed point, where all eigenvalues of the corresponding variable Jacobian Matrix are absolutely larger than 1 in norm, might not imply chaos. Furthermore, we theoretically prove the existence of chaos in a discrete neural networks model in the sense of Marotto with some parameters of the systems entering some regions. And the following numerical simulations and corresponding calculation, as concrete examples, reinforce our theoretical proof.

scholarly journals Krasnoselskii N-Tupled Fixed Point Theorem with Applications to Fractional Nonlinear Dynamical System

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Tamer Nabil
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Evolution Equations ◽  
Nonlinear Dynamical System ◽  
Nonlinear Dynamical ◽  
Fractional Dynamical System ◽  
Theoretical Results

In this paper, N-tupled fixed point theorems for two monotone nondecreasing mappings in complete normed linear space are established. The extension of Krasnoseskii fixed point theorem for a version of N-tupled fixed point is given. Our theoretical results are applied to prove the existence of a mild solution of the system of N-nonlinear fractional evolution equations. Finally, an example of a nonlinear fractional dynamical system is given to illustrate the results.

OPTIMAL TRANSITION RATE AND STOCHASTIC RESONANCE IN A BISTABLE SYSTEM DRIVEN BY POWER-LAW NOISE

2003 ◽  
Vol 14 (03) ◽  
pp. 303-310 ◽  
Author(s):  
J. F. L. FREITAS ◽  
M. L. LYRA
Keyword(s):  
Stochastic Resonance ◽  
Power Law ◽  
Transition Rate ◽  
Noise Intensity ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Resonance Phenomenon ◽  
Periodic Signal ◽  
Nonlinear Dynamical ◽  
Decay Exponent

In this work, we study the stochastic resonance phenomenon in a bistable nonlinear dynamical system in the presence of an uncorrelated noise source whose distribution decays asymptotically as P(ξ) ∝ 1/ξ2α. We investigate the influence of the decay exponent α on the transition rate and on the optimal noise intensity giving the maximum signal-to-noise ratio when a weak periodic signal is superposed to the external noise. We find that the transition rate achieves a maximum for a finite decay exponent α. However, the optimal noise intensity for stochastic resonance depicts a monotonic power-law correction relative to the usual behavior of nonlinear dynamical systems driven by Gaussian noises.

On Global Transversality and Tangency in a Periodically Forced, Damped Duffing Oscillator

2006 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical System ◽  
Melnikov Method ◽  
Nonlinear Dynamical ◽  
Chaotic Flows ◽  
Numerical Predictions ◽  
Small Parameters

In this paper, the analytical conditions for global transversality and tangency of a 2-D nonlinear dynamical system are derived. Further, a periodically forced, damped Duffing oscillator with a separatrix is investigated as an application example. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and the global and tangential flows are illustrated to verify the analytical conditions. The illustrations show that the analytical and numerical predictions of the global transversality and tangency exactly are agreed very well. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters. Therefore, chaotic flows based on the global transversality will be further investigated, and the corresponding results will be presented in a sequel.

Transient Behaviors in Noise-Induced Bifurcations Captured by Generalized Cell Mapping Method with an Evolving Probabilistic Vector

2015 ◽  
Vol 25 (08) ◽  
pp. 1550109 ◽  
Author(s):  
Zigang Li ◽  
Jun Jiang ◽  
Ling Hong
Keyword(s):  
Probability Distribution ◽  
Multiplicative Noise ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical System ◽  
Response Probability ◽  
Mapping Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Insight Into

In this paper, an idea of evolving probabilistic vector (EPV) is introduced into the Generalized Cell Mapping (GCM) method to replace the classical fix-sized probabilistic vector in order to efficiently capture the transient behaviors in noise-induced bifurcations, by which an initial localized probability distribution around a deterministic attracting set of a nonlinear dynamical system may expand abruptly or escape with a jump as the noise intensity increases and exceeds some critical values. A Mathieu–Duffing oscillator under excitation of both additive and multiplicative noise is studied as an example of application to show the validity of the proposed method and the interesting phenomena in noise-induced explosive and dangerous bifurcations of the oscillator that are characterized respectively by an abrupt enlargement and a sudden fast jump of the response probability distribution. The insight into the roles of deterministic global structure and noise as well as their interplay is gained.

Approximation of the Frequency-Energy Dependence in the Nonlinear Dynamical Systems

2016 ◽  
Author(s):  
Mohammad A. Al-Shudeifat
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical System ◽  
Nonlinear Energy Sink ◽  
High Energy ◽  
Initial Energy ◽  
Nonlinear Dynamical ◽  
High Energy Level

In this work, a method is introduced for extracting the approximate backbone branches of the frequency-energy plot from the numerical simulation response of the nonlinear dynamical system. The duffing oscillator is firstly considered to describe the method and later a linear oscillator (LO) coupled with a nonlinear energy sink (NES) is also considered for further demonstration. The systems of concern are numerically simulated at an arbitrary high level of initial input energy. Accordingly, the obtained responses of these systems are employed via the proposed method to extract an approximation for the fundamental backbone branches of the frequency-energy plot. The obtained backbones have been found in excellent agreement with the exact backbones of the considered systems. Even though these approximate backbones have been obtained for only one high energy level, they are still valid for any other initial energy below that level. In addition, they are not affected by the damping variations in the considered systems. Unlike other existing methods, the proposed approach is applicable to well-approximate the backbone branches of the large-scale nonlinear dynamical systems.

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