scholarly journals Lyapunov-Schmidt reduction and singularity analysis of a high-dimensional relative-rotation nonlinear dynamical system

2012 ◽  
Vol 61 (19) ◽  
pp. 194501
Author(s):  
Shi Pei-Ming ◽  
Han Dong-Ying ◽  
Li Ji-Zhao ◽  
Jiang Jin-Shui ◽  
Liu Bin
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical System ◽  
Singularity Analysis ◽  
High Dimensional ◽  
Relative Rotation ◽  
Nonlinear Dynamical

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.

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.

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