Oscillation Suppression in a Delay-Coupled Flexible-Joint System

2013 ◽  
Vol 477-478 ◽  
pp. 123-127
Author(s):  
Shan Ying Jiang
Keyword(s):  
Time Delay ◽  
Periodic Oscillation ◽  
Slow Manifold ◽  
Fast Subsystem ◽  
Singular Perturbation Method ◽  
Joint System ◽  
Flexible Joint ◽  
The Stability ◽  
Slow Subsystem ◽  
Geometric Singular Perturbation Method

A delay-coupled flexible-joint system is investigated in this paper. Because of the different time scales, the flexible-joint system could be transformed into a fast subsystem and a slow subsystem. The geometric singular perturbation method is used to obtain the slow manifold defining as the equilibrium of the fast subsystem. The eigenvalue analysis of the fast subsystem reveals a relation between the stability of the slow manifold and the time delay. The analysis results provide an idea of suppressing the small amplitude periodic oscillation via adjusting the time delay. Numerical simulations are performed to display the effectiveness of this method.

Delay-Induced Bogdanov–Takens Bifurcation and Dynamical Classifications in a Slow-Fast Flexible Joint System

2015 ◽  
Vol 25 (09) ◽  
pp. 1550121 ◽  
Author(s):  
Jian Xu ◽  
Shanying Jiang
Keyword(s):  
Time Delay ◽  
Homoclinic Orbits ◽  
Singular Perturbation Method ◽  
Joint System ◽  
Fast System ◽  
Flexible Joint ◽  
Melnikov Theory ◽  
The Stability ◽  
Delay Differential ◽  
Geometric Singular Perturbation Method

A slow-fast delay-coupled flexible joint system is investigated in this paper. To understand the effects of time delay on the stability and oscillation of the manipulator, the geometric singular perturbation method is extended in dealing with delay differential equations. Bogdanov–Takens (BT) bifurcation of the fast subsystem is obtained, which leads to the existence of homoclinic orbits and is proved to be related to the formation of spiking. After the break of homoclinic orbits, Melnikov theory is introduced to predict the threshold curve indicating the occurrence of chaos. Numerical results show that with the increase of time delay, the stability of the system gets worse, and complicated oscillations including bursting, chaotic-bursting and complete chaos turn up. Besides, it is briefly summarized that the effect of the small parameter in the slow-fast system is to influence the convergence rate of solution trajectories, which is widely neglected in previous works.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Penghe Ge ◽  
Hongjun Cao
Keyword(s):  
Neuron Model ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Fast Subsystem ◽  
One Dimensional ◽  
Sensitive Dependence ◽  
The Stability ◽  
Slow Subsystem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

Homoclinic Orbits of a Buckled Beam Subjected to Transverse Uniform Harmonic Excitation

2019 ◽  
Vol 29 (05) ◽  
pp. 1950061
Author(s):  
D. M. Zhang ◽  
Y. L. Jin ◽  
F. Li
Keyword(s):  
Melnikov Method ◽  
Homoclinic Orbits ◽  
Harmonic Excitation ◽  
Slow Manifold ◽  
Singular Perturbation Method ◽  
Smale Horseshoe ◽  
Buckled Beam ◽  
Resonance Band ◽  
The One ◽  
Geometric Singular Perturbation Method

Homoclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation are investigated in the case of 1:1 internal resonance. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect the equilibria in a resonance band of the slow manifold. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The results obtained imply the existence of the amplitude modulated chaos for the Smale horseshoe sense in the class of buckled beam systems.

On the stability of a class of nonlinear time delay systems

1999 ◽  
Author(s):  
Dan Ivancscu ◽  
Silviu-Iulian Niculcscu ◽  
Jcan-Michcl Dion ◽  
Luc Dugard
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Time Delay Systems ◽  
The Stability

Global stability analysis of fractional-order gene regulatory networks with time delay

2019 ◽  
Vol 12 (06) ◽  
pp. 1950067 ◽  
Author(s):  
Zhaohua Wu ◽  
Zhiming Wang ◽  
Tiejun Zhou
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Gene Regulatory Networks ◽  
Existence And Uniqueness ◽  
Regulatory Networks ◽  
Lyapunov Method ◽  
Regulation Mechanism ◽  
Global Stability Analysis ◽  
Gene Regulatory ◽  
The Stability

Fractional-order gene regulatory networks with time delay (DFGRNs) have proven that they are more suitable to model gene regulation mechanism than integer-order. In this paper, a novel DFGRN is proposed. The existence and uniqueness of the equilibrium point for the DFGRN are proved under certain conditions. On this basis, the conditions on the global asymptotic stability are established by using the Lyapunov method and comparison theorem for the DFGRN, and the stability conditions are dependent on the fractional-order [Formula: see text]. Finally, numerical simulations show that the obtained results are reasonable.

scholarly journals Global Exponential Robust Stability of Static Interval Neural Networks with Time Delay in the Leakage Term

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Guiying Chen ◽  
Linshan Wang
Keyword(s):  
Neural Networks ◽  
Time Delay ◽  
Robust Stability ◽  
Sufficient Conditions ◽  
Leakage Term ◽  
Interval Neural Networks ◽  
Global Exponential Robust Stability ◽  
The Stability ◽  
Delay Differential ◽  
Delay Differential Inequality

The stability of a class of static interval neural networks with time delay in the leakage term is investigated. By using the method ofM-matrix and the technique of delay differential inequality, we obtain some sufficient conditions ensuring the global exponential robust stability of the networks. The results in this paper extend the corresponding conclusions without leakage delay. An example is given to illustrate the effectiveness of the obtained results.

Chaos Control for a Fractional-Order Jerk System via Time Delay Feedback Controller and Mixed Controller

2021 ◽  
Vol 5 (4) ◽  
pp. 257
Author(s):  
Changjin Xu ◽  
Maoxin Liao ◽  
Peiluan Li ◽  
Lingyun Yao ◽  
Qiwen Qin ◽  
...  
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Feedback Controller ◽  
Research Approach ◽  
Controlling Chaos ◽  
The Stability ◽  
Jerk System

In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PDσ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system.

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