scholarly journals Rank One Strange Attractors in Periodically Kicked Lorenz System with Time-Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Wenjie Yang ◽  
Yiping Lin ◽  
Yunxian Dai
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Strange Attractor ◽  
Lorenz System ◽  
Strange Attractors ◽  
Delayed Systems ◽  
Rank One ◽  
Delayed System ◽  
System With Time Delay ◽  
Chaotic Theory

Rank one strange attractor in periodically kicked Lorenz system with time-delay is investigated. Our discussion is based on the theory of rank one maps formulated by Wang and Young. First, we develop the rank one chaotic theory to delayed systems. It is shown that strange attractors occur when periodically kicked delayed system undergoes a generic Hopf bifurcation. Then we use the theory to the periodically kicked Lorenz system with delay, and derivation of conditions for Hopf bifurcation and rank one chaos along with the results of numerical simulations are presented.

Rank One Strange Attractors in Periodically Kicked Predator–Prey System with Time-Delay

2016 ◽  
Vol 26 (07) ◽  
pp. 1640114 ◽  
Author(s):  
Wenjie Yang ◽  
Yiping Lin ◽  
Yunxian Dai ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Strange Attractors ◽  
Periodic Force ◽  
Delayed Systems ◽  
Predator Prey ◽  
Prey System ◽  
Rank One ◽  
Delayed System ◽  
System With Time Delay

This paper is devoted to the study of the problem of rank one strange attractor in a periodically kicked predator–prey system with time-delay. Our discussion is based on the theory of rank one maps formulated by Wang and Young. Firstly, we develop the rank one chaotic theory to delayed systems. It is shown that strange attractors occur when the delayed system undergoes a Hopf bifurcation and encounters an external periodic force. Then we use the theory to the periodically kicked predator–prey system with delay, deriving the conditions for Hopf bifurcation and rank one chaos along with the results of numerical simulations.

Rank-One Chaos in a Periodically Kicked Three-Species Food Chain with Time-Delay

2020 ◽  
Vol 30 (03) ◽  
pp. 2050038
Author(s):  
Ping Yang ◽  
Juan Fang ◽  
Yunxian Dai ◽  
Yiping Lin
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Food Chain ◽  
Strange Attractor ◽  
Positive Equilibrium ◽  
Rank One ◽  
Delayed System

This paper is devoted to studying the problem of rank-one strange attractor in a three-species food chain with time-delay. The conditions for the existence of positive equilibrium and Hopf bifurcation are presented. By using the theory of rank-one maps formulated by Wang and Young in 2001, and then developed by us to the time-delayed system, the conditions for the system having rank-one strange attractor are obtained under periodically kicked system. Numerical simulations are presented to demonstrate the analytic results.

scholarly journals Double Hopf Bifurcation with Strong Resonances in Delayed Systems with Nonlinerities

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
J. Xu ◽  
K. W. Chung
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Nonlinear Systems ◽  
Efficient Method ◽  
Cubic Nonlinearity ◽  
Van Der Pol ◽  
Delayed Systems ◽  
Double Hopf Bifurcation ◽  
Codimension Two ◽  
System With Time Delay

An efficient method is proposed to study delay-induced strong resonant double Hopf bifurcation for nonlinear systems with time delay. As an illustration, the proposed method is employed to investigate the 1 : 2 double Hopf bifurcation in the van der Pol system with time delay. Dynamics arising from the bifurcation are classified qualitatively and expressed approximately in a closed form for either square or cubic nonlinearity. The results show that 1 : 2 resonance can lead to codimension-three and codimension-two bifurcations. The validity of analytical predictions is shown by their consistency with numerical simulations.

Rank One Chaos in Periodically-Kicked Time-Delayed Chen System

2015 ◽  
Vol 25 (08) ◽  
pp. 1550097 ◽  
Author(s):  
Yunxian Dai ◽  
Yiping Lin ◽  
Wenjie Yang ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Chen System ◽  
Periodic Force ◽  
Center Manifold Theorem ◽  
Delayed Differential Equations ◽  
Rank One ◽  
Delayed System ◽  
Normal Form Method ◽  
System With Time Delay

In this paper, we study the existence of rank one strange attractor in time-delayed system. First, we try to develop rank one theory for delayed differential equations. Then, we consider Chen system with time-delay, the conditions under which a supercritical Hopf bifurcation occurs are given by using the normal form method and center manifold theorem. Then, we add an external periodic force as an input and observe rank one strange attractors. Finally, several numerical simulations supporting the theoretical analysis are also given.

scholarly journals Rank one strange attractors in periodically kicked Chua’s system with time delay

2015 ◽  
Vol 2015 (1) ◽  
Author(s):  
Wenjie Yang ◽  
Yiping Lin ◽  
Yunxian Dai ◽  
Yusheng Jia
Keyword(s):  
Time Delay ◽  
Strange Attractors ◽  
Rank One ◽  
System With Time Delay

STABILITY AND HOPF BIFURCATION ON A TWO-NEURON SYSTEM WITH TIME DELAY IN THE FREQUENCY DOMAIN

2007 ◽  
Vol 17 (04) ◽  
pp. 1355-1366 ◽  
Author(s):  
WENWU YU ◽  
JINDE CAO
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Neuron System ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach ◽  
System With Time Delay

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis.

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