A NEW CLASS OF CHAOTIC ATTRACTORS IN MURALI–LAKSHMANAN–CHUA CIRCUIT

2006 ◽  
Vol 16 (09) ◽  
pp. 2659-2670 ◽  
Author(s):  
ALI OKSASOGLU ◽  
QIUDONG WANG
Keyword(s):  
Limit Cycles ◽  
Short Pulse ◽  
Chaotic Attractors ◽  
Hopf Bifurcations ◽  
Stable Limit ◽  
New Class ◽  
Chua Circuit ◽  
Rank One ◽  
Weakly Stable ◽  
Stable Limit Cycles

In this paper, we study the existence of a new class of chaotic attractors, namely the rank-one attractors, in the MLC (Murali–Lakshmanan–Chua) circuit [Murali et al., 1994] by numerical simulations based on a theory of rank-one maps developed in [Wang & Young, 2005]. With the guidance of the theory in [Wang & Young, 2005], weakly stable limit cycles, found through Hopf bifurcations and other numerical means, are subjected to periodic pulses with long relaxation periods to produce rank-one attractors. The periodic pulses are applied directly as an input. Periodic pulses have been used before in various schemes of chaos. However, for this scheme of creating rank-one attractors to work, the applied periodic pulses must have short pulse widths and long relaxation periods. This is one of the key components in creating this new class of chaotic attractors.

RANK ONE CHAOS IN SWITCH-CONTROLLED MURALI–LAKSHMANAN–CHUA CIRCUIT

2006 ◽  
Vol 16 (11) ◽  
pp. 3207-3234 ◽  
Author(s):  
ALI OKSASOGLU ◽  
DONGSHENG MA ◽  
QIUDONG WANG
Keyword(s):  
Numerical Simulations ◽  
Mathematical Theory ◽  
Stable Limit Cycle ◽  
Strange Attractors ◽  
Stable Limit ◽  
Chua Circuit ◽  
Switch Control ◽  
Control Schemes ◽  
Rank One ◽  
Weakly Stable

In this paper, we investigate the creation of strange attractors in a switch-controlled MLC (Murali–Lakshmanan–Chua) circuit. The design and use of this circuit is motivated by a recent mathematical theory of rank one attractors developed by Wang and Young. Strange attractors are created by periodically kicking a weakly stable limit cycle emerging from the center of a supercritical Hopf bifurcation, and are found in numerical simulations by following a recipe-like algorithm. Rigorous conditions for chaos are derived and various switch control schemes, such as synchronous, asynchronous, single-, and multi-pulse, are investigated in numerical simulations.

STABILITY, SYMMETRY-BREAKING BIFURCATIONS AND CHAOS IN DISCRETE DELAYED MODELS

2008 ◽  
Vol 18 (05) ◽  
pp. 1477-1501 ◽  
Author(s):  
MINGSHU PENG ◽  
YUAN YUAN
Keyword(s):  
Limit Cycles ◽  
Bifurcation Theory ◽  
Invariant Tori ◽  
Pitchfork Bifurcation ◽  
Chaotic Attractors ◽  
Stable Limit ◽  
Stable Equilibria ◽  
Multiple Stable Equilibria ◽  
Stable Limit Cycles ◽  
Stable Invariant

In this paper, we use the standard bifurcation theory to study rich dynamics of time-delayed coupling discrete oscillators. Equivariant bifurcations including equivariant Neimark–Sacker bifurcation, equivariant pitchfork bifurcation and equivariant periodic doubling bifurcation are analyzed in detail. In the application, we consider a ring of identical discrete delayed Ikeda oscillators. Multiple oscillation patterns, such as multiple stable equilibria, stable limit cycles, stable invariant tori and multiple chaotic attractors, are shown.

scholarly journals Coexistence of chaotic attractor and unstable limit cycles in a 3D dynamical system

2021 ◽  
Vol 1 ◽  
pp. 50
Author(s):  
Dana Constantinescu ◽  
Gheorghe Tigan ◽  
Xiang Zhang
Keyword(s):  
Dynamical System ◽  
Limit Cycles ◽  
Algebraic Surfaces ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
Stable Limit ◽  
Invariant Algebraic Surfaces ◽  
Stable Limit Cycles ◽  
T System

The coexistence of stable limit cycles and chaotic attractors has already been observed in some 3D dynamical systems. In this paper we show, using the T-system, that unstable limit cycles and chaotic attractors can also coexist. Moreover, by completing the characterization of the existence of invariant algebraic surfaces and their associated global dynamics, we give a better understanding on the disappearance of the strange attractor and the limit cycles of the studied system.

BIFURCATIONS IN A HOST-PARASITE MODEL WITH NONLINEAR INCIDENCE

2006 ◽  
Vol 16 (11) ◽  
pp. 3291-3307 ◽  
Author(s):  
WENDI WANG ◽  
YI LI ◽  
H. W. HETHCOTE
Keyword(s):  
Limit Cycles ◽  
Hopf Bifurcations ◽  
Reproduction Rate ◽  
Stable Limit ◽  
Nonlinear Incidence Rate ◽  
Saddle Node Bifurcation ◽  
Nonlinear Incidence ◽  
Initial States ◽  
Host Parasite ◽  
Stable Limit Cycles

A host-parasite model is proposed that incorporates a nonlinear incidence rate. Under the influence of multiple infectious attacks, the model admits bistable regions such that the infection dies out if initial states lie in one region, and the population and parasites coexist if initial states lie in the other region. It is also found that parasites can drive the population to extinction for suitable parameters. It is verified that the model has a saddle-node bifurcation, Hopf bifurcations and a cusp of codimension 2 or higher codimension. Stable limit cycles and unstable limit cycles are examined as the infection-reduced reproduction rate varies. It is shown that the model goes through the change of stages of infection extinction, infection persistence, infection extinction, and the extinction of both parasites and the population as the contact coefficient increases.

Response : Stable Limit Cycles in Prey-Predator Populations

Science ◽  
1973 ◽  
Vol 181 (4104) ◽  
pp. 1074-1074
Author(s):  
Robert M. May
Keyword(s):  
Limit Cycles ◽  
Stable Limit ◽  
Stable Limit Cycles

On the Bifurcation Structure of a Leslie–Tanner Model with a Generalist Predator

2020 ◽  
Vol 30 (06) ◽  
pp. 2050088
Author(s):  
Luis Miguel Valenzuela ◽  
Gamaliel Blé ◽  
Manuel Falconi
Keyword(s):  
Generalist Predator ◽  
Hopf Bifurcations ◽  
Stable Limit ◽  
Previous Knowledge ◽  
Bifurcation Structure ◽  
Remarkable Feature ◽  
Predator Prey ◽  
Predator Prey Model ◽  
The Subject ◽  
Stable Limit Cycles

In this work, the dynamical properties of a Leslie–Tanner predator–prey model are analyzed. A method is provided to find the regions in the parameter space where Bautin, Hopf and simultaneous supercritical Hopf bifurcations exist. A remarkable feature of the dynamics of the model is the existence of tristability. In fact, the system simultaneously presents three stable limit cycles. These results generalize some previous knowledge on the subject since both prey defense and a generalist predator are incorporated in our analysis.

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