scholarly journals Unstable Limit Cycles and Singular Attractors in a Two-Dimensional Memristor-Based Dynamic System

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 415 ◽  
Author(s):  
Hui Chang ◽  
Qinghai Song ◽  
Yuxia Li ◽  
Zhen Wang ◽  
Guanrong Chen
Keyword(s):  
Hopf Bifurcation ◽  
Dynamic System ◽  
Numerical Simulations ◽  
Limit Cycles ◽  
Two Dimensional ◽  
Local Activity ◽  
Period 2 ◽  
Subcritical Hopf Bifurcation ◽  
Dimensional Dynamical System ◽  
Nested Intervals

This paper reports the finding of unstable limit cycles and singular attractors in a two-dimensional dynamical system consisting of an inductor and a bistable bi-local active memristor. Inspired by the idea of nested intervals theorem, a new programmable scheme for finding unstable limit cycles is proposed, and its feasibility is verified by numerical simulations. The unstable limit cycles and their evolution laws in the memristor-based dynamic system are found from two subcritical Hopf bifurcation domains, which are subdomains of twin local activity domains of the memristor. Coexisting singular attractors are discovered in the twin local activity domains, apart from the two corresponding subcritical Hopf bifurcation domains. Of particular interest is the coexistence of a singular attractor and a period-2 or period-3 attractor, observed in numerical simulations.

scholarly journals Limit Cycles in a Model of Olfactory Sensory Neurons

2019 ◽  
Vol 29 (03) ◽  
pp. 1950038 ◽  
Author(s):  
Yonghui Xia ◽  
Mateja Grašič ◽  
Wentao Huang ◽  
Valery G. Romanovski
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Sensory Neurons ◽  
Limit Cycles ◽  
Center Manifold ◽  
Calcium Oscillations ◽  
Olfactory Sensory Neurons ◽  
Generalized Hopf Bifurcation ◽  
Subcritical Hopf Bifurcation ◽  
Unstable Cycle

We propose an approach to study small limit cycle bifurcations on a center manifold in analytic or smooth systems depending on parameters. We then apply it to the investigation of limit cycle bifurcations in a model of calcium oscillations in the cilia of olfactory sensory neurons and show that it can have two limit cycles: a stable cycle appearing after a Bautin (generalized Hopf) bifurcation and an unstable cycle appearing after a subcritical Hopf bifurcation.

scholarly journals Bifurcation Analysis of Three-Strategy Imitative Dynamics with Mutations

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Wenjun Hu ◽  
Haiyan Tian ◽  
Gang Zhang
Keyword(s):  
Social Networks ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Evolutionary Game ◽  
Critical Value ◽  
Important Research ◽  
Subcritical Hopf Bifurcation ◽  
Cooperative Behaviors ◽  
Theoretical Results

Evolutionary game dynamics is an important research, which is widely used in many fields such as social networks, biological systems, and cooperative behaviors. This paper focuses on the Hopf bifurcation in imitative dynamics of three strategies (Rock-Paper-Scissors) with mutations. First, we verify that there is a Hopf bifurcation in the imitative dynamics with no mutation. Then, we find that there is a critical value of mutation such that the system tends to an unstable limit cycle created in a subcritical Hopf bifurcation. Moreover, the Hopf bifurcation exists for other kinds of the considered mutation patterns. Finally, the theoretical results are verified by numerical simulations through Rock-Paper-Scissors game.

scholarly journals From Hopf Bifurcation to Limit Cycles Control in Underactuated Mechanical Systems

2017 ◽  
Vol 27 (07) ◽  
pp. 1750104 ◽  
Author(s):  
Nahla Khraief Haddad ◽  
Safya Belghith ◽  
Hassène Gritli ◽  
Ahmed Chemori
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Control Method ◽  
Mechanical Systems ◽  
Control Law ◽  
Underactuated Mechanical Systems ◽  
Center Manifold Theorem ◽  
Stability And Instability ◽  
Subcritical Hopf Bifurcation ◽  
The Stability

This paper deals with the problem of obtaining stable and robust oscillations of underactuated mechanical systems. It is concerned with the Hopf bifurcation analysis of a Controlled Inertia Wheel Inverted Pendulum (C-IWIP). Firstly, the stabilization was achieved with a control law based on the Interconnection, Damping, Assignment Passive Based Control method (IDA-PBC). Interestingly, the considered closed-loop system exhibits both supercritical and subcritical Hopf bifurcation for certain gains of the control law. Secondly, we used the center manifold theorem and the normal form technique to study the stability and instability of limit cycles emerging from the Hopf bifurcation. Finally, numerical simulations were conducted to validate the analytical results in order to prove that with IDA-PBC we can control not only the unstable equilibrium but also some trajectories such as limit cycles.

Dynamic Analysis of a Bistable Bi-Local Active Memristor and Its Associated Oscillator System

2018 ◽  
Vol 28 (08) ◽  
pp. 1850105 ◽  
Author(s):  
Hui Chang ◽  
Zhen Wang ◽  
Yuxia Li ◽  
Guanrong Chen
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Theory ◽  
Hysteresis Loops ◽  
Oscillator System ◽  
Local Activity ◽  
Edge Of Chaos ◽  
Bifurcation Phenomenon ◽  
Subcritical Hopf Bifurcation ◽  
New Type ◽  
Pinched Hysteresis

This paper proposes a new type of memristor with two distinct stable pinched hysteresis loops and twin symmetrical local activity domains, named as a bistable bi-local active memristor. A detailed and comprehensive analysis of the memristor and its associated oscillator system is carried out to verify its dynamic behaviors based on nonlinear circuit theory and Hopf bifurcation theory. The local-activity domains and the edge-of-chaos domains of the memristor, which are both symmetric with respect to the origin, are confirmed by utilizing the mathematical cogent theory. Finally, the subcritical Hopf bifurcation phenomenon is identified in the subcritical Hopf bifurcation region of the memristor.

scholarly journals Control of Hopf Bifurcation Type of a Neuron Model Using Washout Filter

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Chunhua Yuan ◽  
Xiangyu Li
Keyword(s):  
Hopf Bifurcation ◽  
Dynamic Characteristics ◽  
Neuron Model ◽  
Analytical Determination ◽  
Model Parameters ◽  
Two Dimensional ◽  
Single Neurons ◽  
Point Distribution ◽  
Subcritical Hopf Bifurcation ◽  
Washout Filter

A quantitative mathematical model of neurons should not only include enough details to consider the dynamics of single neurons but also minimize the complexity of the model so that the model calculation is convenient. The two-dimensional Prescott model provides a good compromise between the authenticity and computational efficiency of a neuron. The dynamic characteristics of the Prescott model under external electrical stimulation are studied by combining analytical and numerical methods in this paper. Through the analysis of the equilibrium point distribution, the influence of model parameters and external stimulus on the dynamic characteristics is described. The occurrence conditions and the type of Hopf bifurcation in the Prescott model are analyzed, and the analytical determination formula of the Hopf bifurcation type in the neuron model is obtained. Washout filter control is used to change the Hopf bifurcation type, so that the subcritical Hopf bifurcation transforms to supercritical Hopf bifurcation, so as to realize the change of the dynamic characteristics of the model.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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