Sliding Dynamics and Bifurcations of a Filippov System with Nonlinear Threshold Control

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Biao Tang ◽  
Wuqiong Zhao
Keyword(s):  
Numerical Methods ◽  
Saddle Node Bifurcation ◽  
Threshold Control ◽  
Change Rate ◽  
Viral Loads ◽  
Modeling Methods ◽  
Dynamical Behaviors ◽  
The Rich ◽  
Sliding Dynamics ◽  
Threshold Setting

Considering the effectiveness of introducing the change rate of viral loads into the threshold setting policy for triggering interventions, we propose an immune-virus Filippov system with a nonlinear threshold. By developing new analytical and numerical methods, we systematically studied the rich dynamical behaviors and bifurcations of the proposed system, including the existence of three sliding segments and three pseudo-equilibria, boundary-center bifurcation, boundary-saddle bifurcation, pseudo-saddle-node bifurcation and tangency bifurcation. We further showed that the proposed system can exhibit virous structures in the coexistence of multiple steady states. Phenomena include bistability of two pseudo-equilibria, tristability and multiplestability of two pseudo-equilibria with regular equilibria or touching cycles. The modeling methods, as well as the analytical and numerical methods, can be widely applied to many other fields.

A Predator–Prey Model with Prey Population Guided Anti-Predator Behavior

2017 ◽  
Vol 27 (07) ◽  
pp. 1750099 ◽  
Author(s):  
Xiaodan Sun ◽  
Yingping Li ◽  
Yanni Xiao
Keyword(s):  
Limit Cycle ◽  
Threshold Value ◽  
Prey Population ◽  
Nonlinear Functional ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Piecewise Model ◽  
The Rich ◽  
Predator Behavior

We consider a predator–prey system with prey population guided anti-predator behavior, in which anti-predator behaviors happen only when the population size of the prey is greater than a threshold. We investigate the rich dynamics of the proposed piecewise model as well as both subsystems without and with nonlinear functional response. In particular, the subsystem with anti-predator behaviors exhibits rich dynamical behaviors including saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Further, besides the dynamical properties of subsystems the piecewise system shows some new complicated dynamical behaviors as the threshold value varies, including unstable limit cycle, semistable limit cycle, bistability of equilibrium and limit cycle, and tristability of three equilibria. From the switching system we can conclude that a great anti-predator rate induces the prey population to persist more likely, but whether the prey and predator populations coexist depends further on the threshold that triggers anti-predator behavior. Especially, a large threshold not only makes coexistence of the prey and predator populations as an equilibrium more likely, but also damps the predator–prey oscillations.

scholarly journals Bifurcation and Chaotic Behavior of a Discrete-Time SIS Model

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Junhong Li ◽  
Ning Cui
Keyword(s):  
Hopf Bifurcation ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Euler Method ◽  
Flip Bifurcation ◽  
Bifurcation Diagrams ◽  
Bifurcation And Chaos ◽  
Sis Model ◽  
Dynamical Behaviors ◽  
The Rich

The discrete-time epidemic model is investigated, which is obtained using the Euler method. It is verified that there exist some dynamical behaviors in this model, such as transcritical bifurcation, flip bifurcation, Hopf bifurcation, and chaos. The numerical simulations, including bifurcation diagrams and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors.

BIFURCATIONS IN A FREQUENCY-WEIGHTED KURAMOTO OSCILLATORS NETWORK

2012 ◽  
Vol 22 (09) ◽  
pp. 1250230 ◽  
Author(s):  
HANQING WANG ◽  
XIANG LI
Keyword(s):  
Hopf Bifurcation ◽  
Human Population ◽  
Activity Patterns ◽  
Dimensional System ◽  
Human Beings ◽  
Saddle Node Bifurcation ◽  
Kuramoto Oscillators ◽  
Oscillator Network ◽  
The Rich ◽  
Low Dimensional

This paper focuses on the origin of rich dynamics in a frequency-weighted Kuramoto-oscillator network, which embeds the periodical activity patterns of human beings to understand the collective behaviorial dynamics in human population. We present analytical results of the dynamics of the model by reducing the N-dimensional system to solvable low-dimensional equations. The bifurcation analysis reveals that there exist supercritical Hopf bifurcation and infinite-period saddle-node bifurcation, which explain the rich dynamics including the incoherent, oscillatory, and synchronous states observed in this model.

Chaos in a Circuit System with Ferroelectric Capacitor

2012 ◽  
Vol 268-270 ◽  
pp. 1308-1312
Author(s):  
Wei Qiao
Keyword(s):  
Numerical Methods ◽  
Orthogonal Polynomial ◽  
Numerical Simulations ◽  
Random Parameter ◽  
Chaotic Attractors ◽  
Deterministic System ◽  
Random Parameters ◽  
Nonlinear Circuit ◽  
Ferroelectric Capacitor ◽  
The Rich

Chaos depends on the random parameters are explored in circuit system with ferroelectric capacitor in this article. A random parameter is considered to characterize the random factors in this kind of capacitor. After we reduce the nonlinear circuit system with random parameter into the equivalent deterministic system by orthogonal polynomial approximation, the available effective numerical methods is applied to obtain the responses of the equivalent deterministic circuit system. Then by numerical simulations, we systematically explore the effect of the random parameter on chaos, and find the intensity of random parameter can present the rich and colorful structures of chaotic attractors.

ON THE DYNAMICS OF SOME NUMERICAL METHODS

2000 ◽  
Vol 11 (07) ◽  
pp. 1481-1487 ◽  
Author(s):  
E. AHMED ◽  
A. S. HEGAZI
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Numerical Methods ◽  
Differential Equations ◽  
Reaction Diffusion ◽  
Discrete Dynamical Systems ◽  
Point Of View ◽  
Partial Differential ◽  
Dynamical Behaviors ◽  
Telegraph Equations

From numerical methods point of view of dynamical systems, we have determined dynamical behaviors of the corresponding systems (i.e., chaotic, stable, bifurcations possibility, etc.). New versions of numerical methods are derived and we have compared the dynamical behaviors of the continuous dynamical systems with their corresponding discrete dynamical systems. An application of partial differential equations is given for reaction-diffusion and telegraph equations.

A NEW CNN-BASED CHAOTIC CIRCUIT: EXPERIMENTAL RESULTS

2009 ◽  
Vol 19 (08) ◽  
pp. 2609-2617 ◽  
Author(s):  
A. BUSCARINO ◽  
L. FORTUNA ◽  
M. FRASCA
Keyword(s):  
Nonlinear Oscillator ◽  
Electronic Devices ◽  
Dynamical Behavior ◽  
Chaotic Regime ◽  
Nonlinear Networks ◽  
Dynamical Behaviors ◽  
Experimental Implementation ◽  
Wide Range ◽  
The Rich

In this note the dissipative form of the nonlinear oscillator proposed in [Pokrovskii et al., 2007] is introduced. The behavior of such system is investigated through an experimental implementation of the circuit realized using electronic devices. The proposed implementation of the dissipative nonlinear oscillator, based on Cellular Nonlinear Networks, is fully described and the characterization of the rich dynamical behavior of the circuit performed, disclosing the wide range of dynamical behaviors assumed with respect to different values of the forcing frequency. Moreover, a master–slave scheme has been proposed to synchronize two dissipative oscillators in the chaotic regime. The onset of synchronization is proved both theoretically and experimentally.

scholarly journals Application of Multistage Homotopy Perturbation Method to the Chaotic Genesio System

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
M. S. H. Chowdhury ◽  
I. Hashim ◽  
S. Momani ◽  
M. M. Rahman
Keyword(s):  
Numerical Methods ◽  
Perturbation Method ◽  
Fourth Order ◽  
Chaotic Systems ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Homotopy Perturbation ◽  
Promising Tool ◽  
Dynamical Behaviors

Finding accurate solution of chaotic system by using efficient existing numerical methods is very hard for its complex dynamical behaviors. In this paper, the multistage homotopy-perturbation method (MHPM) is applied to the Chaotic Genesio system. The MHPM is a simple reliable modification based on an adaptation of the standard homotopy-perturbation method (HPM). The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chaotic Genesio system. Numerical comparisons between the MHPM and the classical fourth-order Runge-Kutta (RK4) solutions are made. The results reveal that the new technique is a promising tool for the nonlinear chaotic systems of ordinary differential equations.

Image Analysis of Intramembrane Particles in Freeze-Fractured Biomembranes Using Computer Simulation Techniques

1975 ◽  
Vol 33 ◽  
pp. 214-215
Author(s):  
D.J. Benefiel ◽  
R.S. Weinstein
Keyword(s):  
Membrane Proteins ◽  
Freeze Fracture ◽  
Integral Membrane Proteins ◽  
Systematic Evaluation ◽  
Intramembrane Particles ◽  
Modeling Methods ◽  
Simulation Techniques ◽  
Freeze Fracture Electron Microscopy ◽  
Fracture Electron ◽  
Computer Simulation Techniques

Intramembrane particles (IMP or MAP) are components of most biomembranes. They are visualized by freeze-fracture electron microscopy, and they probably represent replicas of integral membrane proteins. The presence of MAP in biomembranes has been extensively investigated but their detailed ultrastructure has been largely ignored. In this study, we have attempted to lay groundwork for a systematic evaluation of MAP ultrastructure. Using mathematical modeling methods, we have simulated the electron optical appearances of idealized globular proteins as they might be expected to appear in replicas under defined conditions. By comparing these images with the apearances of MAPs in replicas, we have attempted to evaluate dimensional and shape distortions that may be introduced by the freeze-fracture technique and further to deduce the actual shapes of integral membrane proteins from their freezefracture images.

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