BIFURCATIONS AND CHAOS IN TIME DELAYED PIECEWISE LINEAR DYNAMICAL SYSTEMS

We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of the associated bifurcations and chaos as a function of delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. The significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for low values of time delay evidenced by multiple positive Lyapunov exponents are brought out. The study is extended to the case of two coupled systems, one with delay and the other one without delay.

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Two-Parameter Dynamics of An Autonomous Mechanical Governor System With Time Delay

10.21203/rs.3.rs-564675/v1 ◽

2021 ◽

Author(s):

Shuning Deng ◽

Jinchen Ji ◽

Guilin Wen ◽

Huidong Xu

Keyword(s):

Time Delay ◽

Dynamical Behavior ◽

Evolutionary Process ◽

Basins Of Attraction ◽

Poincaré Section ◽

Poincare Section ◽

Interesting Phenomenon ◽

Computing Technique ◽

Intersection Points ◽

Two Parameter

Abstract Understanding of dynamical behavior in the parameter-state space plays a vital role in the optimal design and motion control of mechanical governor systems. By combining the GPU parallel computing technique with two determinate indicators, namely, the Lyapunov exponents and Poincaré section, this paper presents a detailed study on the two-parameter dynamics of a mechanical governor system with different time delays. By identifying different system responses in two-parameter plane, it is shown that the complexity of evolutionary process can increase significantly with the increase of time delay. The path-following strategy and the time domain collocation method are used to explore the details of the evolutionary process. An interesting phenomenon is found in the dynamical behavior of the delayed governor system, which can cause the inconsistency between the number of intersection points of certain periodic response on Poincaré section and the actual period characteristic. For example, the commonly exhibited period-1 orbit may have two or more intersection points on the Poincaré section instead of one point. Furthermore, the variations of basins of attraction are also discussed in the plane of initial history conditions to demonstrate the observed multistability phenomena and chaotic transitions.

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Control strategies of avian influenza pandemic model with time delay

International Journal of Biomathematics ◽

10.1142/s1793524517501042 ◽

2017 ◽

Vol 10 (07) ◽

pp. 1750104 ◽

Cited By ~ 1

Author(s):

U. Roman ◽

Z. Gul ◽

I. Saeed ◽

U. Hakeem ◽

S. Shafie

Keyword(s):

Optimal Control ◽

Time Delay ◽

Avian Influenza ◽

Control Strategies ◽

Influenza Pandemic ◽

Dynamical Behavior ◽

Controlled System ◽

Exponential Factor ◽

Discrete Time Delay ◽

Delay Differential

A mathematical model for avian influenza with optimal control strategies is presented as a system of discrete time delay differential equations (DDEs) and its important mathematical features are analyzed. In alignment to manage this, we develop an optimally controlled pandemic model of avian influenza and insert a time delay with exponential factor. Then we apply two controlled functions in the form of biosecurity of poultry and the education campaign against avian influenza to control the disperse of the disease. Our optimal control strategies will minimize the number of contaminated humans and contaminated birds. We also derive the basic reproduction number to examine the dynamical behavior of the model and demonstrate the existence of the controlled system. For the justification of our work, we present numerical simulations.

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On oscillation of a food-limited population model with time delay

Abstract and Applied Analysis ◽

10.1155/s1085337503209040 ◽

2003 ◽

Vol 2003 (1) ◽

pp. 55-66 ◽

Cited By ~ 8

Author(s):

Leonid Berezansky ◽

Elena Braverman

Keyword(s):

Differential Equation ◽

Time Delay ◽

Delay Differential Equation ◽

Population Model ◽

Delay Differential ◽

Nonlinear Delay Differential Equation ◽

Nonlinear Delay ◽

Oscillation And Nonoscillation

For a scalar nonlinear delay differential equationṄ(t) = r(t)N(t)(K − N(h(t)))/(K + s(t)N(g(t))),r(t) ≥ 0, h(t) ≤ t, g(t) ≤ tand some generalizations of this equation, we establish explicit oscillation and nonoscillation conditions. Coefficientr(t)and delays are not assumed to be continuous.

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BIFURCATION IN ASYMMETRICALLY COUPLED BVP OSCILLATORS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403007199 ◽

2003 ◽

Vol 13 (05) ◽

pp. 1319-1327 ◽

Cited By ~ 10

Author(s):

TETSUSHI UETA ◽

HIROSHI KAWAKAMI

Keyword(s):

Large Scale ◽

Laboratory Experiments ◽

Dynamical Behavior ◽

Internal Resistance ◽

Coupled Systems ◽

Chaotic Attractors ◽

Rich Variety ◽

Coupling Structure ◽

Bifurcation Phenomena ◽

Two Parameter

BVP oscillator is the simplest mathematical model describing dynamical behavior of neural activity. Large scale neural network can often be described naturally by coupled systems of BVP oscillators. However, even if two BVP oscillators are merely coupled by a linear element, the whole system exhibits complicated behavior. In this letter, we analyze coupled BVP oscillators with asymmetrical coupling structure, besides, each oscillator has different internal resistance. The system shows a rich variety of bifurcation phenomena and strange attractors. We calculate bifurcation diagrams in two-parameter plane around which the chaotic attractors mainly appear and confirm relaxant phenomena in the laboratory experiments. We also briefly report a conspicuous strange attractor.

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Uniformly Convergent Numerical Method for Singularly Perturbed Two Parameter Time Delay Parabolic Problem

International Journal of Applied and Computational Mathematics ◽

10.1007/s40819-019-0672-5 ◽

2019 ◽

Vol 5 (3) ◽

Author(s):

L. Govindarao ◽

J. Mohapatra ◽

S. R. Sahu

Keyword(s):

Time Delay ◽

Numerical Method ◽

Parabolic Problem ◽

Singularly Perturbed ◽

Uniformly Convergent ◽

Two Parameter

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Stability analysis of time-delay differential systems with impulsive effect suffered by logic choice

Results in Control and Optimization ◽

10.1016/j.rico.2021.100026 ◽

2021 ◽

pp. 100026

Author(s):

Rong Li ◽

Zhenhua He

Keyword(s):

Stability Analysis ◽

Time Delay ◽

Impulsive Effect ◽

Differential Systems ◽

Delay Differential Systems ◽

Delay Differential

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The MID property for a second-order neutral time-delay differential equation

2020 24th International Conference on System Theory, Control and Computing (ICSTCC) ◽

10.1109/icstcc50638.2020.9259685 ◽

2020 ◽

Author(s):

Amina Benarab ◽

Islam Boussaada ◽

Karim Trabelsi ◽

Guilherme Mazanti ◽

Catherine Bonnet

Keyword(s):

Differential Equation ◽

Time Delay ◽

Delay Differential Equation ◽

Second Order ◽

Delay Differential

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Dynamical analysis of a stochastic rumor-spreading model with Holling II functional response function and time delay

Advances in Difference Equations ◽

10.1186/s13662-020-03096-9 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Liang’an Huo ◽

Xiaomin Chen

Keyword(s):

Time Delay ◽

White Noise ◽

Functional Response ◽

Response Function ◽

Rapid Development ◽

Dynamical Behavior ◽

Deterministic System ◽

Rumor Spreading ◽

Spreading Model ◽

Holling Ii Functional Response

AbstractWith the rapid development of information society, rumor plays an increasingly crucial part in social communication, and its spreading has a significant impact on human life. In this paper, a stochastic rumor-spreading model with Holling II functional response function considering the existence of time delay and the disturbance of white noise is proposed. Firstly, the existence of a unique global positive solution of the model is studied. Then the asymptotic behavior of the global solution around the rumor-free and rumor-local equilibrium nodes of the deterministic system is discussed. Finally, through some numerical results, the validity and availability of theoretical analysis is verified powerfully, and it shows that some factors such as the transmission rate, the intensity of white noise, and the time delay have significant relationship with the dynamical behavior of rumor spreading.

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Dynamical analysis of a giving up smoking model with time delay

Advances in Difference Equations ◽

10.1186/s13662-019-2450-4 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 4

Author(s):

Zizhen Zhang ◽

Ruibin Wei ◽

Wanjun Xia

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Local Stability ◽

Center Manifold ◽

Dynamical Behavior ◽

Sufficient Conditions ◽

Dynamical Analysis ◽

Normal Form Theory ◽

Center Manifold Theorem ◽

Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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