Subcritical Neimark–Sacker bifurcation and hybrid control in a discrete-time Phytoplankton–Zooplankton model

2021 ◽  
Author(s):  
A. Q. Khan ◽  
M. B. Javaid
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Strategy ◽  
Hybrid Control ◽  
Dynamical Behavior ◽  
Flip Bifurcation ◽  
Two Dimensional ◽  
Theoretical Results

In this paper, we explore the local dynamical behavior with different topological classifications around fixed points, Neimark–Sacker bifurcation and hybrid control in the discrete-time Phytoplankton–Zooplankton model. More precisely, we have investigated the local dynamical behavior with different topological classifications around trivial, semitrivial and interior fixed points of the two-dimensional Phytoplankton–Zooplankton model, respectively. The existence of possible bifurcations around fixed points is also investigated, and it is proved that there exists no flip bifurcation around trivial and semitrivial fixed points but around interior fixed point, the model undergoes Neimark–Sacker bifurcation only. Moreover, hybrid control strategy is utilized for controlling Neimark–Sacker bifurcation in the Phytoplankton–Zooplankton model. Lastly, theoretical results are verified numerically.

scholarly journals Discrete-time phytoplankton–zooplankton model with bifurcations and chaos

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. Q. Khan ◽  
M. B. Javaid
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Chaotic Behavior ◽  
Discrete Analogue ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Time Model ◽  
Local Dynamics

AbstractThe local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model.

scholarly journals Discrete-Time Predator-Prey Model with Bifurcations and Chaos

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
K. S. Al-Basyouni ◽  
A. Q. Khan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Fixed Point

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

Discrete-Time Memristor Model for Enhancing Chaotic Complexity and Application in Secure Communication

2022 ◽  
Author(s):  
Wenhao Yan ◽  
Zijing Jiang ◽  
Qun Ding
Keyword(s):  
Fixed Points ◽  
Discrete Time ◽  
Dynamic Characteristics ◽  
Secure Communication ◽  
Chaotic Systems ◽  
Two Dimensional ◽  
Initial State ◽  
One Dimensional ◽  
Backward Euler ◽  
Short Period

Abstract The physical implementation of continuoustime memristor makes it widely used in chaotic circuits, whereas discrete-time memristor has not received much attention. In this paper, the backward-Euler method is used to discretize TiO2 memristor model, and the discretized model also meets the three fingerprinter characteristics of the generalized memristor. The short period phenomenon and uneven output distribution of one-dimensional chaotic systems affect their applications in some fields, so it is necessary to improve the dynamic characteristics of one-dimensional chaotic systems. In this paper, a two-dimensional discrete-time memristor model is obtained by linear coupling the proposed TiO2 memristor model and one-dimensional chaotic systems. Since the two-dimensional model has infinite fixed points, the stability of these fixed points depends on the coupling parameters and the initial state of the discrete TiO2 memristor model. Furthermore, the dynamic characteristics of one-dimensional chaotic systems can be enhanced by the proposed method. Finally, we apply the generated chaotic sequence to secure communication.

scholarly journals Dynamical behaviors in a discrete fractional-order predator-prey system

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5857-5874 ◽  
Author(s):  
Yao Shi ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Fractional Order ◽  
Control Strategies ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Theoretical Results

This paper is related to the dynamical behaviors of a discrete-time fractional-order predatorprey model. We have investigated existence of positive fixed points and parametric conditions for local asymptotic stability of positive fixed points of this model. Moreover, it is also proved that the system undergoes Flip bifurcation and Neimark-Sacker bifurcation for positive fixed point. Various chaos control strategies are implemented for controlling the chaos due to Flip and Neimark-Sacker bifurcations. Finally, numerical simulations are provided to verify theoretical results. These results of numerical simulations demonstrate chaotic behaviors over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behaviors in the model.

scholarly journals Bifurcation analysis of a discrete-time four-dimensional cubic autocatalator chemical reaction model with coupling through uncatalysed reactant

2021 ◽  
Vol 87 (2) ◽  
pp. 415-439
Author(s):  
Muhammad Salman Khan ◽  
Keyword(s):  
Fixed Point ◽  
Chemical Reaction ◽  
Discrete Time ◽  
Hybrid Control ◽  
Continuous System ◽  
Reaction Model ◽  
Dimensional System ◽  
Chemical Reaction Model ◽  
Cubic Autocatalator ◽  
Positive Fixed Point

In this manuscript, we discuss a four-dimensional cubic autocatalator chemical reaction model in continuous form. We investigate the existence of one and only positive fixed point and then we have obtained some parametric conditions for local stability of continuous system by using Routh-Hurwitz stability criteria. Moreover, we discretize the four-dimensional continuous cubic autocatalator chemical reaction model by using Euler’s forward method and then by using a nonstandard difference scheme we obtained a consistent discrete-time counterpart of four-dimensional cubic autocatalator chemical reaction model. Parametric conditions for local asymptotic stability of one and only positive fixed point of obtained system are also discussed. It is shown that the obtained system experiences the Neimark-Sacker bifurcation at one and only positive fixed point by using a general standard for Neimark-Sacker bifurcation. The discrete-time counterpart of genuine four-dimensional system displays chaotic dynamics at different standards of bifurcation parameter. Furthermore, the control of Neimark-Sacker bifurcation and chaos is also deliberated by using a generalized hybrid control scheme, which is based on parameter perturbation and feedback control. Finally, some numerical examples are given to strengthen our theoretical results.

A new class of two-dimensional rational maps with self-excited and hidden attractors

2021 ◽  
Author(s):  
Li-Ping Zhang ◽  
Yang Liu ◽  
Zhou-Chao Wei ◽  
Hai-Bo Jiang ◽  
Qin-Sheng Bi
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Complex Dynamics ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Rational Maps ◽  
Two Dimensional ◽  
Hidden Attractors ◽  
New Class ◽  
Rational Term

Abstract This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stabilities of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan-Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

Chaotic Mixing of Highly Viscous Liquids With Rectangular or Elliptical Rotors

2005 ◽  
Author(s):  
M. Erol Ulucakli
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Fixed Points ◽  
Stokes Flow ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Rectangular Cell ◽  
Viscous Liquids ◽  
Unstable Manifolds ◽  
Time Periodic

The objective of this research is to experimentally investigate various mixing regions in a two-dimensional Stokes flow driven by a rectangular or elliptical rotor. Flow occurs in a rectangular cell filled with a very viscous fluid. The Reynolds number based on rotor size is in the order of 0.5. The flow is time-periodic and can be analyzed, both theoretically and experimentally, by considering the Poincare map that maps the position of a fluid particle to its position one period later. The mixing regions of the flow are determined, theoretically, by the fixed points of this map, either hyperbolic or degenerate, and their stable and unstable manifolds. Experimentally, the mixing regions are visualized by releasing a blob of a passive dye at one of these fixed points: as the flow evolves, the blob stretches to form a streak line that lies on the unstable manifold of the fixed point.

scholarly journals Supercritical Neimark–Sacker Bifurcation and Hybrid Control in a Discrete-Time Glycolytic Oscillator Model

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
A. Q. Khan ◽  
E. Abdullah ◽  
Tarek F. Ibrahim
Keyword(s):  
Discrete Time ◽  
Hybrid Control ◽  
Oscillator Model ◽  
Positive Equilibrium ◽  
Time Model ◽  
Discrete Time Model ◽  
Prime Period ◽  
Dynamical Properties ◽  
Positive Equilibrium Point ◽  
Theoretical Results

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior of ℝ+2. It is proved that, for all parametric values, Pxy+α/β+α2,α is the unique positive equilibrium point of the glycolytic oscillator model. Further local dynamical properties along with different topological classifications about the equilibrium Pxy+α/β+α2,α have been investigated by employing the method of linearization. Existence of prime period and periodic points of the model under consideration are also investigated. It is proved that, about the fixed point Pxy+α/β+α2,α, the discrete-time glycolytic oscillator model undergoes no bifurcation, except Neimark–Sacker bifurcation. A further hybrid control strategy is applied to control Neimark–Sacker bifurcation in the discrete-time model. Finally, theoretical results are verified numerically.

scholarly journals Stability of the Discrete Stage–Structure Prey–Predator Model

2020 ◽  
Vol 55 (1) ◽  
Author(s):  
Rehab Noori Shalan ◽  
Shireen R. Jawad ◽  
Alaa Hussien Lafta
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Three Dimensional ◽  
Stage Structure ◽  
Prey Population ◽  
Topological Properties ◽  
Proposed Model ◽  
Predator Model ◽  
The Stability ◽  
Theoretical Results

This paper discusses the discrete stage–structure prey-predator model involved in the Beddington–DeAngelis type of functional response described by differential equation systems proposed as three-dimensional systems. Furthermore, the predators are divided into two types of populations, namely, mature and immature, along with the prey population. The stability of all possible fixed points is demonstrated by solving our proposed model analytically using the standard lemma and topological properties, which give all possible properties to each fixed point. In the same manner, we identify three fixed points, which are as follows: the origin fixed point, which means there are no species; the axial fixed point, which means the prey population increases logistically with the absence of a predator one (mature and immature populations); and the positive fixed point, which signifies the coexistence of all species. We show that the numerical simulations part is used not only to plot the time series of fixed values, but also, to find and illustrate the theoretical results.

scholarly journals Dynamical Behavior of Two Dimensional Hénon Maps

2012 ◽  
Vol 47 (1) ◽  
pp. 55-60
Author(s):  
MS Islam ◽  
MS Islam
Keyword(s):  
Fixed Points ◽  
Computer Programming ◽  
Dynamical Behavior ◽  
Periodic Points ◽  
Numerical Results ◽  
Two Dimensional ◽  
Henon Maps ◽  
Hénon Maps ◽  
Non Linear ◽  
Parameter Values

In this article, we study the two dimensional non-linear dynamical behavior of Hénon maps. We investigate the parameter values for which fixed points and periodic points of period two exist and study the dimension of the maps. We also investigate the numerical results of the maps and use computer programming Mathematica for generating graphs and computations. DOI: http://dx.doi.org/10.3329/bjsir.v47i1.10722 Bangladesh J. Sci. Ind. Res. 47(1), 55-60, 2012

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