scholarly journals Neimark-Sacker Bifurcation Analysis for a Discrete-Time System of Two Neurons

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Changjin Xu
Keyword(s):  
Discrete Time ◽  
System Modelling ◽  
Discrete Time System ◽  
Time System ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Time Systems ◽  
The Given ◽  
Theoretical Results

A class of discrete-time system modelling a network with two neurons is considered. First, we investigate the global stability of the given system. Next, we study the local stability by techniques developed by Kuznetsov to discrete-time systems. It is found that Neimark-Sacker bifurcation (or Hopf bifurcation for map) will occur when the bifurcation parameter exceeds a critical value. A formula determining the direction and stability of Neimark-Sacker bifurcation by applying normal form theory and center manifold theorem is given. Finally, some numerical simulations for justifying the theoretical results are also provided.

scholarly journals Bifurcation Analysis for a Two-Dimensional Discrete-Time Hopfield Neural Network with Delays

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Yaping Ren ◽  
Yongkun Li
Keyword(s):  
Neural Network ◽  
Discrete Time ◽  
Bifurcation Analysis ◽  
Hopfield Neural Network ◽  
Discrete Time System ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
The Stability ◽  
Two Parameters

A bifurcation analysis is undertaken for a discrete-time Hopfield neural network with four delays. Conditions ensuring the asymptotic stability of the null solution are obtained with respect to two parameters of the system. Using techniques developed by Kuznetsov to a discrete-time system, we study the Neimark-Sacker bifurcation (also called Hopf bifurcation for maps) of the system. The direction and the stability of the Neimark-Sacker bifurcation are investigated by applying the normal form theory and the center manifold theorem.

Analysis of Oscillatory Patterns of a Discrete-Time Rosenzweig–MacArthur Model

2018 ◽  
Vol 28 (06) ◽  
pp. 1850075
Author(s):  
Xiaoqin P. Wu ◽  
Liancheng Wang
Keyword(s):  
Discrete Time ◽  
Normal Forms ◽  
Critical Values ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Positive Equilibrium Point ◽  
Associated Species ◽  
Theoretical Results

This article investigates the oscillatory patterns of the following discrete-time Rosenzweig–MacArthur model [Formula: see text] The system describes the evolution and interaction of the populations of two associated species (prey and predator) from generation to generation. We show that this system can exhibit co-dimension-1 bifurcations (flip and Neimark–Sacker bifurcations) as [Formula: see text] crosses some critical values and codimension-2 bifurcations (1:2, 1:3, and 1:4 resonances) for certain critical values of [Formula: see text] at the positive equilibrium point. The normal form theory and the center manifold theorem are used to obtain the normal forms. For codimension-2 bifurcations, the bifurcation diagrams are established by using these normal forms along the orbits of differential equations. Numerical simulations are presented to confirm the theoretical results.

scholarly journals Hopf Bifurcation of a Delayed Ecoepidemic Model with Ratio-Dependent Transmission Rate

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Wanjun Xia ◽  
Soumen Kundu ◽  
Sarit Maitra
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Transmission Rate ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Ratio Dependent ◽  
Theoretical Results

A delayed ecoepidemic model with ratio-dependent transmission rate has been proposed in this paper. Effects of the time delay due to the gestation of the predator are the main focus of our work. Sufficient conditions for local stability and existence of a Hopf bifurcation of the model are derived by regarding the time delay as the bifurcation parameter. Furthermore, properties of the Hopf bifurcation are investigated by using the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out in order to validate our obtained theoretical results.

scholarly journals Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao ◽  
Huaqing Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
Virus Model ◽  
Theoretical Results

By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

CHAOTIFICATION OF DISCRETE-TIME SYSTEMS USING NEURONS

2004 ◽  
Vol 14 (04) ◽  
pp. 1405-1411 ◽  
Author(s):  
H. S. KWOK ◽  
WALLACE K. S. TANG
Keyword(s):  
Computer Simulations ◽  
Discrete Time ◽  
Activation Function ◽  
Feedback Signal ◽  
Discrete Time System ◽  
Hyperbolic Tangent ◽  
Time System ◽  
Discrete Time Systems ◽  
Arbitrary Dimensions ◽  
Time Systems

In this paper, a neuron is introduced for chaotifying nonchaotic discrete-time systems with arbitrary dimensions. By modeling the neuron with a hyperbolic tangent activation function, a scalar feedback signal expressed in a linear combination of the neuron outputs is used. Chaos can then be generated from the controlled discrete-time system. The existence of chaos is verified by both theoretical proof and computer simulations.

The MMVC for the Nth Order Linear Discrete-Time Systems under Prospective Strong Intervention

2014 ◽  
Vol 496-500 ◽  
pp. 1630-1633
Author(s):  
Qiu Ju Wang ◽  
Ru Dong Gai
Keyword(s):  
Discrete Time ◽  
System Model ◽  
Discrete Time System ◽  
Time System ◽  
Order Linear ◽  
First Order ◽  
Discrete Time Systems ◽  
Minimal Variance ◽  
Two Parameters ◽  
Time Systems

This paper is devoted to the issue of the modified minimal variance control (MMVC)for the nth linear discrete-time systems under prospective strong intervention (PSI). At fist, establish the Nth order linear discrete time system model. Based on the research of the first-order linear discrete time systems under PSI with the constraint of minimal variance control, the algorithm is extended to the nth order linear discrete time systems, so one can get MMVC of the nth order linear discrete-time systems with constraint under PSI and by introducing two parameters to proof.

scholarly journals HYPERCHAOTIC DYNAMICS OF A NEW FRACTIONAL DISCRETE-TIME SYSTEM

Fractals ◽  
2021 ◽  
pp. 2140034
Author(s):  
AMINA-AICHA KHENNAOUI ◽  
ADEL OUANNAS ◽  
SHAHER MOMANI ◽  
ZOHIR DIBI ◽  
GIUSEPPE GRASSI ◽  
...  
Keyword(s):  
Discrete Time ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Discrete Time System ◽  
Time System ◽  
Discrete Time Systems ◽  
Simulation Results ◽  
Time Systems ◽  
First Time

In recent years, some efforts have been devoted to nonlinear dynamics of fractional discrete-time systems. A number of papers have so far discussed results related to the presence of chaos in fractional maps. However, less results have been published to date regarding the presence of hyperchaos in fractional discrete-time systems. This paper aims to bridge the gap by introducing a new three-dimensional fractional map that shows, for the first time, complex hyperchaotic behaviors. A detailed analysis of the map dynamics is conducted via computation of Lyapunov exponents, bifurcation diagrams, phase portraits, approximated entropy and [Formula: see text] complexity. Simulation results confirm the effectiveness of the approach illustrated herein.

Exponential Stability Analysis of Fixed Point Interfered Nonlinear Digital Systems in the Presence of Quantization and Overflow Constraints

2020 ◽  
Vol 19 (04) ◽  
pp. 2050040
Author(s):  
Saddam Hussain Malik ◽  
Muhammad Tufail ◽  
Muhammad Rehan ◽  
Shakeel Ahmed
Keyword(s):  
Fixed Point ◽  
Stability Analysis ◽  
Exponential Stability ◽  
Discrete Time ◽  
Digital Systems ◽  
Discrete Time System ◽  
Time System ◽  
Discrete Time Systems ◽  
Digital Hardware ◽  
Time Systems

Finite word length is a practical limitation when discrete-time systems are implemented by using digital hardware. This restriction degrades the performance of a discrete-time system and may even lead it toward instability. This paper, addresses the stability and disturbance attenuation performance analysis of nonlinear discrete-time systems under the influence of energy-bounded external interferences when such systems are subjected to quantization and overflow effects of fixed point hardware. The proposed methodology, in comparison with previous paper, describes exponential stability for the nonlinear discrete-time systems by considering composite nonlinearities of digital hardware. The proposed criteria that ensure exponential stability and [Formula: see text] performance index for the digital systems under consideration are presented in the form of a set of linear matrix inequalities (LMIs) by exploiting Lyapunov stability theory, Lipschitz condition and sector conditions for different types of commonly used quantization and overflow arithmetic properties, and the results are validated for recurrent neural networks. Furthermore, novel stability analysis results for a nonlinear discrete-time system under hardware constraints can also be observed as a special case of the proposed criteria.

scholarly journals Bifurcation Analysis for an SEIRS-V Model with Delays on the Transmission of Worms in a Wireless Sensor Network

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Zizhen Zhang ◽  
Yougang Wang
Keyword(s):  
Wireless Sensor Network ◽  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Wireless Sensor ◽  
Bifurcation Parameter ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Theoretical Results

Hopf bifurcation for an SEIRS-V model with delays on the transmission of worms in a wireless sensor network is investigated. We focus on existence of the Hopf bifurcation by regarding the diverse delay as a bifurcation parameter. The results show that propagation of worms in the wireless sensor network can be controlled when the delay is suitably small under some certain conditions. Then, we study properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Finally, we give a numerical example to support the theoretical results.

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