Stability and Hopf Bifurcation of Nearest-Neighbor Coupled Neural Networks With Delays

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Lu Wang ◽  
Min Xiao ◽  
Shuai Zhou ◽  
Yurong Song ◽  
Jinde Cao
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Nearest Neighbor ◽  
Multiple Delays ◽  
Dimensional System ◽  
Practical Applications ◽  
Coupled Neural Networks ◽  
The Stability ◽  
Related Characteristic ◽  
Theoretical Analyses

Abstract In this paper, a high-dimensional system of nearest-neighbor coupled neural networks with multiple delays is proposed. Nowadays, most present researches about neural networks have studied the connection between adjacent nodes. However, in practical applications, neural networks are extremely complicated. This paper further considers that there are still connection relationships between nonadjacent nodes, which reflect the intrinsic characteristics of neural networks more accurately because of the complexity of its topology. The influences of multiple delays on the local stability and Hopf bifurcation of the system are explored by selecting the sum of delays as bifurcation parameter and discussing the related characteristic equations. It is found that the dynamic behaviors of the system depend on the critical value of bifurcation. In addition, the conditions that ensure the stability of the system and the criteria of Hopf bifurcation are given. Finally, the correctness of the theoretical analyses is verified by numerical simulation.

Turing Instability and Hopf Bifurcation in Cellular Neural Networks

2021 ◽  
Vol 31 (08) ◽  
pp. 2150143
Author(s):  
Zunxian Li ◽  
Chengyi Xia
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Cellular Neural Networks ◽  
Bifurcation Theorem ◽  
Decoupling Method ◽  
Dynamical Behaviors ◽  
The Stability ◽  
Approximate Expressions ◽  
Zero Equilibrium

In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.

scholarly journals Applications of fidelity measures to complex quantum systems

2016 ◽  
Vol 374 (2069) ◽  
pp. 20150153 ◽  
Author(s):  
Sandro Wimberger
Keyword(s):  
Cold Atoms ◽  
Dimensional System ◽  
Many Body ◽  
Practical Applications ◽  
Dynamical Regimes ◽  
Many Body Systems ◽  
Fidelity Measures ◽  
The Stability ◽  
Low Dimensional ◽  
Quantum Motion

We revisit fidelity as a measure for the stability and the complexity of the quantum motion of single-and many-body systems. Within the context of cold atoms, we present an overview of applications of two fidelities, which we call static and dynamical fidelity, respectively. The static fidelity applies to quantum problems which can be diagonalized since it is defined via the eigenfunctions. In particular, we show that the static fidelity is a highly effective practical detector of avoided crossings characterizing the complexity of the systems and their evolutions. The dynamical fidelity is defined via the time-dependent wave functions. Focusing on the quantum kicked rotor system, we highlight a few practical applications of fidelity measurements in order to better understand the large variety of dynamical regimes of this paradigm of a low-dimensional system with mixed regular–chaotic phase space.

Fractional-order bidirectional associate memory (BAM) neural networks with multiple delays: The case of Hopf bifurcation

2021 ◽  
Vol 182 ◽  
pp. 471-494
Author(s):  
Changjin Xu ◽  
Zixin Liu ◽  
Maoxin Liao ◽  
Peiluan Li ◽  
Qimei Xiao ◽  
...  
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Fractional Order ◽  
Multiple Delays ◽  
Bam Neural Networks

Stability and Bifurcation in a Diffusive Logistic Population Model with Multiple Delays

2015 ◽  
Vol 25 (08) ◽  
pp. 1550107 ◽  
Author(s):  
Shanshan Chen ◽  
Junjie Wei
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Dirichlet Boundary Condition ◽  
Population Model ◽  
Dirichlet Boundary ◽  
Positive Equilibrium ◽  
Multiple Delays ◽  
Stability And Bifurcation ◽  
The Stability

A diffusive logistic population model with multiple delays and Dirichlet boundary condition is considered in this paper. The stability/instability of the positive equilibrium and delay induced Hopf bifurcation are investigated. Moreover, we show which kind of delay could actually affect the dynamics.

Nonlinear Stability and Bifurcation of Multi-D.O.F. Chatter System in Grinding Process

2006 ◽  
Vol 304-305 ◽  
pp. 141-145 ◽  
Author(s):  
Qing Kai Han ◽  
Tao Yu ◽  
Zhi Wei Zhang ◽  
Bang Chun Wen
Keyword(s):  
Dynamic Properties ◽  
Grinding Wheel ◽  
Dimensional System ◽  
Grinding Process ◽  
Center Manifold Theorem ◽  
Stability And Bifurcation ◽  
The Stability ◽  
Grinding Machine ◽  
Low Dimensional ◽  
Theoretical Analyses

The nonlinear chatter in grinding machine system is discussed analytically in the paper. In higher speed grinding process, the self-excited chatter vibration is mostly induced by the change of grinding speed and grinding wheel shape. Here the grinding machine tool is viewed as a nonlinear multi-D.O.F. autonomous system, in which hysteretic factors of contact surfaces are also introduced. Firstly, the DOFs of the above system are reduced efficiently without changing its dynamic properties by utilizing the center manifold theorem and averaging method. Then, a low dimensional system and corresponding averaging equations are obtained. The stability and bifurcation of chatter system are discussed on the base of deduced averaging equations. It is proved that chatter occurs as a Hopf bifurcation emerging from the steady state at the origin of system. The theoretical analyses on the multi-DOF chattering system will lead to further understanding of the nonlinear mechanisms of higher speed grinding processes.

ON THE NUMERICAL DETECTION OF CODIMENSION-3 BIFURCATIONS OF PERIODIC SOLUTIONS (WITH APPLICATION TO OPTICAL BISTABILITY)

1994 ◽  
Vol 04 (06) ◽  
pp. 1425-1446
Author(s):  
KLAUS-GEORG NOLTE ◽  
IVAN L’HEUREUX
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Optical Bistability ◽  
Continuation Method ◽  
Period Doubling ◽  
Bistable System ◽  
Dimensional System ◽  
Systems Of Equations ◽  
The Stability ◽  
Stationary Behavior

Based upon the combination of the pseudo-arclength continuation method and the Poincaré map defined on a variable return plane, systems of equations are constructed that trace a Takens-Bogdanov bifurcation, a cusp, an isola formation/perturbed bifurcation point and a degenerate period-doubling/secondary Hopf bifurcation of periodic solutions of autonomous ordinary differential equations. The implementation of these ideas into a collection of FORTRAN codes and its application to a five-dimensional system describing an optical bistable system lead to the detection of interesting codimension-3 bifurcations away from the stationary behavior. A winged cusp, a swallow tail, a degenerate hysteresis point, an isola formation point for a codimension-1 loop and two kinds of degenerate Takens-Bogdanov bifurcations of periodic solutions are presented. Finally, based upon the computation of the stability coefficient “a”, attractive tori are found in a systematic way and briefly discussed.

Bifurcations due to different delays of high-order fractional neural networks

2021 ◽  
pp. 2150075
Author(s):  
Chengdai Huang ◽  
Jinde Cao
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Critical Values ◽  
Communication Delay ◽  
Leakage Delay ◽  
Bifurcation Parameter ◽  
Stability Performance ◽  
The Stability

This paper expounds the bifurcations of two-delayed fractional-order neural networks (FONNs) with multiple neurons. Leakage delay or communication delay is viewed as a bifurcation parameter, stability zones and bifurcation conditions with respect to them are commendably established, respectively. It declares that both leakage delay and communication delay immensely influence the stability and bifurcation of the developed FONNs. The explored FONNs illustrate superior stability performance if selecting a lesser leakage delay or communication delay, and Hopf bifurcation generates once they overstep their critical values. The verification of the feasibility of the developed analytic results is implemented via numerical experiments.

scholarly journals Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Juan Liu ◽  
Changwei Sun ◽  
Yimin Li
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

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