Exponential Convergence of Delayed Dynamical Systems

2001 ◽  
Vol 13 (3) ◽  
pp. 621-635 ◽  
Author(s):  
Tianping Chen ◽  
Shun-ichi Amari
Keyword(s):  
Dynamical Systems ◽  
Exponential Convergence ◽  
Stability And Convergence ◽  
Delayed Systems ◽  
Mild Conditions

We discuss some delayed dynamical systems, investigating their stability and convergence. We prove that under mild conditions, these delayed systems are global exponential convergent.

Monotone Semiflows Generated by Neutral Functional Differential Equations With Application to Compartmental Systems

1991 ◽  
Vol 43 (5) ◽  
pp. 1098-1120 ◽  
Author(s):  
Jianhong Wu ◽  
H. I. Freedman
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Stability And Convergence ◽  
Strong Monotonicity ◽  
Neutral Functional Differential Equations ◽  
Neutral Equations ◽  
Functional Differential ◽  
Monotone Dynamical Systems

AbstractThis paper is devoted to the machinery necessary to apply the general theory of monotone dynamical systems to neutral functional differential equations. We introduce an ordering structure for the phase space, investigate its compatibility with the usual uniform convergence topology, and develop several sufficient conditions of strong monotonicity of the solution semiflows to neutral equations. By applying some general results due to Hirsch and Matano for monotone dynamical systems to neutral equations, we establish several (generic) convergence results and an equivalence theorem of the order stability and convergence of precompact orbits. These results are applied to show that each orbit of a closed biological compartmental system is convergent to a single equilibrium.

An Efficient Numerical Simulation for Solving Dynamical Systems With Uncertainty

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Ali Ahmadian ◽  
Soheil Salahshour ◽  
Chee Seng Chan ◽  
Dumitur Baleanu
Keyword(s):  
Dynamical Systems ◽  
Stability And Convergence ◽  
Uncertain Parameters ◽  
Crucial Issue ◽  
Computational Costs ◽  
First Order ◽  
Wide Range ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Fuzzy Dynamical Systems

In a wide range of real-world physical and dynamical systems, precise defining of the uncertain parameters in their mathematical models is a crucial issue. It is well known that the usage of fuzzy differential equations (FDEs) is a way to exhibit these possibilistic uncertainties. In this research, a fast and accurate type of Runge–Kutta (RK) methods is generalized that are for solving first-order fuzzy dynamical systems. An interesting feature of the structure of this technique is that the data from previous steps are exploited that reduce substantially the computational costs. The major novelty of this research is that we provide the conditions of the stability and convergence of the method in the fuzzy area, which significantly completes the previous findings in the literature. The experimental results demonstrate the robustness of our technique by solving linear and nonlinear uncertain dynamical systems.

An almost mixing of all orders property of algebraic dynamical systems

2017 ◽  
Vol 39 (5) ◽  
pp. 1211-1233
Author(s):  
L. ARENAS-CARMONA ◽  
D. BEREND ◽  
V. BERGELSON
Keyword(s):  
Dynamical Systems ◽  
Mild Conditions ◽  
Algebraic Dynamical Systems ◽  
Strongly Mixing

We consider dynamical systems, consisting of $\mathbb{Z}^{2}$-actions by continuous automorphisms on shift-invariant subgroups of $\mathbb{F}_{p}^{\mathbb{Z}^{2}}$, where $\mathbb{F}_{p}$ is the field of order $p$. These systems provide natural generalizations of Ledrappier’s system, which was the first example of a 2-mixing $\mathbb{Z}^{2}$-action that is not 3-mixing. Extending the results from our previous work on Ledrappier’s example, we show that, under quite mild conditions (namely, 2-mixing and that the subgroup defining the system is a principal Markov subgroup), these systems are almost strongly mixing of every order in the following sense: for each order, one just needs to avoid certain effectively computable logarithmically small sets of times at which there is a substantial deviation from mixing of this order.

Semi-Discretization of Delayed Dynamical Systems

2001 ◽  
Author(s):  
Tamás Insperger ◽  
Gábor Stépán
Keyword(s):  
Dynamical Systems ◽  
Mathieu Equation ◽  
Special Kind ◽  
Delayed Systems ◽  
The Past ◽  
Linear Discrete System ◽  
Approximate System ◽  
Discretization Technique ◽  
The Stability ◽  
Time Periodic

Abstract An efficient numerical method is presented for the stability analysis of linear retarded dynamical systems. The method is based on a special kind of discretization technique with respect to the past effect only. The resulting approximate system is delayed and time-periodic in the same time, but still, it can be transformed analytically into a high dimensional linear discrete system. The method is especially efficient for time varying delayed systems, including the case when the time delay itself varies in time. The method is applied to determine the stability charts of the delayed Mathieu equation with damping.

scholarly journals Rejecting Chaotic Disturbances Using a Super-Exponential-Zeroing Neurodynamic Approach for Synchronization of Chaotic Sensor Systems

Sensors ◽  
2018 ◽  
Vol 19 (1) ◽  
pp. 74 ◽  
Author(s):  
Dechao Chen ◽  
Shuai Li ◽  
Qing Wu
Keyword(s):  
Exponential Convergence ◽  
Convergence Property ◽  
Stability And Convergence ◽  
Sensor Systems ◽  
Initial State ◽  
Synchronization Process ◽  
Convergence Performance ◽  
The Stability ◽  
The Impact ◽  
Lower Error Bound

Due to the existence of time-varying chaotic disturbances in complex applications, the chaotic synchronization of sensor systems becomes a tough issue in industry electronics fields. To accelerate the synchronization process of chaotic sensor systems, this paper proposes a super-exponential-zeroing neurodynamic (SEZN) approach and its associated controller. Unlike the conventional zeroing neurodynamic (CZN) approach with exponential convergence property, the controller designed by the proposed SEZN approach inherently possesses the advantage of super-exponential convergence property, which makes the synchronization process faster and more accurate. Theoretical analyses on the stability and convergence advantages in terms of both faster convergence speed and lower error bound within the task duration are rigorously presented. Moreover, three synchronization examples substantiate the validity of the SEZN approach and the related controller for synchronization of chaotic sensor systems. Comparisons with other approaches such as the CZN approach, show the convergence superiority of the proposed SEZN approach. Finally, extensive tests further investigate the impact on convergence performance by choosing different values of design parameter and initial state.

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