scholarly journals On the Number of Periodic Orbits to Odd Order Differential Delay Systems

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1731
Author(s):  
Weigao Ge ◽  
Lin Li
Keyword(s):  
Variational Method ◽  
Periodic Orbits ◽  
Index Theory ◽  
Accurate Method ◽  
Delay System ◽  
Delay Systems ◽  
Differential Delay ◽  
Odd Order

In this paper, we study the periodic orbits of a type of odd order differential delay system with 2k−1 lags via the S1 index theory and the variational method. This type of system has not been studied by others. Our results provide a new and more accurate method for counting the number of periodic orbits.

scholarly journals Stability of Stochastic Differential Delay Systems with Delayed Impulses

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yanlei Wu
Keyword(s):  
Delay System ◽  
Delay Systems ◽  
Stochastic Delay ◽  
Differential Delay ◽  
Almost Sure Exponential Stability ◽  
Continuous Dynamic System ◽  
Continuous Dynamic ◽  
The Stability ◽  
Delay Differential ◽  
Delayed Impulses

We investigate the stability of stochastic delay differential systems with delayed impulses by Razumikhin methods. Some criteria on thepth moment and almost sure exponential stability are obtained. It is shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. Moreover, it is also shown that if a continuous dynamic system is stable, then, under some conditions, the delayed impulses do not destroy the stability of the systems. The effectiveness of the proposed results is illustrated by two examples.

scholarly journals Multiple periodic orbits of high-dimensional differential delay systems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zhongmin Sun ◽  
Weigao Ge ◽  
Lin Li
Keyword(s):  
Periodic Orbits ◽  
Theoretical Basis ◽  
Delay Systems ◽  
High Dimensional ◽  
Differential Delay ◽  
Multiple Periodic Orbits

AbstractIn this paper, we consider differential delay systems of the form $$x'(t)=-\sum_{s=1}^{2k-1}(-1)^{s+1} \nabla F \bigl(x(t-s) \bigr), $$x′(t)=−∑s=12k−1(−1)s+1∇F(x(t−s)), in which the coefficients of the nonlinear terms with different hysteresis have different signs. Such systems have not been studied before. The multiplicity of the periodic orbits is related to the eigenvalues of the limit matrix. The results provide a theoretical basis for the study of differential delay systems.

On linear generalized neutral differential delay systems

2009 ◽  
Vol 346 (7) ◽  
pp. 691-704 ◽  
Author(s):  
Athanasios A. Pantelous ◽  
Grigoris I. Kalogeropoulos
Keyword(s):  
Delay Systems ◽  
Differential Delay

Stabilizing Gain Regions of PID Controller for Time Delay Systems

2013 ◽  
Vol 313-314 ◽  
pp. 432-437
Author(s):  
Fu Min Peng ◽  
Bin Fang
Keyword(s):  
Stability Analysis ◽  
Time Delay ◽  
Pid Controller ◽  
Delay System ◽  
Nyquist Plot ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Frequency Range ◽  
Time Delay System ◽  
Key Points

Based on the inverse Nyquist plot, this paper proposes a method to determine stabilizing gain regions of PID controller for time delay systems. According to the frequency characteristic of the inverse Nyquist plot, it is confirmed that the frequency range is used for stability analysis, and the abscissas of two kind key points are obtained in this range. PID gain is divided into several regions by abscissas of key points. Using an inference and two theorems presented in the paper, the stabilizing PID gain regions are determined by the number of intersections of the inverse Nyquist plot and the vertical line in the frequency range. This method is simple and convenient. It can solve the problem of getting the stabilizing gain regions of PID controller for time delay system.

scholarly journals Enhanced Disturbance-Observer-Based Control for a Class of Time-Delay System with Uncertain Sinusoidal Disturbances

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xinyu Wen
Keyword(s):  
Time Delay ◽  
Flight Control ◽  
Disturbance Observer ◽  
Delay System ◽  
Delay Systems ◽  
Flight Control System ◽  
Control Laws ◽  
Nonlinear Disturbance Observer ◽  
Dynamic Performances ◽  
Uncertain Disturbance

This paper is concerned with disturbance-observer-based control (DOBC) for a class of time-delay systems with uncertain sinusoidal disturbances. The disturbances are decomposed as precise and uncertain parts using nonlinear disturbance observer (DO) after appropriate coordinate transformation. And then the two parts can be compensated by corresponding controller, respectively, such that the classic DOBC method is extended to uncertain disturbance rejection. One novel feature of the proposed method is that even if the precise disturbance parameters are inaccessible, the merits of DOBC can be inherited. By integrating the disturbance observers with feedback control laws with time delay, the disturbances can be rejected and the desired dynamic performances can be guaranteed. Finally, simulations for a flight control system are given to demonstrate the effectiveness of the results.

Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850155 ◽  
Author(s):  
Chengwei Dong
Keyword(s):  
Nonlinear System ◽  
Variational Method ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Building Blocks ◽  
Topological Classification ◽  
One Dimensional ◽  
Sivashinsky Equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

scholarly journals Delay-induced switched states in a slow–fast system

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180118 ◽  
Author(s):  
Stefan Ruschel ◽  
Serhiy Yanchuk
Keyword(s):  
Nonlinear Dynamics ◽  
Positive Feedback ◽  
Delay Time ◽  
Delay System ◽  
Delay Systems ◽  
Round Trip ◽  
Theme Issue ◽  
Feedback Function ◽  
Fast System ◽  
Two Component

We consider the two-component delay system εx ′( t ) = −  x ( t ) −  y ( t ) +  f ( x ( t  − 1)), y ′( t ) =  ηx ( t ) with small para- meters ε , η and positive feedback function f . Previously, such systems have been reported to model switching in optoelectronic experiments, where each switching induces another one after approximately one delay time, related to one round trip of the signal. In this paper, we study these delay-induced switched states. We provide conditions for their existence and show how the formal limits ε  → 0 and/or η  → 0 facilitate our understanding of this phenomenon. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

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