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’.

scholarly journals Nonlinear dynamics of delay systems: an overview

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180389 ◽  
Author(s):  
A. Otto ◽  
W. Just ◽  
G. Radons
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Effects ◽  
Signal Propagation ◽  
Delay System ◽  
Delay Systems ◽  
Climate Modelling ◽  
Theme Issue ◽  
Time Varying Delay ◽  
Interdisciplinary Field ◽  
Laser Systems

Time delays play an important role in many fields such as engineering, physics or biology. Delays occur due to finite velocities of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delay systems can be described by delay differential equations and often include non-negligible nonlinear effects. This overview article introduces the theme issue ‘Nonlinear dynamics of delay systems’, which contains new fundamental results in this interdisciplinary field as well as recent developments in applications. Fundamentally, new results were obtained especially for systems with time-varying delay and state-dependent delay and for delay system with noise, which do often appear in real systems in engineering and nature. The applications range from climate modelling over network dynamics and laser systems with feedback to human balancing and machine tool chatter. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

scholarly journals Complex partial synchronization patterns in networks of delay-coupled neurons

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180128 ◽  
Author(s):  
D. Nikitin ◽  
I. Omelchenko ◽  
A. Zakharova ◽  
M. Avetyan ◽  
A. L. Fradkov ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Temporal Dynamics ◽  
Delay Systems ◽  
Theme Issue ◽  
Partial Synchronization ◽  
Multiplex Network ◽  
Control Scheme ◽  
Non Local ◽  
Spatio Temporal ◽  
Fast Oscillations

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh–Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

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.

scholarly journals Theory of hybrid dynamical systems and its applications to biological and medical systems

2010 ◽  
Vol 368 (1930) ◽  
pp. 4893-4914 ◽  
Author(s):  
Kazuyuki Aihara ◽  
Hideyuki Suzuki
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Mathematical Modelling ◽  
Systems Theory ◽  
Control Systems ◽  
Hybrid Systems ◽  
Hybrid Dynamical Systems ◽  
Medical Systems ◽  
Theme Issue ◽  
And Control

In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to discuss recent developments in understanding nonlinear dynamics of hybrid dynamical systems in the two main theoretical fields of dynamical systems theory and control systems theory. A combined study of the hybrid systems dynamics in the two theoretical fields might contribute to a more comprehensive understanding of hybrid dynamical systems. In addition, mathematical modelling by hybrid dynamical systems is particularly important for understanding the nonlinear dynamics of biological and medical systems as they have many discontinuities such as threshold-triggered firing in neurons, on–off switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer. Hence, the second aim is to discuss recent applications of hybrid dynamical systems in biology and medicine. Thus, this Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine. As the introduction to the topics in this Theme Issue, we provide a brief history of nonlinear dynamics and mathematical modelling, different mathematical models of hybrid dynamical systems, the relationship between dynamical systems theory and control systems theory, examples of complex behaviour in a simple neuron model and its variants, applications of hybrid dynamical systems in biology and medicine as a road map of articles in this Theme Issue and future directions of hybrid systems modelling.

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.

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