Connectionists: an educational review on time delay systems

[Claudius Gros]

Dear All

Time delays are ubiquitous in dynamical networks and
many other systems. To this regard we would like to
point out a new introductory review,

   Chaos in time delay systems, an educational review

by H. Wernecke, B. Sandor, C. Gros
    Physics Reports 824, 1-40 (2019).

https://www.sciencedirect.com/science/article/abs/pii/S0370157319302601

abstract
--------
The time needed to exchange information in the physical world induces a delay 
term when the respective system is modeled by differential equations. Time 
delays are hence ubiquitous, being furthermore likely to induce instabilities 
and with it various kinds of chaotic phases. Which are then the possible types 
of time delays, induced chaotic states, and methods suitable to characterize 
the resulting dynamics? This review presents an overview of the field that 
includes an in-depth discussion of the most important results, of the standard 
numerical approaches and of several novel tests for identifying chaos. Special 
emphasis is placed on a structured representation that is straightforward to 
follow. Several educational examples are included in addition as entry points 
to the rapidly developing field of time delay systems.

table of contents
-----------------

1 Introduction

1.1 Time delays in theory and nature
1.2 Outline
1.3 States and state histories
1.4 Lyapunov exponents
1.4.1 Local Lyapunov exponents
1.4.2 Global and maximal Lyapunov exponents
1.4.3 Global Lyapunov exponents for maps
1.5 Educational example: Stability of a fixed point
1.5.1 Analytic ansatz for local Lyapunov exponents
1.5.2 Euler map

2 Types of time delay systems

2.1 Single constant time delay
2.2 Multiple constant time delays
2.3 Time-varying delay
2.4 State-dependent delay
2.5 Conservative vs. dissipative delay
2.6 Distribution of delays
2.7 Reducible time delay systems
2.8 Neutral delay systems
2.9 Networks with delay coupling
2.10 Long time delays

3 Characterizing the dynamics of time delay systems

3.1 Fixed points
3.2 Types of chaotic motion
3.2.1 Delay induced chaos
3.2.2 Partially predictable chaos
3.2.3 Weak and strong chaos
3.2.4 Intermittent and laminar chaos
3.2.5 Transient chaos
3.3 Lyapunov spectrum
3.4 Lyapunov prediction time
3.5 Phase space contraction rate
3.6 Poincaré section
3.7 The power spectrum of attractors 3.8 The dimension of attractors
3.8.1 Mori dimension
3.8.2 Kaplan-Yorke dimension
3.8.3 Fractal dimension
3.8.4 Correlation dimension
3.9 Binary tests for identifying chaos
3.9.1 Cross-distance scaling exponent
3.9.2 Gottwald-Melbourn test
3.10 Space-time interpretation of time delay systems

4 Numerical treatment

4.1 Numerical integration
4.1.1 Euler algorithm
4.1.2 Euler integration as a discrete map
4.1.3 Explicit Runge-Kutta algorithms
4.2 Lyapunov exponents
4.2.1 Maximal Lyapunov exponent from two diverging trajectories
4.2.2 Benettin’s algorithm
4.2.3 Extracting Lyapunov exponents from the Euler map

5 Conclusions

###
### Prof. Dr. Claudius Gros
### http://itp.uni-frankfurt.de/~gros
###
### Complex and Adaptive Dynamical Systems, A Primer
### A graduate-level textbook, Springer (2008/10/13/15)
###
### Mageia, das Buch der Farben
### http://www.buchderfarben.de
###
### Life for barren exoplanets: The Genesis project
### https://link.springer.com/article/10.1007/s10509-016-2911-0
###