[Comp-neuro] an educational review on time delay systems

Claudius Gros gros at itp.uni-frankfurt.de
Wed Oct 30 09:06:24 CET 2019

Dear All

Time delays are ubiquitous in the brain. 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).


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

More information about the Comp-neuro mailing list