Complexity Science

An Introduction

Mark A Peletier, Rutger A van Santen;Erik Steur

• Publisher: WSPC
• Year: 2019
• ISBN:
  9789813239616
• Topic: Physical Sciences
• Subtopic: Chaotic Behavior in Systems

This book on complexity science comprises a collection of chapters on methods and principles from a wide variety of disciplinary fields — from physics and chemistry to biology and the social sciences.

In this two-part volume, the first part is a collection of chapters introducing different aspects in a coherent fashion, and providing a common basis and the founding principles of the different complexity science approaches; the next provides deeper discussions of the different methods of use in complexity science, with interesting illustrative applications.

The fundamental topics deal with self-organization, pattern formation, forecasting uncertainties, synchronization and revolutionary change, self-adapting and self-correcting systems, and complex networks. Examples are taken from biology, chemistry, engineering, epidemiology, robotics, economics, sociology, and neurology.

Contents:

• Basic Concepts:
  • The Many Facets of Complexity
  • Disguises of Complexity
• Tools:
  • Complex Dynamics of Deterministic Nonlinear Systems (Erik Steur and Henk Nijmeijer)
  • Pattern Formation in Reaction-Diffusion Systems — An Explicit Approach (Arjen Doelman)
  • A Primer on Stochastic Processes (Mark A Peletier)
  • Random Graphs Models for Complex Networks, and the Brain (Remco van der Hofstad)
  • Mesoscale Simulations of Complex Fluids (Johan T Padding)
• Applications:
  • Biorhythms and the Brain (Jos H T Rohling and Johanna H Meijer)
  • Modelling of Collective Motion (Barry W Fitzgerald, Rutger A van Santen and Johan T Padding)
  • Path-Integral Representation of Diluted Pedestrian Dynamics (Alessandro Corbetta and Federico Toschi)
  • Chemical Reaction Kinetic Perspective with Mesoscopic Nonequilibrium Thermodynamics (Hong Qian)
  • Metabolic Pathways and Optimisation (Robert Planqué and Josephus Hulshof)
  • Particle-Based Modelling of Flows Through Obstacles (Emilio N M Cirillo, Adrian Muntean and Rutger A van Santen)

Readership: A wide spectrum of graduate students and researchers interested in complexity ranging from mathematical and chemical to biology and social sciences.Complexity;Self-Organization;Pattern Formation;Synchronization;Networks0 Key Features:

• Provides tutorial introductory chapters; highlighting theories, concepts and methods at bachelor/master level from a unifying perspective
• Includes more advanced chapters that provide deep background on the different matters
• Explains and illustrates application areas

Part B

Tools

Chapter 3

Complex dynamics of deterministic nonlinear systems

Erik Steur and Henk Nijmeijer

We give an introduction to the analysis of the dynamics of deterministic nonlinear systems from a systems and control point of view. In particular, we discuss the stabilizing or destabilizing effect of feedback interconnections in nonlinear dynamical systems. With the help of this machinery we explain two types of complex collective dynamics in networks of nonlinear systems.

1. Introduction

A central topic in complexity science is the study of asymptotic behavior of deterministic nonlinear dynamical systems. In this chapter we consider those nonlinear systems whose dynamics are described by ordinary differential equations (ODEs). As mentioned in Chapter 1, Sec. 3.1, such systems arise from fundamental laws of physics, mean-field approximations, or they can be the result of empirical modeling. However, we do not focus on modeling; we consider the models (in the form of ODEs) to be given. Our contribution lies in the introduction of a number of tools for the analysis of their asymptotic dynamics, i.e. the dynamical behavior of the system as time grows large.

We start with a discussion of a geometric approach to the analysis of the dynamics of deterministic nonlinear systems. Then we present some results from stability theory and we discuss when a loss of stability results in a bifurcation that gives rise to “new” dynamics. Bifurcations are related to tipping points [46], emergent oscillations and pattern formation (this chapter and Chapter 4) and chaos [55]. Subsequently we introduce dynamical systems with inputs and outputs, which allow the systems to interact with their environment or other systems. We discuss the stability of such input-output systems in a feedback configuration. In particular, we highlight the role of the internal dynamics of an input-output system, which are the dynamics that are not directly observed via the output. We show that stability properties of these internal dynamics determine whether of not the whole system is (de)stabilized via the feedback.

Next we introduce a class of nonlinear dynamical systems that interconnect over a network. Such systems, which we call network dynamical systems (NDSs) are omnipresent in society, science and nature. Examples include power networks, biochemical circuits and metabolic networks (see Chapter 12) and the human brain (see Chapters 6 and 8), just to name a few. NDSs may have complex collective dynamics, like the emergence of spontaneous order (i.e. synchronization) or other types of patterned activity, which result from nonlinearities in the systems’ dynamics, the interactions, or both. We present two numerical studies of collective dynamics of NDSs, namely i. synchronization in an NDS with model neurons, and ii. emergent oscillations in an NDS with electrically coupled cells, which are silent in absence of interaction. We analyze these collective dynamics from a systems and control point of view. We consider an NDS as a dynamical feedback system and we apply our previously introduced analysis tools to explain to synchronization and emergent oscillations. In particular, we show that the internal dynamics of the subsystems of the NDS play an important role in the emergence of these collective dynamics.

Notation

We adapt the following (standard) notation. We denote by

the real numbers,

₊ the positive real numbers, and

₊ the non-negative real numbers (i.e.

₊ =

₊ ∪ {0}). For a natural number n (i.e. n ∈ {1, 2, . . .})

ⁿ is the n-dimensional real space, i.e. the space of n-tuples of real numbers. We represent an element x of

ⁿ as a column vector

where x₁, x₂, . . . , x_n ∈

. Transposition of a matrix (or vector) is denoted by superscript T. The n × n-dimensional identity matrix is denoted by I_n. The Euclidean norm of a vector is denoted by || · ||, i.e. ||x|| =

for any ...