Complexity And Control: Towards A Rigorous Behavioral Theory Of Complex Dynamical Systems

Towards a Rigorous Behavioral Theory of Complex Dynamical Systems

Vladimir G Ivancevic, Darryn J Reid

• Publisher: WSPC
• Year: 2014
• Topic: Biological Sciences
• eBook ISBN:
  9789814635882
• Subtopic: Science General
• Index: Biological Sciences

The book Complexity and Control: Towards a Rigorous Behavioral Theory of Complex Dynamical Systems is a graduate-level monographic textbook, intended to be a novel and rigorous contribution to modern Complexity Theory.

This book contains 11 chapters and is designed as a one-semester course for engineers, applied and pure mathematicians, theoretical and experimental physicists, computer and economic scientists, theoretical chemists and biologists, as well as all mathematically educated scientists and students, both in industry and academia, interested in predicting and controlling complex dynamical systems of arbitrary nature.

Contents:

• Introduction
• Local Geometrical Machinery for Complexity and Control
• Global Categorical Framework for Complexity and Control
• Dynamics of Crowd Behaviors: From Complex Plane to Quantum Random Fields
• Hierarchical Self-Similarity in Group and Crowd Behaviors
• Hybrid Topological Lie-Hamiltonian Learning in Evolving Energy Landscapes
• Complexity and Control in Solitary Conductive PDEs
• Quantum-Computation for Perceptual Control Architecture
• Complexity and Control in Entropic and Stochastic Self-Organization
• Crash Simulator: Brain-and-Spine Injury Mechanics
• Conclusion
• Code Samples Used for Complexity and Control

Readership: Professional and researchers in the field of nonlinear science, chaos and dynamical and complex systems.
Key Features:

• White noise analysis is a relatively young mathematical research area. This volume will present and demonstrate how this new, powerful mathematical tool can be effectively utilized and applied to current problems in the interdisciplinary sciences
• It combines white noise analysis with the equally powerful Feynman path integral, thus allowing us to track the evolution even of systems with memory and strong correlations
• It is user-friendly

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Introduction

 

We introduce our motive for writing this book on complexity and control with a popular “complexity myth,” which seems to be quite wide spread among chaos and complexity theory fashionistas:

Low-dimensional systems usually exhibit complex behaviours (which we know from May’s studies of the Logistic map), while high-dimensional systems usually exhibit simple behaviours (which we know from synchronisation studies of the Kuramoto model).

We admit that this naive view on complex (e.g., human) systems versus simple (e.g., physical) systems might seem compelling to various technocratic managers and politicians; indeed, the idea makes for appealing sound-bites. However, it is enough to see both in the equations and computer simulations of pendula of various degree¹ – (i) a single pendulum, (ii) a double pendulum, and (iii) a triple pendulum – that this popular myth is plain nonsense. The only thing that we can learn from it is what every tyrant already knows: by using force as a strong means of control, it is possible to effectively synchronise even hundreds of millions of people, at least for a while.

More generally, the complexity theory, in its current stage of development, is still fascinated by emergence of wonderful behavioural qualities of high-dimensional self-organised systems. For example, the ‘science of fractal images’ is surely very nice, bridging the gap between science and visual arts, and creating illustrations that serve to inspire and fascinate. Yet we are left still wanting for something we contend is fundamentally important, namely a necessary capability for prediction and control². This requirement was introduced into the world of physical sciences, at the end of the seventeenth century, by Sir Isaac Newton, and is the reason for the enduring usefulness of his theory, its empirical refutation notwithstanding. So, we can regard even the world of visually-appealing cellular automata, which are supposed to be able to efficiently model any kind of a complex system, as actually belonging to pre-Newtonian scientific era, because they are not capable of predicting and controlling individual agents’ behaviours.

We are arguing neither for nor against cellular automata and derived ostensibly soft engineering technologies³, which we acknowledge have produced many useful contributions to modern science and technology. In this book, we are proposing an alternative approach to behavioural complexity theory, intended to be especially rigorous, and designed for prediction and control of both crowds – or, indeed, large sets of agents of any kind – and of the individual agents comprising such groupings. We hope that this approach will also find its way to fields such as economics, where, in our view, the need for more sophisticated models such as this is especially pressing. In simple words, our goal is develop a rigorous framework for modelling very complex systems, potentially with thousands of individual agents, while not ignoring the psycho-physical individuality of each included agent. This novel, ‘hard maths’ approach to behavioural complexity theory can be called high-dimensional Kähler manifold complexity. It is a multi-fold mathematical approach, combining manifold geometry, loop topology, complex analysis and gauge theory, and has already been successfully tested in modern physics. From the signals-and-systems perspective, it is a nonlinear generalization of Kalman’s linear state-space control and filter theory; from dynamical perspective it is both Lagrangian and Hamiltonian; from geometrical perspective it is both Riemannian and symplectic; from physical perspective it is both classical and quantum; from ODE/PDE perspective it is both deterministic and stochastic. It is a uniquely promising internal framework for a wide variety of nonlinear systems of ODEs and PDEs, both real- and complex-valued. We submit that it is ready to become available as a new foundation of a rigorous behavioral complexity theory.

The other, external side, a ‘wrapper’, of this new approach to complexity is the commutative flow framework based on functional composition. Briefly, composing maps (or, functions, processes, signals, systems, transformations) is like following directed paths from one object to another (e.g., from set to set). In general, a diagram is commutative iff any two paths along arrows that start at the same point and finish at the same point yield the same ‘homomorphism’ via compositions along successive arrows. Commutativity of the whole diagram follows from commutativity of its triangular components (depicting a commutative flow, see Figure 1).

Fig. 1.1. A commutative flow (including all curved arrows) on a triangulated di-graph, defining the resulting path AB. Note that the commutativity of the whole diagram, and therefore the definition of the path AB, follows from the commutativity of its triangular components.

The result, we hope, is a rigorous theory of complex behaviour and control that can address some of the shortcomings of contemporary ideas in complexity. Our approach is founded on the principle of making definitive predictions, both by which we may act in the world and by which we may subject our models to the harshest responsible logical and empirical scrutiny we can muster. Indeed, it is our enduring hope that our approach will eventually be supplanted by one or more improved theories of greater explanatory and predictive power; such is the nature of the growth of knowledge.

 

¹If the reader needs a bit stronger argument to dismiss the validity of this popular complexity myth, they might recall that the famous Liouville theorem describes the basic characteristic of any n-dimensional Hamiltonian system described by the phase-space coordinates qⁱ ∈ M and the corresponding momenta p_i ∈ M: the time-evolution of this system preserves the phase-space volume (or, Liouville measure) defined by the following ‘wedge’ products on the phase-space manifold M:

vol = dq¹ ∧···∧ dqⁿ ∧ dp₁ ∧ ··· ∧ dp_n.

This preservation of volume vol causes structural instability of the system, i.e., the phase-space spreading effect, by which small phase regions R_t(∈ M) will tend to get distorted from the initial one R_o(∈ M) (during the system evolution). The problem is much more serious in higher dimensions than in lower dimensions, since there are so many ‘directions’ in which the region R_t(∈ M) can locally spread (see, e.g. [Pen89] for the popular explanation).

²In most approaches, a successful predictability is a necessary condition for a successful controllability. In other words, a comprehensive model, reflecting an aspiration of a ‘natural law’, should be capable of making successful definitive predictions; which is the same as saying that it should be refutable. Having such a model, it is just a matter of adding a controller function to make the system both controllable and observable. However, things can be more complicated than this in practice: among the most successful controllers are model-free fuzzy-logic controllers, which use linguistic variables to control the system at hand - without even knowing the system’s model (e.g., to balance a double inverted pendulum using a fuzzy-logic controller without defining the Lagrangian equations of motion, it is necessary only to specify and tune-up a set of fuzzy If-Then rules, such as “If the two angles are medium left and small right, respectively, while the two angular speeds are high and medium, respectively, then push the cart with medium force to the left”). In this book we will focus on the modern and efficient modeling theory, rather than on the model-free technology, including fuzzy logic, as well as neural, Bayesian, Markovian and small-world networks.

³including genetic algorithms, evolutionary computation, artificial life, etc.

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Local Geometrical Machinery for Complexity and Control

In this Chapter, we present local geometrical machinery for studying complexity and control, consisting of dynamics on Kähler manifolds, which combine three geometrical structures – Riemannian, symplectic and complex (Hermitian) – in a mutually compatible way. In other words, every Kähler manifold is simultaneously Riemannian, symplectic and complex (Hermitian). It is well known that Riemannian manifolds represent the stage on which Lagrangian dynamics is set, symplectic manifolds represent the stage for Hamiltonian dynamics, and complex (Hermitian) varieties comprise the stage for quantum dynamics. Therefore, Kähler manifolds represent the richest dynamical stage available where Lagrangian, Hamiltonian, and quantum dynamics all dance together.

2.1 Introduction: Why Kähler Manifolds?

A set of attractor-following dynamics of n agents (for arbitrary n) in the complex plan...

Table of contents

• Cover Page
• Title
• Copyright
• Contents
• 1 Introduction
• 2 Local Geometrical Machinery for Complexity and Control
• 3 Global Categorical Framework for Complexity and Control
• 4 Dynamics of Crowd Behaviors: From Complex Plane to Quantum Random Fields
• 5 Hierarchical Self-Similarity in Group and Crowd Behaviors
• 6 Hybrid Topological Lie-Hamiltonian Learning in Evolving Energy Landscapes
• 7 Complexity and Control in Solitary Conductive PDEs
• 8 Quantum-Computation for Perceptual Control Architecture
• 9 Complexity and Control in Entropic and Stochastic Self-Organization
• 10 Crash Simulator: Brain-and-Spine Injury Mechanics
• 11 Conclusion
• 12 Code Samples Used for Complexity and Control
• References
• Index