Complexity and Control

The following sections are included:

• Preface
• Acknowledgments
• Contents
• Glossary of Frequently Used Symbols

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 fromMay's studies of the Logisticmap), 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 degree – (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.

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.

In this Chapter, we present a global structural framework for complexity and control, given by modern algebraic and computational machinery of categories, functors and derived natural structures. Category theory provides us with what amounts to a powerful abstraction that hides and unifies the details of various individual theories and models. The central concept is that of transformations between objects that preserve structure.

The following sections are included:

• Complex Plane Dynamics of Crowds and Groups
  • Introduction
  • Complex-Valued Dynamics of Crowd and Group Behaviors
  • Kähler Geometry of Crowd and Group Dynamics
  • Computer Simulations of Crowds and Croups Dynamics
  • Braids of Agents' Behaviors in the Complex Plane
  • Hilbert-Space Control of Crowds and Groups Dynamics
• Quantum Random Fields: A Unique Framework for Simulation, Optimization, Control and Learning
  • Introduction
  • Adaptive Quantum Oscillator
  • Optimization and Learning on Banach and Hilbert Spaces
• Appendix
  • Complex-Valued Image Processing
  • Linear Integral Equations
  • Riemann-Liouville Fractional Calculus
  • Rigorous Geometric Quantization
  • Supervised Machine-Learning Methods
  • First-Order Logic and Quantum Random Fields

In this Chapter, a nonlinear, complex, Hamiltonian description of socio–cognio–physical dynamics at the macroscopic, classical, inter-personal crowd level and microscopic, quantum, intra-personal agent level, is presented, uniquely, in the form of the open Liouville equation. At the microscopic level, this can be considered to be a nonlinear extension of the linear correlation and factor dynamics. This implies the arrow of time in both microscopic and macroscopic processes and shows the existence of the formal crowd-agent space-time self-similarity. This in itself shows the existence of a unique control law, which acts on different scales of agent functioning. This self–similar socio–cognio–physical control law enables us to use the crowd dynamics simulator (previously developed at Defence Science & Technology Organisation, Australia), for recursive simulation of individual agents' representation spaces on a cluster of computers.

In this Chapter, a novel bidirectional algorithm for hybrid (discrete + continuous-time) Lie-Hamiltonian evolution in adaptive energy landscape-manifold is designed and its topological representation is proposed. The algorithm is developed within a geometrically and topologically extended framework of Hopfield's neural nets and Haken's synergetics (it is currently designed in Mathematica, although with small changes it could be implemented in Symbolic C++ or any other computer algebra system). The adaptive energy manifold is determined by the Hamiltonian multivariate cost function H, based on the user-defined vehicle-fleet configuration matrix W, which represents the pseudo-Riemannian metric tensor of the energy manifold. Search for the global minimum of H is performed using random signal differential Hebbian adaptation. This stochastic gradient evolution is driven (or, pulled-down) by ‘gravitational forces’ defined by the 2nd Lie derivatives of H. Topological changes of the fleet matrix W are observed during the evolution and its topological invariant is established. The evolution stops when the W–topology breaks down into several connectivity-components, followed by topology-breaking instability sequence (i.e., a series of phase transitions).

In this Chapter, we review and analyze models of controlled complexity in nonlinear pulse conduction, ranging from the Hodgkin-Huxley action potentials propagating along neural fibers to rogue waves in optical fibers. The novel model proposed is an alternative to the Hodgkin-Huxley neural model in the form of the sine-Gordon wave equation. This new alternative explains pulse conduction in terms of general wave phenomena (such as kinks, solitons and breathers).

In this Chapter, we propose a quantum-dynamical modeling approach to Perceptual Control Architecture, using large networks of Josephson Junctions and their category-theoretic generalizations with fuzzy associative functors

Our approach provides the basis for composing modular multi-layered perceptual control architectures using Josephson Junction Networks, employing intuitively appealing category-theoretic abstractions to hide the algebraic details from the designer while nonetheless being able to rigorously ensure functional composition correctness. That is, our approach ensures that Josephson Junction Networks, as a modeling primitive, can be composed into formally correct multi-layered perceptual control architectures, while hiding the underlying algebraic systems of equations from the designer under a blanket of category-theoretic abstraction.

In this Chapter, we are primarily interested in developing advanced entropic and stochastic models of military command and control (C2). The underpinning thesis here stands in sharp contrast to much grand historical theorizing based on the idea of discovering some inexorable laws of historical destiny; this belief shows up as efforts to predict the future by uncovering ‘trends’ and treating them as ‘laws’ as well as in narratives attributing causes to anything and everything except randomness that are constructed – of course – with the full benefits of hindsight. Instead, our approach expressly allows for the fact that significant combinations of events might arise purely randomly, and the observed realized outcome is merely one path of events among potentially innumerable unrealized possibilities. Entropy – an order parameter by which we may quantify uncertainty, information content, complexity, or relative inadequacy of historical determinism, depending on our point of view at the time – is therefore a central concern in our conceptualization.

Recently, the first author has proposed a new coupled loading-rate hypothesis as a unique cause of both brain and spinal injuries, which states that they are both caused by a Euclidean jolt, an impulsive loading that strikes head and spine (or, any other part of the human body)- in several coupled degrees-of-freedom simultaneously. Injury never happens in a single direction only, nor is it ever caused by a static force. It is always an impulsive translational plus rotational force. The Euclidean jolt causes two basic forms of brain, spine and other musculo-skeletal injuries: (i) localized translational dislocations; and (ii) localized rotational disclinations. In the present Chapter, we first review this unique mechanics of a general human mechanical injury, and then describe how it can be predicted and controlled by a crash simulator toolbox. This rigorous Matlab toolbox has been developed using an existing thirdparty toolbox DiffMan, for accurately solving differential equations on smooth manifolds and mechanical Lie groups. The present crash simulator toolbox performs prediction/control of brain and spinal injuries within the framework of the Euclidean group SE(3) of rigid motions in our natural 3-dimensional space.

It is often held that things should always be made simple, which presumes that either that they can always be made simple or that all the jetisoned logic doesn't matter anyway. Alledgedly, anything should be explainable so that anyone can understand it. Don't get bogged down in dreary details. It should be effortless for the reader: low-dimensional systems exhibit complex behaviour while high-dimensional systems exhibit simple behaviour (to return to our prolegomonal opening), competition is a universal solution, demand must increase as price falls, and everything under the sun neatly fits a power law. Or so the story goes…

The following sections are included:

• Mathematica^Ⓡ Code
  • Generic Chaotic Simulator
  • Vector Differential Operators
  • NLS Explorer
• 2C++ Code
  • C++ Lambda Functions for Real Calculus
  • Accelerometer Data Processor
  • Simple Predictor-Corrector Integrator
  • Solving the BVP with the Shooting Method
  • Linear Hyperbolic PDE Solver
  • Linear Elliptic PDE Solver
  • Method of Lines for a Set of the NLS Equations
• C# Code
  • Iterative Equation Solver
  • Simulated Annealing: A Function Minimum
  • Simple Nonlinear Dynamics
  • Nonlinear Pendulum Simulator
  • Lagrangian Dynamics Simulator
  • Complex-Valued Crowd Attractor Dynamics
• Freeform Fortran Code
  • Lorenz Attractor Simulator
  • Complex Lorenz Attractor
  • Simple SGE Soliton
  • Complex Signal Presentation
  • Gaussian Wave Packet
  • Hermitian Matrices
  • Euclidean L2-Norm
  • Vector/Matrix Operations
• Plain C-Code: Levenberg-Marquardt Optimizer
• Free Basic Code: 2D Crowd Dynamics with 3000 Agents

The following sections are included:

• References
• Index