Practical Numerical Algorithms for Chaotic Systems

One of the basic tenets of science is that deterministic systems are completely predictable-given the initial condition and the equations describing a system, the behavior of the system can be predicted 1 for all time. The discovery of chaotic systems has eliminated this viewpoint. Simply put, a chaotic system is a deterministic system that exhibits random behavior. Though identified as a robust phenomenon only twenty years ago, chaos has almost certainly been encountered by scientists and engi­ neers many times during the last century only to be dismissed as physical noise. Chaos is such a wide-spread phenomenon that it has now been reported in virtually every scientific discipline: astronomy, biology, biophysics, chemistry, engineering, geology, mathematics, medicine, meteorology, plasmas, physics, and even the social sci­ ences. It is no coincidence that during the same two decades in which chaos has grown into an independent field of research, computers have permeated society. It is, in fact, the wide availability of inex­ pensive computing power that has spurred much of the research in chaotic dynamics. The reason is simple: the computer can calculate a solution of a nonlinear system. This is no small feat. Unlike lin­ ear systems, where closed-form solutions can be written in terms of the system's eigenvalues and eigenvectors, few nonlinear systems and virtually no chaotic systems possess closed-form solutions.

1 Steady-State Solutions.- 1.1 Systems.- 1.1.1 Autonomous continuous-time dynamical systems.- 1.1.2 Non-autonomous continuous-time dynamical systems.- 1.1.3 Relationship between autonomous and non-autonomous systems.- 1.1.4 Useful facts regarding continuous-time dynamical systems.- 1.1.5 Discrete-time systems.- 1.2 Limit sets.- 1.2.1 Equilibrium points.- 1.2.2 Periodic solutions.- 1.2.3 Quasi-periodic solutions.- 1.2.4 Chaos.- 1.2.5 Predictive power.- 1.3 Summary.- 2 Poincaré Maps.- 2.1 Definitions.- 2.1.1 The Poincaré map for non-autonomous systems.- 2.1.2 The Poincaré map for autonomous systems.- 2.2 Limit Sets.- 2.2.1 Equilibrium points.- 2.2.2 Periodic solutions.- 2.2.3 Quasi-periodic solutions.- 2.2.4 Chaos.- 2.3 Higher-order Poincaré maps.- 2.4 Algorithms.- 2.4.1 Choosing the hyperplane ?.- 2.4.2 Locating hyperplane crossings.- 2.5 Summary.- 3 Stability.- 3.1 Eigenvalues.- 3.2 Characteristic multipliers.- 3.2.1 Characteristic multipliers.- 3.2.2 Characteristic multipliers and the variational equation.- 3.2.3 Characteristic multipliers and equilibrium points.- 3.3 Lyapunov exponents.- 3.3.1 Definition.- 3.3.2 Lyapunov exponents of an equilibrium point.- 3.3.3 Lyapunov numbers of a fixed point.- 3.3.4 Perturbation subspaces.- 3.3.5 Lyapunov exponents of non-chaotic limit sets.- 3.3.6 Lyapunov exponents of chaotic attractors.- 3.4 Algorithms.- 3.4.1 Eigenvalues at an equilibrium point.- 3.4.2 Characteristic multipliers.- 3.4.3 Lyapunov exponents.- 3.5 Summary.- 4 Integration.- 4.1 Types.- 4.2 Integration error.- 4.2.1 Local errors.- 4.2.2 Global errors.- 4.2.3 Numerical stability.- 4.3 Stiff equations.- 4.4 Practical considerations.- 4.4.1 Variable step-size and order.- 4.4.2 Output points.- 4.4.3 Solving implicit equations.- 4.4.4 Error considerations.- 4.4.5 Integrating chaotic systems.- 4.4.6 Start-up costs.- 4.5 Summary.- 5 Locating Limit Sets.- 5.1 Introduction.- 5.1.1 Brute-force approach.- 5.1.2 Newton-Raphson approach.- 5.2 Equilibrium points.- 5.3 Fixed points.- 5.4 Closed orbits.- 5.5 Periodic solutions.- 5.5.1 The non-autonomous case.- 5.5.2 The autonomous case.- 5.6 Two-periodic solutions.- 5.6.1 Finite differences.- 5.6.2 Spectral balance.- 5.7 Chaotic solutions.- 5.8 Summary.- 6 Manifolds.- 6.1 Definitions and theory.- 6.1.1 Continuous-time systems.- 6.1.2 Discrete-time systems.- 6.2 Algorithms.- 6.2.1 Continuous-time systems.- 6.2.2 Discrete-time systems.- 6.3 Summary.- 7 Dimension.- 7.1 Dimension.- 7.1.1 Definitions.- 7.1.2 Algorithms.- 7.2 Reconstruction.- 7.3 Summary.- 8 Bifurcation Diagrams.- 8.1 Definitions.- 8.2 Algorithms.- 8.2.1 Brute force.- 8.2.2 Continuation.- 8.3 Summary.- 9 Programming.- 9.1 The user interface.- 9.1.1 The dynamical system interface.- 9.1.2 The program initialization interface.- 9.1.3 The interactive interface.- 9.2 Languages.- 9.2.1 Modular design.- 9.3 Library definitions.- 9.3.1 RKF―Runge-Kutta-Fehlberg integration.- 9.3.2 PARSE―input parsing routines.- 9.3.3 BINFILE―binary data files.- 9.3.4 GRAF―graphics.- 10 Phase Portraits.- 10.1 Trajectories.- 10.1.1 Selection of initial conditions.- 10.1.2 Calculating the trajectory.- 10.1.3 Arrowheads.- 10.1.4 Drawing the vector field.- 10.2 Limit sets.- 10.2.1 Equilibrium points.- 10.2.2 Limit cycles.- 10.2.3 Index.- 10.3 Basins.- 10.3.1 Definitions.- 10.3.2 Examples.- 10.3.3 Calculating boundaries of basins of attraction.- 10.4 Programming tips.- 10.4.1 Consistency checking.- 10.4.2 History files.- 10.5 Summary.- A The Newton-Raphson Algorithm.- B The Variational Equation.- C Differential Topology.- C.1 Differential topology.- C.2 Structural stability.- D The Poincaré Map.- E One Lyapunov Exponent Vanishes.- F Cantor Sets.- G List of Symbols.