Computational Methods in Bifurcation Theory and Dissipative Structures

"Dissipative structures" is a concept which has recently been used in physics to discuss the formation of structures organized in space and/or time at the expense of the energy flowing into the system from the outside. The space-time structural organization of biological systems starting from the subcellular level up to the level of ecological systems, coherent structures in laser and of elastic stability in mechanics, instability in hydro- plasma physics, problems dynamics leading to the development of turbulence, behavior of electrical networks and chemical reactors form just a short list of problems treated in this framework. Mathematical models constructed to describe these systems are usually nonlinear, often formed by complicated systems of algebraic, ordinary differ- ential, or partial differential equations and include a number of character- istic parameters. In problems of theoretical interest as well as engineering practice, we are concerned with the dependence of solutions on parameters and particularly with the values of parameters where qualitatively new types of solutions, e.g., oscillatory solutions, new stationary states, and chaotic attractors, appear (bifurcate). Numerical techniques to determine both bifurcation points and the depen- dence of steady-state and oscillatory solutions on parameters are developed and discussed in detail in this text. The text is intended to serve as a working manual not only for students and research workers who are interested in dissipative structures, but also for practicing engineers who deal with the problems of constructing models and solving complicated nonlinear systems.

1. Introduction.- 1.1 General Introduction.- 1.2 Dissipative Structures in Physical, Chemical, and Biological Systems.- 1.2.1 The problems of elastic stability.- 1.2.2 Bifurcations to divergence and flutter in flow-induced oscillations.- 1.2.3 Wheelset nonlinear hunting problem.- 1.2.4 Buckling of a shallow elastic arch.- 1.2.5 Dissipative structures in fluid mechanics.- 1.2.6 Biological systems.- 1.2.7 Reaction-diffusion problems.- 1.3 Basic Concepts and Properties of Nonlinear Systems.- 1.3.1 Steady-state solutions.- 1.3.2 Stability of solutions.- 1.3.3 Evolution systems.- 1.4 Examples.- 2. Multiplicity and Stability in Lumped-Parameter Systems (LPS).- 2.1 Steady-State Solutions.- 2.2 Dependence of Steady-State Solutions on a Parameter—Solution Diagram.- 2.3 Stability of Steady-State Solutions.- 2.4 Branch Points—Real Bifurcation.- 2.4.1 Evaluation of limit and bifurcation points.- 2.4.2 Direction of branches at a bifurcation point.- 2.4.2.1 Selecting starting points for the continuation algorithm.- 2.4.2.2 Illustrative examples.- 2.4.2.3 Bifurcation points with higher degeneration.- 2.4.3 Occurrence of isolas, isola formation.- 2.5 Branch Points—Complex Bifurcations.- 2.5.1 The Hopf bifurcation theorem.- 2.5.2 Direct decomposition technique for location of the complex bifurcation point.- 2.5.3 Direct iteration techniques.- 2.6 Bifurcation Diagram.- 2.7 Transient Behavior of LPS—Numerical Methods.- 2.7.1 Runge-Kutta methods.- 2.7.2 Multistep methods.- 2.7.3 Integration along the solution arc.- 2.7.4 Integration of phase trajectories for autonomous systems.- 2.7.5 Numerical methods for stiff systems of ODE.- 2.7.6 Systems of differential and algebraic equations.- 2.7.7 Integration of differential equations with time delay.- 2.8 Computation of Periodic Solutions.- 2.8.1 Transformation into an initial-value problem—the shooting method..- 2.8.2 Stability of periodic solutions.- 2.8.3 Continuation of periodic solutions.- 2.8.4 Bifurcation of periodic solutions.- 2.9 Chaotic Attractors.- 2.9.1 Characterization of chaotic attractors.- 2.9.2 Liapunov exponents.- 2.9.3 Power spectra.- 2.9.4 The Poincaré map.- 3. Multiplicity and Stability in Distributed-Parameter Systems (DPS).- 3.1 Steady-State Solutions—Methods for Solving Nonlinear Boundary-Value Problems.- 3.1.1 Finite-difference methods.- 3.1.2 Quasi-linearization.- 3.1.3 Shooting methods.- 3.2 Dependence of Steady-State Solutions on a Parameter.- 3.3 Branch Points—Methods for Evaluating Real and Complex Bifurcation Points.- 3.3.1 Primary bifurcation.- 3.3.2 Secondary real bifurcation.- 3.3.3 Secondary complex bifurcation.- 3.4 Methods for Transient Simulation of Parabolic Equations—Finite-Difference Methods.- 3.4.1 Nonlinearity approximation.- 3.4.2 Automatic control of time step k.- 3.4.3 Automatic control of spatial step size h, equidistant net.- 3.4.4 Adaptive nonequidistant net.- 4. Development of Quasi-stationary Patterns with Changing Parameter.- 4.1 Quasi-stationary Behavior in LPS—Examples.- 4.2 Quasi-stationary Behavior in DPS—Examples.- 5. Perspectives.- Appendix A DERPAR—A Continuation Algorithm.- Appendix B SHOOT—An Algorithm for Solving Nonlinear Boundary-Value Problems by the Shooting Method.- Appendix C Bifurcation and Stability Theory.- C. 1 Invariant Manifolds and the Center-Manifold Theorem (Reduction of Dimension).- C.2 Normal Forms.- C.3 Bifurcation of Singular Points of Vector Fields.- C.4 Codimension of a Vector Field. Unfolding of a Vector Field.- C.5 Construction of a Versal Deformation.- C.6 Bifurcations of Codimension 2.- C.7 Bifurcations from Limit Cycles.- References.