"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.

Inhalt

1. Introduction.- 1.1 General Introduction.- 1.2 Dissipative Structures in Physical, Chemical, and Biological Systems.- 1.3 Basic Concepts and Properties of Nonlinear 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 ParameterSolution Diagram.- 2.3 Stability of Steady-State Solutions.- 2.4 Branch PointsReal Bifurcation.- 2.5 Branch PointsComplex Bifurcations.- 2.6 Bifurcation Diagram.- 2.7 Transient Behavior of LPSNumerical Methods.- 2.8 Computation of Periodic Solutions.- 2.9 Chaotic Attractors.- 3. Multiplicity and Stability in Distributed-Parameter Systems (DPS).- 3.1 Steady-State SolutionsMethods for Solving Nonlinear Boundary-Value Problems.- 3.2 Dependence of Steady-State Solutions on a Parameter.- 3.3 Branch PointsMethods for Evaluating Real and Complex Bifurcation Points.- 3.4 Methods for Transient Simulation of Parabolic EquationsFinite-Difference Methods.- 4. Development of Quasi-stationary Patterns with Changing Parameter.- 4.1 Quasi-stationary Behavior in LPSExamples.- 4.2 Quasi-stationary Behavior in DPSExamples.- 5. Perspectives.- Appendix A DERPARA Continuation Algorithm.- Appendix B SHOOTAn 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.