Scientific Computation

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 Poincare 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.