The study of nonlinear dynamics is a fascinating question which is at the very heart of the understanding of many important problems of the natural sciences. Two of the oldest and most notable classes of problems in nonlinear dynamics are the problems of celestial mechanics, especially the study of the motion of bodies in the solar system, and the problems of turbulence in fluids. Both phenomena have attracted the interest of scientists for a long time; they are easy to observe, and lead to the formation and development of complicated patterns that we would like to understand. The first class of problems are of finite dimensions, the latter problems have infinite dimensions, the dimensions here being the number of parameters which is necessary to describe the configuration of the system at a given instant of time. Besides these problems, whose observation is accessible to the layman as well as to the scientist, there is now a broad range of nonlinear turbulent phenomena (of either finite or infinite dimensions) which have emerged from recent developments in science and technology, such as chemical dynamics, plasma physics and lasers, nonlinear optics, combustion, mathematical economy, robotics, . . . . In contrast to linear systems, the evolution of nonlinear systems obeys complicated laws that, in general, cannot be arrived at by pure intuition or by elementary calculations.
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ISBN-13
978-1-4684-0313-8 (9781468403138)
DOI
10.1007/978-1-4684-0313-8
Schweitzer Klassifikation
General Introduction. The User's Guide.- 1. Mechanism and Description of Chaos. The Finite-Dimensional Case.- 2. Mechanism and Description of Chaos. The Infinite-Dimensional Case.- 3. The Global Attractor. Reduction to Finite Dimension.- 4. Remarks on the Computational Aspect.- 5. The User's Guide.- I General Results and Concepts on Invariant Sets and Attractors.- 1. Semigroups, Invariant Sets, and Attractors.- 2. Examples in Ordinary Differential Equations.- 3. Fractal Interpolation and Attractors.- II Elements of Functional Analysis.- 1. Function Spaces.- 2. Linear Operators.- 3. Linear Evolution Equations of the First Order in Time.- 4. Linear Evolution Equations of the Second Order in Time.- III Attractors of the Dissipative Evolution Equation of the First Order in Time: Reaction-Diffusion Equations. Fluid Mechanics and Pattern Formation Equations.- 1. Reaction-Diffusion quations.- 2. Navier-Stokes Equations (n = 2).- 3. Other Equations in Fluid Mechanics.- 4. Some Pattern Formation Equations.- 5. Semilinear Equations.- 6. Backward Uniqueness.- IV Attractors of Dissipative Wave Equations.- 1. Linear Equations: Summary and Additional Results.- 2. The Sine-Gordon Equation.- 3. A Nonlinear Wave Equation of Relativistic Quantum Mechanics.- 4. An Abstract Wave Equation.- 5. A Nonlinear SchrÖdinger Equation.- 6. Regularity of Attractors.- 7. Stability of Attractors.- V Lyapunov Exponents and Dimension of Attractors.- 1. Linear and Multilinear Algebra.- 2. Lyapunov Exponents and Lyapunov Numbers.- 3. Hausdorff and Fractal Dimensions of Attractors.- VI Explicit Bounds on the Number of Degrees of Freedom and the Dimension of Attractors of Some Physical Systems.- 1. The Lorenz Attractor.- 2. Reaction-Diffusion quations.- 3. Navier-Stokes Equations (n =2).- 4. OtherEquations in Fluid Mechanics.- 5. Pattern Formation quations.- 6. Dissipative Wave quations.- 7. A Nonlinear chrÖdinger Equation.- 8. Differentiability of the emigroup.- VII Non-Well-Posed Problems, Unstable Manifolds, Lyapunov Functions, and Lower Bounds on Dimensions.- A: NON-WELL-POSED PROBLEMS.- 1. Dissipativity and Well Posedness.- 2. Estimate of Dimension for Non-Well-Posed Problems: Examples in Fluid Dynamics.- B: UNSTABLE MANIFOLDS, LYAPUNOV FUNCTIONS, AND LOWER BOUNDS ON DIMENSIONS.- 3. Stable and Unstable Manifolds.- 4. The Attractor of a Semigroup with a Lyapunov Function.- 5. Lower Bounds on imensions of Attractors: An Example.- VIII The Cone and Squeezing Properties. Inertial Manifolds.- 1. The Cone Property.- 2. Construction of an Inertial Manifold: Description of the Method.- 3. Existence of an Inertial Manifold.- 4. Examples.- 5. Approximation and Stability of the Inertial Manifold with Respect to Perturbations.- APPENDIX Collective Sobolev Inequalities.- 1. Notations and Hypotheses.- 1.1. The Operator 31.- 1.2. The SchrÖdinger-Type Operators.- 2. Spectral Estimates for SchrÖdinger-Type Operators.- 2.1. The Birman-Schwinger Inequality.- 2.2. The Spectral Estimate.- 3. Generalization of the Sobolev-Lieb-Thirring Inequality (I).- 4. Generalization of the Sobolev-Lieb-Thirring Inequality (II).- 4.1. The Space-Periodic Case.- 4.2. The General Case.- 4.3. Proof of Theorem 4.1.- 5. Examples.
