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Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. This book provides the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom).

Hamiltonian systems. --- Degree of freedom. --- Arnold’s paper. --- Hamiltonian system. --- KAM theorem. --- action variables. --- autonomous Hamiltonian system. --- celestial mechanics. --- conservation of action variables. --- instability of dynamical systems. --- integrable Hamiltonian systems. --- linearly stable. --- magnetic fields. --- motion of charged particles. --- negligible friction. --- non integrable. --- perturb. --- several degrees of freedom. --- stable solution.

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This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art."

Schrödinger operator. --- Green's functions. --- Hamiltonian systems. --- Evolution equations. --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Functions, Green's --- Functions, Induction --- Functions, Source --- Green functions --- Induction functions --- Source functions --- Operator, Schrödinger --- Differential equations --- Differentiable dynamical systems --- Potential theory (Mathematics) --- Differential operators --- Quantum theory --- Schrödinger equation --- Almost Mathieu operator. --- Analytic function. --- Anderson localization. --- Betti number. --- Cartan's theorem. --- Chaos theory. --- Density of states. --- Dimension (vector space). --- Diophantine equation. --- Dynamical system. --- Equation. --- Existential quantification. --- Fundamental matrix (linear differential equation). --- Green's function. --- Hamiltonian system. --- Hermitian adjoint. --- Infimum and supremum. --- Iterative method. --- Jacobi operator. --- Linear equation. --- Linear map. --- Linearization. --- Monodromy matrix. --- Non-perturbative. --- Nonlinear system. --- Normal mode. --- Parameter space. --- Parameter. --- Parametrization. --- Partial differential equation. --- Periodic boundary conditions. --- Phase space. --- Phase transition. --- Polynomial. --- Renormalization. --- Self-adjoint. --- Semialgebraic set. --- Special case. --- Statistical significance. --- Subharmonic function. --- Summation. --- Theorem. --- Theory. --- Transfer matrix. --- Transversality (mathematics). --- Trigonometric functions. --- Trigonometric polynomial. --- Uniformization theorem.

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The description for this book, An Introduction to Linear Transformations in Hilbert Space. (AM-4), Volume 4, will be forthcoming.

Transformations (Mathematics) --- Hyperspace. --- Additive function. --- Adjoint. --- Affine space. --- Axiom. --- Banach space. --- Boundary value problem. --- Cartesian coordinate system. --- Characteristic function (probability theory). --- Combination. --- Commutative property. --- Compact space. --- Contradiction. --- Coordinate system. --- Corollary. --- Countable set. --- Differential operator. --- Dimension (vector space). --- Effective method. --- Equation. --- Existential quantification. --- Fourier transform. --- Function (mathematics). --- Group algebra. --- Hamiltonian mechanics. --- Hamiltonian system. --- Hilbert space. --- Integral transform. --- Jacobi matrix. --- Lebesgue integration. --- Limit point. --- Linear combination. --- Linear map. --- Linear space (geometry). --- Mathematics. --- Measure (mathematics). --- Metric space. --- Moment problem. --- Mutual exclusivity. --- Necessity and sufficiency. --- Normal operator. --- Operational calculus. --- Orthonormality. --- Rational number. --- Self-adjoint operator. --- Self-adjoint. --- Statistic. --- Step function. --- Subset. --- Summation. --- System of linear equations. --- Theorem. --- Topological group. --- Topology. --- Unit sphere. --- Unitary transformation. --- Vector space. --- Weak convergence (Hilbert space). --- Weak convergence.

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The description for this book, Contributions to the Theory of Nonlinear Oscillations (AM-41), Volume IV, will be forthcoming.

Nonlinear oscillations. --- Algebraic curve. --- Analytic continuation. --- Analytic function. --- Asymptotic analysis. --- Banach space. --- Big O notation. --- Boundary value problem. --- Calculation. --- Canonical transformation. --- Cartesian coordinate system. --- Change of variables. --- Characteristic exponent. --- Coefficient. --- Computation. --- Conic section. --- Continuous function. --- Convex set. --- Counterexample. --- Curvature. --- Curve. --- Degrees of freedom (statistics). --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differential equation. --- Dimension. --- Dimensional analysis. --- Division by zero. --- Eigenvalues and eigenvectors. --- Elementary proof. --- Equation. --- Essential singularity. --- Existence theorem. --- Existential quantification. --- Exterior (topology). --- Fixed-point theorem. --- Forcing (mathematics). --- Forcing (recursion theory). --- Function space. --- Functional equation. --- Hamiltonian system. --- Hyperplane. --- Inflection point. --- Initial condition. --- Initial value problem. --- Integral equation. --- Inverse function. --- Iteration. --- Lagrangian mechanics. --- Lefschetz fixed-point theorem. --- Limit cycle. --- Limit of a sequence. --- Limit point. --- Limit set. --- Line segment. --- Linearity. --- Line–line intersection. --- Lipschitz continuity. --- Lyapunov stability. --- Mathematical optimization. --- Mathematics. --- Monotonic function. --- Newton polygon. --- Nonlinear system. --- Orbital stability. --- Ordinary differential equation. --- Ordinate. --- Parameter. --- Parametrization. --- Parity (mathematics). --- Partial derivative. --- Periodic function. --- Periodic point. --- Perturbation theory (quantum mechanics). --- Phase space. --- Power series. --- Principal part. --- Proportionality (mathematics). --- Quadratic. --- Real variable. --- Scalar (physics). --- Scientific notation. --- Sign (mathematics). --- Significant figures. --- Solomon Lefschetz. --- Special case. --- Sturm–Liouville theory. --- Subset. --- Surface of revolution. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Transversal (geometry). --- Unification (computer science). --- Upper half-plane. --- Variable (mathematics). --- Variational method (quantum mechanics). --- Vector field. --- Vector notation. --- Zero of a function.

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The leading scientists who gave these papers under the sponsorship of the Royal Society in early 1987 provide reviews of facets of the subject of chaos ranging from the practical aspects of mirror machines for fusion power to the pure mathematics of geodesics on surfaces of negative curvature. The papers deal with systems in which chaotic conditions arise from initial value problems with unique solutions, as opposed to those where chaos is produced by the introduction of noise from an external source. Table of Contents Diagnosis of Dynamical Systems with Fluctuating Parameters D. Ruelle Nonlinear Dynamics, Chaos, and Complex Cardiac Arrhythmias L. Glass, A. L. Goldberger, M. Courtemanche, and A. Shrier Chaos and the Dynamics of Biological Populations R. M. May Fractal Bifurcation Sets, Renormalization Strange Sets, and Their Universal Invariants D. A. Rand From Chaos to Turbulence in Bnard Convection A. Libchaber Dynamics of Convection N. O. Weiss Chaos: A Mixed Metaphor for Turbulence E. A. Spiegel Arithmetical Theory of Anosov Diffeomorphisms F. Vivaldi Chaotic Behavior in the Solar System J. Wisdom Chaos in Hamiltonian Systems I. C. Percival Semi-Classical Quantization, Adiabatic Invariants, and Classical Chaos W. P. Reinhardt and I. Dana Particle Confinement and Adiabatic Invariance B. V. Chirikov Some Geometrical Models of Chaotic Dynamics C. Series The Bakerian Lecture: Quantum Chaology M. V. BerryOriginally published in 1989.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Chaotic behavior in systems --- Accuracy and precision. --- Action potential. --- Adiabatic invariant. --- Adiabatic theorem. --- Amplitude. --- Approximation. --- Attractor. --- Belousov–Zhabotinsky reaction. --- Bifurcation diagram. --- Bifurcation theory. --- Big O notation. --- Boolean function. --- Boundary value problem. --- Calculation. --- Cardiac arrhythmia. --- Chaos theory. --- Characteristic exponent. --- Classical limit. --- Computation. --- Convection. --- Correlation dimension. --- Degrees of freedom (statistics). --- Diagram (category theory). --- Differential equation. --- Dimension. --- Dirac delta function. --- Dynamical system. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Equation. --- Equations of motion. --- Estimation. --- Exponential growth. --- Factorization. --- Fractal dimension. --- Hamiltonian mechanics. --- Hamiltonian system. --- Heteroclinic bifurcation. --- Homoclinic bifurcation. --- Homoclinic orbit. --- Hopf bifurcation. --- Information dimension. --- Initial condition. --- Instability. --- Integrable system. --- Internal heating. --- Kirkwood gap. --- Libration. --- Limit cycle. --- Lorenz system. --- Mathematics. --- Non-Euclidean geometry. --- Nonlinear resonance. --- Nonlinear system. --- Orbital eccentricity. --- Orbital resonance. --- Orrery. --- Parameter. --- Parasystole. --- Partial differential equation. --- Perturbation theory (quantum mechanics). --- Phase space. --- Pitchfork bifurcation. --- Plane wave. --- Poisson point process. --- Power series. --- Probability. --- Proportionality (mathematics). --- Quantum chaos. --- Quantum mechanics. --- Quasiperiodic motion. --- Radius of convergence. --- Rate of convergence. --- Rayleigh number. --- Regime. --- Renormalization. --- Resonance. --- Rotation around a fixed axis. --- Rotation number. --- Saddle-node bifurcation. --- Scattering. --- Separatrix (mathematics). --- Sinus rhythm. --- Soliton. --- Special case. --- Stable manifold. --- Standard map. --- Statistic. --- Statistical mechanics. --- Stochastic. --- Symbolic dynamics. --- Test particle. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Tidal locking. --- Time evolution. --- Two-dimensional space. --- Universality class. --- Winding number.