Controlling chaos

Nonlinear stability of fluid and plasma equilibria

Stability in dynamical systems subject to some law of force is considered. This leads to a set of differential equations which govern the motion. (AIP)

Stability in dynamical systems

We prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. The proof is based on a combination of normal form theory and rigorous numerical computations.

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The Lorenz attractor exists

The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to an improved long-time behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Stormer/Verlet method. It thus presents a cross-section of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton-Stormer-Verlet-leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent long-time behaviour of the method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, preservation of adiabatic invariants.

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Geometric numerical integration illustrated by the Störmer-Verlet method

The Three-Body Problem: Covarinace of Lagarangian Derivatives.- Canonical Transformation.- The Hamilton-Jacobi Equation.- The Cauchy-Existence Theorem.- The n-Body Poblem.- Collision.- The Regularizing Transformation.- Application to the Three-Bdy Problem.- An Estimate of the Perimeter.- An Estimate of the Velocity.- Sundman's Theorem.- Triple Collision.- Triple-Collision Orbits.- Periodic Solutions: The Solutions of Lagrange.- Eigenvalues.- An Existence Theorem.- The Convergence Proof.- An Application to the Solution of Lagrange.- Hill's Problem.- A Generalization of Hill's Problem.- The Continuation Method.- The Fixed-Point Theorem.- Area-Preserving Analytic Transformations.- The Birkhoff Fixed-Point Theorem.- Stability: The Function-Theoretic Center Problem.- The Convergence Proof.- The Poincare Center Problem.- The Theorem of Liapunov.- The Theorem of Dirichlet.- The Normal Form of Hamiltonian Systems.- Area-Preserving Transformations.- Existence of Invariant Curves.- Proof of Lemma.- Application to the Stability Problem.- Stability of Equilibrium Solutions.- Quasi-Periodic Motion and Systems of Several Degrees of Freedom.- The Recurrence Theorem.

C. I. Kalme

Papers

Lectures on Celestial Mechanics