Rigorous Enclosures of a Slow Manifold

Rigorous Enclosures of Slow Manifolds

Slow-fast dynamical systems have two time scales and an explicit parameter representing the ratio of these time scales. Locally invariant slow manifolds along which motion occurs on the slow time scale are a prominent feature of slow-fast systems. This paper introduces a rigorous numerical method to compute enclosures of the slow manifold of a slow-fast system with one fast and two slow variabl...

Asymptotics of a Slow Manifold

Approximately invariant elliptic slow manifolds are constructed for the Lorenz– Krishnamurthy model of fast-slow interactions in the atmosphere. As is the case for many other twotime-scale systems, the various asymptotic procedures that may be used for this construction diverge, and there are no exactly invariant slow manifolds. Valuable information can however be gained by capturing the detail...

Rigorous Enclosures of Ellipsoids and Directed Cholesky Factorizations

This paper discusses the rigorous enclosure of an ellipsoid by a rectangular box, its interval hull, providing a convenient preprocessing step for constrained optimization problems. A quadratic inequality constraint with a positive definite Hessian defines an ellipsoid. The Cholesky factorization can be used to transform a strictly convex quadratic constraint into a norm inequality, for which t...

Slow Manifold of a Neuronal Bursting Model

Slow invariant manifold of heartbeat model

A new approach called Flow Curvature Method has been recently developed in a book entitled Differential Geometry Applied to Dynamical Systems. It consists in considering the trajectory curve, integral of any n-dimensional dynamical system as a curve in Euclidean n-space that enables to analytically compute the curvature of the trajectory or the flow. Hence, it has been stated on the one hand th...