Description
Book SynopsisThis book aims to present a new approach called Flow Curvature Method that applies Differential Geometry to Dynamical Systems. Hence, for a trajectory curve, an integral of any n-dimensional dynamical system as a curve in Euclidean n-space, the curvature of the trajectory — or the flow — may be analytically computed. Then, the location of the points where the curvature of the flow vanishes defines a manifold called flow curvature manifold. Such a manifold being defined from the time derivatives of the velocity vector field, contains information about the dynamics of the system, hence identifying the main features of the system such as fixed points and their stability, local bifurcations of codimension one, center manifold equation, normal forms, linear invariant manifolds (straight lines, planes, hyperplanes).In the case of singularly perturbed systems or slow-fast dynamical systems, the flow curvature manifold directly provides the slow invariant manifold analytical equation associated with such systems. Also, starting from the flow curvature manifold, it will be demonstrated how to find again the corresponding dynamical system, thus solving the inverse problem.
Table of ContentsDifferential Equations; Hartman-Grobman Theorem; Liapounoff Stability Theorem; Phase Portraits; Poincare-Bendixson Theorem; Attractors; Strange Attractors; Hamiltonian and Integrable Systems; K A M Theorem; Invariant Sets; Global/Local Invariance; Center Manifold Theorem; Normal Form Theorem; Local Bifurcations of Codimension 1; Hopf Bifurcation, Slow-Fast Dynamical Systems; Geometric Singular Perturbation Theory; Darboux Theory of Integrability; Differential Geometry; Generalized Frenet Moving Frame; Curvatures of Trajectory Curves; Flow Curvature Manifold; Flow Curvature Method; Van der Pol Model; FitzHugh-Nagumo Model; Pikovskii-Rabinovich-Trakhtengerts Model; Rikitake Model; Chua's Model; Lorenz Model.
