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More Compactification for Differential Systems

Author(s): Harry Gingold | Daniel Solomon

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 03;
Issue: 01;
Start page: 190;
Date: 2013;
Original page

Keywords: Nonlinear; Polynomial; Compactification; Ultra Extended Euclidean Space; Critical Point; Equilibrium Point; Critical Point at Infinity; Critical Direction at Infinity; Basin of Divergence; Basin of Convergence; Ideal Solutions; Asymptotic; Stability; Global; Globally Asymptotically Stable; Jacobian; Painleve Analysis | Competing Species; Model; Lorenz Equations; Periodic Surface; Differential Geometry; Attractor; Repeller

This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.

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