Geometry of cycles in quadratic systems
Copyright policy )Geometry of cycles in quadratic systems
The paper is devoted to study of the affine and Euclidean differential geometric properties of cycles occurring in the phase plane of a quadratic system of differential equations. It is shown that every quadratic cycle has exactly six affine vertices and the number of Euclidean vertices can vary, not just from cycle to cycle, but for the same cycle under a linear coordinate transformation. The authors show that an upper bound for the number of Euclidean vertices over all noncircular quadratic cycles is twelve and that a sharp upper bound is six. Moreover, the article contains a general argument that yields the uniform bound \(2(6n-2)(7n-3)\) on the number of vertices and \(2(21n-9)(22n-10)\) on the number of affine vertices for any cycle appearing in the phase plane of a polynomial system of degree \(n\).
- University of North Carolina at Chapel Hill United States
- University of North Carolina at Charlotte United States
Geometric methods in ordinary differential equations, cycle, curvature, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations
Geometric methods in ordinary differential equations, cycle, curvature, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations
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