Abstract
In this paper we study the existence and uniqueness of limit cycles for so-called quadratic systems with a symmetrical solution: dx(t)/dt = P2(x, y) ≡ a00+ a10x + a01y + a20x2+ a11xy + a02y2dy(t)/dt = Q2(x, y) ≡ b00+ b10x + b01y + b20x2+ b11xy + b02y2where (x, y) ∈ ℝ2t ∈ ℝ, aijbij∈ ℝ, i.e. a real planar system of autonomous ordinary differential equations with linear and quadratic terms in the two independent variables. We prove that a quadratic system with a solution symmetrical with respect to a line can be of two types only. Either the solution is an algebraic curve of degree at most 3 or all solutions of the quadratic system are symmetrical with respect to this line. For completeness we give a new proof of the uniqueness of limit cycles for quadratic systems with a cubic algebraic invariant, a result previously only available in Chinese literature. Together with known results about quadratic systems with algebraic invariants of degree 2 and lower, this implies the main result of this paper, i.e. that quadratic systems with a symmetrical solution have at most one limit cycle which if it exists is hyperbolic.
Highlights
We will study the existence and uniqueness of limit cycles for so-called quadratic systems with a symmetrical solution: +a10 x a01y a20 x2a11 xy a02y2 dy(t) dt b00b10 x b01y b20 x2 b11 xy b02y2 (1.1)A
We prove that a quadratic system with a solution symmetrical with respect to a line can be of two types only
We provide the main result about the structure of systems (1.2) with a symmetrical solution in two steps
Summary
We will study the existence and uniqueness of limit cycles for so-called quadratic systems with a symmetrical solution:. A solution is symmetrical with respect to this line when for a solution curve (x(t), y(t)) the reflected curve (x(t), −y(t)) is a solution as well or when the reflected curve with reversed time traversal is a solution (x(−t), −y(−t)) These two distinct types are covered by writing the system of differential equations without explicit time-parametrization: dy dx. In principle there could be solutions which are symmetrical with respect to a line, where there more than two y-values for a given x These cases will automatically be included in our study of the restricted form y2 = φ(x) if we just take one branch of this multivalued function and ignore possible other symmetrical branches. A simple example shows that this case can occur proving the sharp upper bound on the degree of the algebraic curve
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