Quadratic systems with a symmetrical solution

A. Zegeling, Robert E. Kooij

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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.

Original languageEnglish
Article number32
Pages (from-to)1-18
Number of pages18
JournalElectronic Journal of Qualitative Theory of Differential Equations
Publication statusPublished - 2018


  • Algebraic curve
  • Limit cycle
  • Ordinary differential equations

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