## 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 = P_{2}(x, y) ≡ a_{00}+ a_{10}x + a_{01}y + a_{20}x^{2}+ a_{11}xy + a_{02}y^{2}dy(t)/dt = Q_{2}(x, y) ≡ b_{00}+ b_{10}x + b_{01}y + b_{20}x^{2}+ b_{11}xy + b_{02}y^{2}where (x, y) ∈ ℝ^{2}t ∈ ℝ, a_{ij}b_{ij}∈ 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 language | English |
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Article number | 32 |

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Electronic Journal of Qualitative Theory of Differential Equations |

Volume | 2018 |

DOIs | |

Publication status | Published - 2018 |

## Keywords

- Algebraic curve
- Limit cycle
- Ordinary differential equations