### Periodic solutions of periodically perturbed planar autonomous systems: a topological approach

#### Abstract

The aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle $x_0$ of least period $T_0>0$ when it is perturbed by a small parameter, $T_1-$periodic perturbation. In the case when $T_0/T_1$ is a rational number $l/k$, with $l, k$ prime numbers, we provide conditions to guarantee, for the parameter perturbation $\varepsilon>0$ sufficiently small, the existence of $klT_0-$ periodic solutions $x_\varepsilon$ of the perturbed system which converge to the trajectory $\tilde x_0$ of the limit cycle as $\varepsilon\to 0$. Moreover, we state conditions under which $T=klT_0$ is the least period of the periodic solutions $x_\varepsilon$. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when $T_0/T_1$ is an irrational number we show the nonexistence, whenever $T>0$ and $\varepsilon>0$, of $T-$periodic solutions $x_\varepsilon$ of the perturbed system converging to $\tilde x_0$. The employed methods are based on the topological degree.

#### Article information

Source
Adv. Differential Equations Volume 11, Number 4 (2006), 399-418.

Dates
First available in Project Euclid: 18 December 2012

Mathematical Reviews number (MathSciNet)
MR2215621

Zentralblatt MATH identifier
1107.34035

Subjects
Primary: 34C25: Periodic solutions
Secondary: 34D10: Perturbations

#### Citation

Kamenskii, Mikhail; Makarenkov, Oleg; Nistri, Paolo. Periodic solutions of periodically perturbed planar autonomous systems: a topological approach. Adv. Differential Equations 11 (2006), no. 4, 399--418. https://projecteuclid.org/euclid.ade/1355867702.