# American Institute of Mathematical Sciences

January  2008, 7(1): 49-61. doi: 10.3934/cpaa.2008.7.49

## On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes

 1 Research Institute of Mathematics, Voronezh State University, Ul. Universitetskaja pl. 1, 394006, Voronezh, Russian Federation 2 Dipartimento di Ingegneria dell' Informazione, Università di Siena, Via Roma 56, 53100, Siena, Italy

Received  January 2007 Revised  June 2007 Published  October 2007

We consider an autonomous system in $\mathbb R^n$ having a limit cycle $x_0$ of period $T>0$ which is nondegenerate in a suitable sense, (see Definition 2.1). We then consider the perturbed system obtained by adding to the autonomous system a $T$-periodic, not necessarily differentiable, term whose amplitude tends to $0$ as a small parameter $\varepsilon>0$ tends to $0.$ Assuming the existence of a $T$-periodic solution $x_\varepsilon$ of the perturbed system and its convergence to $x_0$ as $\varepsilon\to 0$, the paper establishes the existence of $\Delta_\varepsilon\to 0$ as $\varepsilon\to 0$ such that $||x_\varepsilon(t+\Delta_\varepsilon)-x_0(t)||\le\varepsilon M$ for some $M>0$ and any $\varepsilon>0$ sufficiently small. This paper completes the work initiated by the authors in [4] and [11]. Indeed, in [4] the existence of a family of $T$-periodic solutions $x_\varepsilon$ of the perturbed system considered here was proved. While in [11] for perturbed systems in $\mathbb R^2$ the rate of convergence was investigated by means of the method considered in this paper.
Citation: Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure & Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49
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