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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

Oleg Makarenkov 1, and  Paolo Nistri 2,

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.
Keywords: rate of convergence., Nondegenerate limit cycles, perturbed autonomous systems, periodic solutions.
Mathematics Subject Classification: 34E10, 37G15; Secondary: 34C2.
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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