Averaging methods for piecewise-smooth ordinary differential equations

Michal Fečkan; Július Pačuta; Michal Pospíśil; Pavol Vidlička; Michal Fečkan; Július Pačuta; Michal Pospíśil; Pavol Vidlička

doi:10.3934/math.2019.5.1466

AIMS Mathematics

2019, Volume 4, Issue 5: 1466-1487. doi: 10.3934/math.2019.5.1466

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Averaging methods for piecewise-smooth ordinary differential equations

1.
Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia
2.
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia

Received: 11 July 2019 Accepted: 30 August 2019 Published: 23 September 2019
MSC : 34A36, 34C29

The averaging method is developed for periodic piecewise-smooth systems. We discuss the behavior of solutions intersecting the discontinuity boundary and the problems it introduces. We illustrate these difficulties on specific examples. In the case of transversal and sliding solutions, we introduce conditions that allow us to prove averaging theorems for piecewise-smooth periodic differential equations.
- piecewise-smooth differential equations,
- averaging method,
- Filippov regularization
Citation: Michal Fečkan, Július Pačuta, Michal Pospíśil, Pavol Vidlička. Averaging methods for piecewise-smooth ordinary differential equations[J]. AIMS Mathematics, 2019, 4(5): 1466-1487. doi: 10.3934/math.2019.5.1466

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Abstract

The averaging method is developed for periodic piecewise-smooth systems. We discuss the behavior of solutions intersecting the discontinuity boundary and the problems it introduces. We illustrate these difficulties on specific examples. In the case of transversal and sliding solutions, we introduce conditions that allow us to prove averaging theorems for piecewise-smooth periodic differential equations.

References

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[8]	J. A. Sanders, F. Verhulst, J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, 2 Eds, New York: Springer, 2007.

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