×

On continuous approximation of discontinuous systems. (English) Zbl 1136.34302

In this paper by using Tikhonov’s theorem for singularly perturbed differential equations, the authors study the relationship between dynamics of discontinuous differential equations and their continuous approximations along periodic solutions.

MSC:

34A36 Discontinuous ordinary differential equations
34A60 Ordinary differential inclusions
34C25 Periodic solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Awrejcewicz, J.; Delfs, J., Dynamics of a self-excited stick-slip oscillator with two degrees of freedom, Part I: Investigation of equilibria, European J. Mech. A/Solids, 9, 4, 269-282 (1990) · Zbl 0712.70043
[2] Awrejcewicz, J.; Delfs, J., Dynamics of a self-excited stick-slip oscillator with two degrees of freedom, Part II: Slip-stick, slip-slip, stick-slip transitions, periodic and chaotic orbits, European J. Mech. A/Solids, 9, 5, 397-418 (1990) · Zbl 0732.70014
[3] Awrejcewicz, J.; Lamarque, C.-H., Bifurcation and Chaos in Nonsmooth Mechanical Systems (2003), World Scientific: World Scientific Singapore · Zbl 1067.70001
[4] Awrejcewicz, J.; Olejnik, P., Stick-slip dynamics of a two-degree-of-freedom system, Int. J. Bifurcat. Chaos, 13, 4, 843-861 (2003) · Zbl 1067.70021
[5] Brogliato, B., Nonsmooth Mechanics (1999), Springer: Springer London · Zbl 0917.73002
[6] Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations (1959), Springer: Springer Berlin · Zbl 0082.07602
[7] Deimling, K., Multivalued Differential Equations (1992), Walter de Gruyter: Walter de Gruyter Berlin · Zbl 0760.34002
[8] M. Farkas, Periodic motions, Applied Mathematical Sciences, vol. 104, Springer, New York, 1994.; M. Farkas, Periodic motions, Applied Mathematical Sciences, vol. 104, Springer, New York, 1994. · Zbl 0805.34037
[9] Filippov, A. F., Differential equations with discontinuous right-hand side, Mathematical Applications (1988), Kluwer: Kluwer Dordrecht · Zbl 0664.34001
[10] R.I. Leine, D.H. van Campen, B.L. van de Vrande, Bifurcations in nonlinear discontinuous systems, Nonlinear Dynam. 23 (2000) 105-164.; R.I. Leine, D.H. van Campen, B.L. van de Vrande, Bifurcations in nonlinear discontinuous systems, Nonlinear Dynam. 23 (2000) 105-164. · Zbl 0980.70018
[11] Vasileva, A. B.; Butuzov, V. F., Asymptotic Expansion of Solutions of Singularly Perturbed Equations (1973), Nauka: Nauka Moscow, (in Russian) · Zbl 0364.34028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.