Proceedings of the American Mathematical Society

Stability of periodic solutions for Lipschitz systems obtained via the averaging method

Author(s): Adriana Buica; Aris Daniilidis
Journal: Proc. Amer. Math. Soc. 135 (2007), 3317-3327.
MSC (2000): Primary 34C29, 34C25; Secondary 49J52
Posted: May 17, 2007
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Abstract: Existence and asymptotic stability of the periodic solutions of the Lipschitz system $x^{\prime}(t)=\varepsilon F(t,x,\varepsilon )$ is hereby studied via the averaging method. The traditional $C^{1}$ dependence of $F(s,\cdot,\varepsilon)$ on

is relaxed to the mere strict differentiability of $F(s,\cdot,0)$ at $z=z_{0}$ for $\varepsilon=0$ , giving room to potential applications for structured nonsmooth systems.

1.: D. Aussel, A. Daniilidis & L. Thibault, Subsmooth sets: functional characterizations and related concepts, Trans. Amer. Math. Soc. 357 (2005), 1275-1301. MR 2115366
2.: J. Bolte, A. Daniilidis & A. Lewis, The $\L$ ojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim. 17 (2006), 1205-1223. MR 2274510
3.: A. Buica & J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math. 128 (2004), 7-22. MR 2033097 (2004j:34102)
4.: A. Buica & J. Llibre, Bifurcation of limit cycles from a -dimensional center in control systems, International Journal of Bifurcation and Chaos 15 (2005), 2653-2662. MR 2174572 (2006e:34070)
5.: F. Clarke, Optimization and nonsmooth analysis, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983. MR 709590 (85m:49002)
6.: J. Guckenheimer & P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, Springer-Verlag New York, Heidelberg, Berlin, 1983. MR 709768 (85f:58002)
7.: J. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Co., 1980. MR 587488 (82e:34001)
8.: Y. Hartono & A.H.P. van der Burgh, Higher-order averaging: periodic solutions, linear systems and an application, Nonlinear Analysis 52 (2003), 1727-1744. MR 1956172 (2003m:34105)
9.: J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conference Series in Mathematics 40, AMS, Providence, R.I., 1979. MR 525202 (80c:47055)
10.: R.T. M'Closkey, An averaging theorem for time-periodic degree zero homogeneous differential equations, Systems Control Lett. 32 (1999), 179-183. MR 1492438 (99b:34089)
11.: B. Mordukhovich, Variational analysis and generalized differentiation. (Vol. I & II) Grundlehren der Mathematischen Wissenschaften 331, Springer-Verlag, Berlin, 2006.
12.: T. Rockafellar & R. Wets, Variational analysis, Grundlehren der Mathematischen Wissenschaften 317, Springer-Verlag, Berlin, 1998. MR 1491362 (98m:49001)
13.: J. Sanders & F. Verhulst, Averaging methods in nonlinear dynamical systems, Appl. Math. Sci. 59, Springer, 1985. MR 810620 (87d:34065)
14.: S. Schecter, Stable manifolds in the method of averaging, Trans. Amer. Math. Soc. 308 (1988), 159-176. MR 946437 (89i:34061)
15.: A.R. Teel, J. Peuteman & D. Aeyels, Semi-global practical asymptotic stability and averaging, Systems Control Lett. 37 (1999), 329-334. MR 1753260 (2001b:34098)

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Adriana Buica
Affiliation: Department of Applied Mathematics, Babes-Bolyai University, Cluj-Napoca 400084, Romania
Email: abuica@math.ubbcluj.ro

Aris Daniilidis
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra 08193, Spain
Email: arisd@mat.uab.es

DOI: 10.1090/S0002-9939-07-08929-0
PII: S 0002-9939(07)08929-0
Keywords: Periodic solution, averaging method, nonsmooth Lipschitz system, Poincar\'{e}--Andronov mapping, fixed point
Received by editor(s): August 1, 2006
Posted: May 17, 2007
Additional Notes: The first author was supported by the ``Agence universitaire de la Francophonie'' (France)
The second author was supported by the MEC Grant No. MTM2005-08572-C03-03 (Spain)
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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