An averaging result for periodic solutions of Carathéodory differential equations

Novaes, Douglas

doi:10.1090/proc/15810

An averaging result for periodic solutions of Carathéodory differential equations
HTML articles powered by AMS MathViewer

by Douglas D. Novaes PDF

Proc. Amer. Math. Soc. 150 (2022), 2945-2954 Request permission

Abstract:

This paper is concerned with the problem of existence of periodic solutions for perturbative Carathéodory differential equations. The main result provides sufficient conditions on the averaged equation that guarantee the existence of periodic solutions. Additional conditions are also provided to ensure the uniform convergence of a periodic solution to a constant function. The proof of the main theorem is mainly based on an abstract continuation result for operator equations.

References

N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Translated from the second revised Russian edition, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, Inc., New York, 1961. MR 0141845
N. Bogolyubov, O Nekotoryh Statističeskih Metodah v Matematičeskoĭ Fizike, Akademiya Nauk Ukrainskoĭ SSR, 1945 (Russian). MR 0016575
Adriana Buică and Jaume Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math. 128 (2004), no. 1, 7–22. MR 2033097, DOI 10.1016/j.bulsci.2003.09.002
Anna Capietto, Jean Mawhin, and Fabio Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992), no. 1, 41–72. MR 1042285, DOI 10.1090/S0002-9947-1992-1042285-7
José Ángel Cid, Jean Mawhin, and Mirosława Zima, An abstract averaging method with applications to differential equations, J. Differential Equations 274 (2021), 231–250. MR 4186657, DOI 10.1016/j.jde.2020.11.051
P. Fatou, Sur le mouvement d’un système soumis à des forces à courte période, Bull. Soc. Math. France 56 (1928), 98–139 (French). MR 1504928, DOI 10.24033/bsmf.1131
Robert E. Gaines and Jean L. Mawhin, Coincidence degree, and nonlinear differential equations, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin-New York, 1977. MR 0637067, DOI 10.1007/BFb0089537
Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
N. Krylov and N. Bogolyubov, Prilozhenie metodov nelineinoi mekhaniki k teorii statsionarnykh kolebanii (The Application of Methods of Nonlinear Mechanics to the Theory of Stationary oscillations). Kiev: Akademiya Nauk Ukrainskoĭ SSR, Kiev, 1934.
Jaroslav Kurzweil, Ordinary differential equations, Studies in Applied Mechanics, vol. 13, Elsevier Scientific Publishing Co., Amsterdam, 1986. Introduction to the theory of ordinary differential equations in the real domain; Translated from the Czech by Michal Basch. MR 929466
Jaume Llibre, Ana C. Mereu, and Douglas D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations 258 (2015), no. 11, 4007–4032. MR 3322990, DOI 10.1016/j.jde.2015.01.022
Jaume Llibre, Douglas D. Novaes, and Camila A. B. Rodrigues, Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones, Phys. D 353/354 (2017), 1–10. MR 3669353, DOI 10.1016/j.physd.2017.05.003
Jaume Llibre, Douglas D. Novaes, and Marco A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity 27 (2014), no. 3, 563–583. MR 3177572, DOI 10.1088/0951-7715/27/3/563
Jaume Llibre, Douglas D. Novaes, and Marco A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math. 139 (2015), no. 3, 229–244. MR 3336682, DOI 10.1016/j.bulsci.2014.08.011
Jean Mawhin, Degré topologique et solutions périodiques des systèmes différentiels non linéaires, Bull. Soc. Roy. Sci. Liège 38 (1969), 308–398 (French). MR 594965
J. Mawhin, Équations intégrales et solutions périodiques des systèmes différentiels non linéaires, Acad. Roy. Belg. Bull. Cl. Sci. (5) 55 (1969), 934–947 (French, with English summary). MR 265691
J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, Providence, R.I., 1979. Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, 1977. MR 525202, DOI 10.1090/cbms/040
Yu. A. Mitropolskii and Nguyen Van Dao, Applied asymptotic methods in nonlinear oscillations, Solid Mechanics and its Applications, vol. 55, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1490834, DOI 10.1007/978-94-015-8847-8
Douglas D. Novaes and Francisco B. Silva, Higher order analysis on the existence of periodic solutions in continuous differential equations via degree theory, SIAM J. Math. Anal. 53 (2021), no. 2, 2476–2490. MR 4249059, DOI 10.1137/20M1346705
J. A. Sanders, F. Verhulst, and J. Murdock, Averaging methods in nonlinear dynamical systems, 2nd ed., Applied Mathematical Sciences, vol. 59, Springer, New York, 2007. MR 2316999
Ferdinand Verhulst, Nonlinear differential equations and dynamical systems, Universitext, Springer-Verlag, Berlin, 1990. Translated from the Dutch. MR 1036522, DOI 10.1007/978-3-642-97149-5
Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732, DOI 10.1007/978-1-4612-4838-5