Poincaré index and periodic solutions of perturbed autonomous systems

Makarenkov, O.

doi:10.1090/S0077-1554-09-00176-9

Poincaré index and periodic solutions of perturbed autonomous systems
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by O. Yu. Makarenkov
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2009, 1-30

DOI: https://doi.org/10.1090/S0077-1554-09-00176-9

Published electronically: December 3, 2009

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Abstract:

Classical conditions for the bifurcation of periodic solutions in perturbed auto-oscillating and conservative systems go back to Malkin and Mel’nikov, respectively. These authors’ papers were based on the Lyapunov–Schmidt reduction and the implicit function theorem, which lead to the requirement that both the cycles and the zeros of the bifurcation functions be simple. In this paper a geometric approach is put forward which does not assume these requirements, but imposes a certain condition on the Poincaré index of a generating cycle with respect to some auxiliary vector field. The approach is based on calculating the topological degree of the Poincaré operator of the perturbed system with respect to interior and exterior neighbourhoods of a generating cycle, as a consequence of which the conclusion of the main theorem guarantees bifurcation of a certain number of periodic solutions towards the interior of the cycle, and of a certain number of periodic solutions towards the exterior of the cycle. Concrete examples are given, where this approach either establishes bifurcation of a greater number of periodic solutions compared with the known classical results, or provides additional information on the location of these solutions.

References

P. S. Alexandrov, Combinatorial topology. Vol. 1, 2 and 3, Dover Publications, Inc., Mineola, NY, 1998. Translated from the Russian; Reprint of the 1956, 1957 and 1960 translations. MR 1643155
A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maĭer, Qualitative theory of second-order dynamic systems, Halsted Press [John Wiley & Sons], New York-Toronto; Israel Program for Scientific Translations, Jerusalem-London, 1973. Translated from the Russian by D. Louvish. MR 0350126
K. Borsuk, Drei Sätze über die $n$-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177–190.
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, A continuation theorem for periodic boundary value problems with oscillatory nonlinearities, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 2, 133–163. MR 1328574, DOI 10.1007/BF01295308
Carmen Chicone and Marc Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989), no. 2, 433–486. MR 930075, DOI 10.1090/S0002-9947-1989-0930075-2
S.-N. Chow and J. A. Sanders, On the number of critical points of the period, J. Differential Equations 64 (1986), no. 1, 51–66. MR 849664, DOI 10.1016/0022-0396(86)90071-9
Lubomir Gavrilov, Remark on the number of critical points of the period, J. Differential Equations 101 (1993), no. 1, 58–65. MR 1199482, DOI 10.1006/jdeq.1993.1004
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original. MR 1139515
B. P. Demidovich, Lectures on mathematical stability theory, Moscow University, Moscow, 1998. (Russian)
Mikhail Kamenskii, Oleg Makarenkov, and Paolo Nistri, A continuation principle for a class of periodically perturbed autonomous systems, Math. Nachr. 281 (2008), no. 1, 42–61. MR 2376467, DOI 10.1002/mana.200610586
Mikhail Kamenskii, Valeri Obukhovskii, and Pietro Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter & Co., Berlin, 2001. MR 1831201, DOI 10.1515/9783110870893
Mikhail Kamenskii, Oleg Makarenkov, and Paolo Nistri, Small parameter perturbations of nonlinear periodic systems, Nonlinearity 17 (2004), no. 1, 193–205. MR 2023439, DOI 10.1088/0951-7715/17/1/012
Mikhail Kamenski, Oleg Makarenkov, and Paolo Nistri, Periodic solutions for a class of singularly perturbed systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 11 (2004), no. 1, 41–55. MR 2033323
M. I. Kamenskiĭ, O. Yu. Makarenkov, and P. Nistri, An approach to the theory of ordinary differential equations with a small parameter, Dokl. Akad. Nauk 388 (2003), no. 4, 439–442 (Russian). MR 2004140
M. A. Krasnosel′skiĭ and P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 263, Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. MR 736839, DOI 10.1007/978-3-642-69409-7
M. A. Krasnosel′skiĭ, Operator sdviga po traktoriyam differentsial′nykh uravneniĭ, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0203159
M. A. Krasnosel’skiĭ, A. I. Perov, A. I. Povolotskiĭ, and P. P. Zabreĭko, Vector fields on the plane, Fizmatgiz, Moscow, 1963. (Russian)
A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32 (1990), no. 4, 537–578. MR 1084570, DOI 10.1137/1032120
Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78 (French). MR 1509338
Solomon Lefschetz, Differential equations: geometric theory, Pure and Applied Mathematics, Vol. VI, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0094488
Oleg Makarenkov and Paolo Nistri, Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations, J. Math. Anal. Appl. 338 (2008), no. 2, 1401–1417. MR 2386507, DOI 10.1016/j.jmaa.2007.05.086
O. Yu. Makarenkov, On one approach in the theory of ordinary differential equations with a small parameter, Diploma thesis, Math. Fac., Voronezh Univ., 2003. (Russian)
I. G. Malkin, On Poincaré’s theory of periodic solutions, Akad. Nauk SSSR. Prikl. Mat. Meh. 13 (1949), 633–646 (Russian). MR 0036384
J. Mawhin, Le problème des solutions périodiques en mécanique non linéaire, Thèse de doctorat en sciences, Université de Liège, 1969.
Jean Mawhin, Degré topologique et solutions périodiques des systèmes différentiels non linéaires, Bull. Soc. Roy. Sci. Liège 38 (1969), 308–398 (French). MR 594965
V. K. Mel′nikov, On the stability of a center for time-periodic perturbations, Trudy Moskov. Mat. Obšč. 12 (1963), 3–52 (Russian). MR 0156048
Yu. A. Mitropol′skiĭ, Metod usredneniya v nelineĭnoĭ mekhanike, “Naukova Dumka”, Kiev, 1971 (Russian). MR 0664945
Yu. A. Mitropol′skiĭ, On periodic solutions of systems of nonlinear differential equations with non-differentiable right-hand sides, Ukrain. Mat. Ž. 11 (1959), 366–379 (Russian, with English summary). MR 0141846
Albert D. Morozov, Degenerate resonances in Hamiltonian systems with $3/2$ degress of freedom, Chaos 12 (2002), no. 3, 539–548. MR 1939451, DOI 10.1063/1.1484275
A. D. Morozov, The complete qualitative investigation of Duffing’s equation, Differencial′nye Uravnenija 12 (1976), no. 2, 241–255, 378 (Russian). MR 0409973
A. D. Morozov and L. P. Shil′nikov, On nonconservative periodic systems close to two-dimensional Hamiltonian, Prikl. Mat. Mekh. 47 (1983), no. 3, 385–394 (Russian); English transl., J. Appl. Math. Mech. 47 (1983), no. 3, 327–334 (1984). MR 760165, DOI 10.1016/0021-8928(83)90058-8
Fumio Nakajima and George Seifert, The number of periodic solutions of $2$-dimensional periodic systems, J. Differential Equations 49 (1983), no. 3, 430–440. MR 715695, DOI 10.1016/0022-0396(83)90005-0
Rafael Ortega, A criterion for asymptotic stability based on topological degree, World Congress of Nonlinear Analysts ’92, Vol. I–IV (Tampa, FL, 1992) de Gruyter, Berlin, 1996, pp. 383–394. MR 1389089
Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
Oskar Perron, Die Ordnungszahlen linearer Differentialgleichungssysteme, Math. Z. 31 (1930), no. 1, 748–766 (German). MR 1545146, DOI 10.1007/BF01246445
L. S. Pontryagin, Ordinary differential equations, ADIWES International Series in Mathematics, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1962. Translated from the Russian by Leonas Kacinskas and Walter B. Counts. MR 0140742
H. Poincaré, Sur les courbes définies par les équations différentielles. I, II, Jordan J. (4) I (1885), 167–244; ibid. II (1886) 151–211.
M. B. H. Rhouma and C. Chicone, On the continuation of periodic orbits, Methods Appl. Anal. 7 (2000), no. 1, 85–104. MR 1796007, DOI 10.4310/MAA.2000.v7.n1.a5
A. M. Samoĭlenko, On periodic solutions of differential equations with nondifferentiable right-hand sides, Ukrain. Mat. Ž. 15 (1963), 328–332 (Russian). MR 0159990
Russell A. Smith, Some applications of Hausdorff dimension inequalities for ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), no. 3-4, 235–259. MR 877904, DOI 10.1017/S030821050001920X
Kazuyuki Yagasaki, The Melnikov theory for subharmonics and their bifurcations in forced oscillations, SIAM J. Appl. Math. 56 (1996), no. 6, 1720–1765. MR 1417478, DOI 10.1137/S0036139995281317
Yulin Zhao, On the monotonicity of the period function of a quadratic system, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 795–810. MR 2153144, DOI 10.3934/dcds.2005.13.795