A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS

doi:10.1007/s10473-024-0609-x

Abstract

Abstract: We consider the persistence of affine periodic solutions for perturbed affine periodic systems. Such $(Q,T)$-affine periodic solutions have the form $x(t+T)=Qx(t)$ for all $t\in\mathbf{R}$, where $T>0$ is fixed and $Q$ is a nonsingular matrix. These are a kind of spatiotemporal symmetric solutions, e.g. spiral waves. We give the averaging method for the existence of affine periodic solutions in two situations: one in which the initial values of the affine periodic solutions of the unperturbed system form a manifold, and another that does not rely on the structure of the initial values of the unperturbed system's affine periodic solutions. The transversal condition is determined using the Brouwer degree. We also present a higher order averaging method for general degenerate systems by means of the Brouwer degree and a Lyapunov-Schmidt reduction.

Key words: affine-periodic solution, perturbed system, Brouwer degree, averaging method

CLC Number:

34C29

Xue YANG, Jiamin XING, Yong LI. A GENERAL AVERAGING METHOD FOR AFFINE PERIODIC SOLUTIONS[J].Acta mathematica scientia,Series B, 2024, 44(6): 2207-2224.

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References

[1] Buică A, Llibre J.Averaging methods for finding periodic orbits via Brouwer degree. Bull Sci Math, 2004, 128: 7-22
[2] Buică A, Llibre J, Makarenkov O.Bifurcations from nondegenerate families of periodic solutions in Lipschitz systems. J Differential Equations, 2012, 252(6): 3899-3919
[3] Cândido M R, Llibre J, Novaes D D.Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction. Nonlinearity, 2017, 30: 3560-3586
[4] Chang X, Li Y.Rotating periodic solutions of second order dissipative dynamical systems. Discrete Contin Dyn Syst, 2016, 36: 643-652
[5] Cheng C, Huang F, Li Y.Affine-periodic solutions and pseudo affine-periodic solutions for differential equations with exponential dichotomy and exponential trichotomy. J Appl Anal Comput, 2016, 6: 950-967
[6] Ekeland I.Hamilton-Jacobi on the symplectic group. Rend Istit Mat Univ Trieste, 2017, 49: 137-146
[7] Evard J Cl, Jafari F.The set of all $m\times n$ rectangular real matrices of rank $r$ is connected by analytic regular arcs. Proc Amer Math Soc, 1994, 120: 413-419
[8] Giné J, Llibre J, Wu K, Zhang X.Averaging methods of arbitrary order, periodic solutions and integrability. J Differential Equations, 2016, 260: 4130-4156
[9] Hale J K. Ordinary Differential Equations.2nd ed. Huntington, NY: Robert E Krieger Publishing Co, Inc, 1980
[10] Han M, Sun H, Balanov Z.Upper estimates for the number of periodic solutions to multi-dimensional systems. J Differential Equations, 2019, 266: 8281-8293
[11] Hirsch M W. Differential Topology.Graduate Texts in Mathematics, No 33. New York, Heidelberg: Springer-Verlag, 1976
[12] Jiang X, Yang X, Li Y.Affine periodic solutions of stochastic differential equations. arXiv:1908.11499
[13] Krylov N, Bogolyubov N.Prilozhenie metodov nelineinoi mekhaniki k teorii statsionarnykh kolebanii (The Application of Methods of Nonlinear Mechanics to the Theory of Stationary oscillations). Kiev: Akademiya Nauk UkrainskoĭSSR, 1934
[14] Li Y, Huang F.Levinson's problem on affine-periodic solutions. Adv Nonlinear Stud, 2015, 15: 241-252
[15] Liang F, Han M, Jiang C.Limit cycle bifurcations of a planar near-integrable system with two small parameters. Acta Math Sci, 2021, 41B: 1034-1056
[16] Liu G, Li Y, Yang X.Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems. J Differential Equations, 2018, 265: 1324-1352
[17] Liu G, Li Y, Yang X.Infinitely many rotating periodic solutions for second-order Hamiltonian systems. J Dyn Control Syst, 2019, 25: 159-174
[18] Liu G, Li Y, Yang X.Rotating periodic solutions for asymptotically linear second-order Hamiltonian systems with resonance at infinity. Math Methods Appl Sci, 2017, 40: 7139-7150
[19] Liu G, Li Y, Yang X.Rotating periodic solutions for super-linear second order Hamiltonian systems. Appl Math Lett, 2018, 79: 73-79
[20] Liu S, Han M, Li J.Bifurcation methods of periodic orbits for piecewise smooth systems. J Differential Equations, 2021, 275: 204-233
[21] Llibre J, Novaes D D.Improving the averaging theory for computing periodic solutions of the differential equations. Z Angew Math Phys, 2015, 66(4): 1401-1412
[22] Llibre J, Novaes D D, Teixeira M A.Higher order averaging theory for finding periodic solutions via Brouwer degree. Nonlinearity, 2014, 27: 563-583
[23] Moser J.Is the solar system stable? Math Intelligencer, 1978, 1(2): 65-71
[24] Novaes D D, Silva F B.Higher order analysis on the existence of periodic solutions in continuous differential equations via degree theory. SIAM J Math Anal, 2021, 53: 2476-2490
[25] Palais B, Palais R.Euler's fixed point theorem: the axis of a rotation. J Fixed Point Theory Appl, 2007, 2(2): 215-220
[26] Rhouma M B H, Chicone C. On the continuation of periodic orbits. Methods Appl Anal, 2000, 7(1): 85-104
[27] Sanders J, Verhulst F, Murdock J.Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences. New York: Springer, 2007
[28] Verhulst F.Nonlinear Differential Equations and Dynamical Systems. New York: Springer Science & Business Media, 2006
[29] Wang C, Yang X, Li Y.Affine-periodic solutions for nonlinear differential equation. Rocky Mountain J Math, 2016, 46: 1717-1737
[30] Wang H, Yang X, Li Y.Rotating-symmetric solutions for nonlinear systems with symmetry. Acta Math Appl Sin Engl Ser, 2015, 31: 307-312
[31] Wang S, Yang X, Li Y. The mechanism of rotating waves in a ring of unidirectionally coupled Lorenz systems. Commun Nonlinear Sci Numer Simul, 2020, 90: Art 105370
[32] Xing J, Yang X, Li Y.Affine-periodic solutions by averaging methods. Sci China Math, 2018, 61: 439-452
[33] Xing J, Yang X, Li Y.Lyapunov center theorem on rotating periodic orbits for Hamiltonian systems. J Differential Equations, 2023, 363: 170-194
[34] Xing J, Yang X, Li Y.Rotating periodic solutions for convex Hamiltonian systems. App Math Lett, 2019, 89: 91-96
[35] Xu F, Yang X.Affine-periodic solutions by asymptotic method. J Dyn Control Syst, 2021, 27: 271-281
[36] Xu F, Yang X, Li Y, Liu M.Existence of affine-periodic solutions to Newton affine-periodic systems. J Dyn Control Syst, 2019, 25: 1-19
[37] Yang X, Zhang Y, Li Y.Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions. Rocky Mountain J Math, 2017, 47: 2423-2438
[38] Zhang Y, Yang X, Li Y. Affine-periodic solutions for dissipative systems. Abstr Appl Anal, 2013, 2013: Art 157140