Journal of Advances in Applied Mathematics

Download PDF (264.9 KB) PP. 37 - 44 Pub. Date: April 1, 2019

DOI: 10.22606/jaam.2019.42001

• Le Huy Tien¹, Nguyen Minh Man^{2
  1}Department of Mathematics, Mechanics and Informatics, Vietnam National University at Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam
• Le Duc Nhien^{1*
  2}Faculty of Basic Sciences, Ha Noi University of Mining and Geology, 18 Pho Vien, Ha Noi, Viet Nam

This paper is concerned with the existence of invariant manifolds for dynamical equations on a periodic time scale when the nonlinear perturbation has a small global Lipschitz constant. Particularly, for time-varying non-regressive dynamical equations, which have exponential dichotomies on a periodic time scale with bounded graininess, we use the method of graph transforms as in [1] to prove that there exists a unique integral manifold of that systems.

Integral manifold, graph transforms, time scales, linear dynamic equation.

[1] N. V. Minh and J. Wu, “Invariant manifolds of partial functional differential equations,” J. Differential Equations, vol. 198, no. 2, pp. 381–421, 2004.

[2] W. A. Coppel, “Stability and Asymptotic Behavior of Differential Equations,” Heath Math. Monogr., Heath & Co., Boston, 1965.

[3] W. A. Coppel, “Dichotomies in Stability Theory,” Lecture Notes in Math., Springer-Verlag, vol. 629, 1978.

[4] K. J. Palmer, “Exponential Dichotomies and Fredholm,” Proceedings of the American Mathematical Society, vol. 104, no. 1, pp. 149–156, 1988.

[5] K. J. Palmer, “Exponential Dichotomies and Transversal Homoclinic Points,” J. Differential Equations, vol. 55, no. 2, pp. 225–256, 1984.

[6] K. J. Palmer, “A perturbation theorem for exponential dichotomies,” Proc. Roy. Soc. Edinburgh Sect. A, vol. 106, no. 1–2, pp. 25–37, 1987.

[7] K. J. Palmer, “Exponential dichotomies for almost periodic equations,” Proc. Amer. Math. Soc., vol. 101, no. 2, pp. 293–298, 1987.

[8] K. J. Palmer, “Shadowing in Dynamical Systems: Theory and Applications,” Kluwer Academic Publishers, Dordrecht, Boston, 2000.

[9] R. J. Sacker and G. R. Sell, “A spectral theory for linear differential systems,” J. Differential Equations, vol. 27, no. 3, pp. 320–358, 1978.

[10] R. J. Sacker and G. R. Sell, “Dichotomies for linear evolutionary equations in Banach spaces,” J. Differential Equations, vol. 113, no. 1, pp. 17–67, 1994.

[11] N. Fenichel, “Geometric singular perturbation theory for ordinary differential equations,” J. Differential Equations, vol. 31, no. 1, pp. 53–98, 1979.

[12] N. Fenichel, “Persistence and smoothness of invariant manifolds for flows,” Indiana Univ. Math. J., vol. 21, pp. 193–226, 1971.

[13] W. Beyn, “On well-posed problems for connecting orbits in dynamical systems,” Contemp. Math., vol. 172, pp. 131–168, 1994.

[14] W. J. Beyn, “The numerical computation of connecting orbits in dynamical systems,” IMA J. Numer. Anal., vol. 10, pp. 379–405, 1990.

[15] W. J. Beyn and J. Lorenz, “Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals,” Numer. Funct. Anal. Optim., vol. 20, pp. 201–244, 1999.

[16] U. Ascher, R. M. Mattheij and R. D. Russell, “Solution of Boundary Value Problems for ODEs,” Prentice Hall, Englewood Cliffs, NJ, 1988.

[17] H. W. Broer, H. M. Osinga and G. Vegter, “Algorithms for computing normally hyperbolic invariant manifolds,” Z. Angew. Math. Phys., vol. 48, pp. 480–524, 1997.

[18] L. Dieci and J. Lorenz, “Computation of invariant tori by the method of characteristics,” SIAM J. Numer. Anal., vol. 32, pp. 1436–1474, 1995.

[19] L. Barreira, D. Dragicevic and C. Valls, “Fredholm operators and nonuniform exponential dichotomies,” Chaos, Solitons and Fractals, vol. 85, pp. 120–127, 2016.

[20] L. Barreira, D. Dragicevic and C. Valls, “Nonuniform exponential dichotomies and Fredholm operators for flows,” Aequat. Math., vol. 91, no. 2, pp.301–316, 2017.

[21] S. Hilger, “Analysis on measure chains - a unified approach to continuous and discrete calculus,” Results Math., vol. 18, pp. 18–56, 1990.

[22] J. Zhang, M. Fan and H. Zhu, “Existence and roughness of exponential dichotomies of linear dynamic equations on time scales,” Comp. and Math. with Appl., vol. 59, pp. 2658–2675, 2009.

[23] J. Zhang, Y. Song and Z. Zhao, “General exponential dichotomies on time scales and parameter dependence of roughness,” Adva. Diff. Equations., 2013.

[24] L. Yang, J. Zhang, X. Chang and Z. Liu, “Exponential dichotomy on time scales and admissibility of the pair (Cb rd(T+,X),Lp(T+,X)),” Adv. Diff. Equations, 2015.

[25] M. Bohner and A. Peterson, “Dynamic equations on time scales: an introduction with applications,” Birkhäuser Boston, Inc., 2001.

[26] M. Bohner and A. Peterson, “Advances in dynamic equations on time scales,” Birkhäuser Boston, Inc., Boston, 2003.

[27] C. Pðtzsche, “Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients,” J. Anal. Math. Appl., vol 289, pp. 317–335, 2004.

[28] J. Zhang, M. Fan and H. Zhu, “Necessary and sufficient criteria for the existence of exponential dichotomy on time scales,” Comp. and Math. with Appl., vol. 60, pp. 2387–2398, 2010.

[29] E. R. Kaufmann and Y. N. Raffoul, “Periodic solutions for a neutral nonlinear dynamical equations on a time scales,” J. Math. Anal. Appl., vol. 319, no. 1, pp. 315–325, 2006.

[30] R. Martin, “Nonlinear Operators and Differential Equations in Banach Spaces,” Wiley Interscience, New York, 1976.

[31] Z. Nitecki, “An introduction to the orbit structure of diffeomorphisms,” MIT Press, Cambridge, MA, 1971.

[32] P. Duarte and S. Klein, “Lyapunov Exponents of Linear Cocycles,” Atlantis Press, 2016.