Our aim in this paper is to establish stable manifolds near hyperbolic equilibria of fractional differential equations in arbitrary finite-dimensional spaces.
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Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models. J. Math. Anal. Appl. 325(1), 542–553 (2007)
Baleanu, D., Mustafa, O.G., Agarwal, R.P.: Asymptotic integration of \((1 + \alpha )\)-order fractional differential equations. Comput. Math. Appl. 62, 1492–1500 (2011)
Baleanu, D., Agarwal, R.P., Mustafa, O.G., Cosulschi, M.: Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A: Math. Theor 44, 055203 (2011). (9 pp.)
Bonilla, B., Rivero, M., Trujillo, J.J.: On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187, 68–78 (2007)
Cong, N.D., Doan, T.S., Tuan, H.T.: On fractional Lyapunov exponent for solutions of linear fractional differential equations. Fract. Calc. Appl. Anal. 17(2), 285–306 (2014)
Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifolds for planar fractional differential equations. Appl. Math. Comput. 226, 157–168 (2014)
Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: Structure of the fractional Lyapunov spectrum for linear fractional differential equations. Adv. Dyn. Syst. Appl. 9, 149–159 (2014)
Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifold for fractional differential equations in high dimensional space. Preprint IMH2015/03/01, Institute of Mathematics, Vietnam Academy of Science and Technology, Mar 2015. http://math.ac.vn/en/component/staff/?task=showPrePrint&year=2015
Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: Linearized asymptotic stability for fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 39, 1–13 (2016)
Deshpande, A., Daftardar-Gejji, V.: Local stable manifold theorem for fractional systems. Nonlinear Dyn. 83, 2435–2452 (2016)
Deng, W.: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. 72(3–4), 1768–1777 (2010)
Diethelm, K.: The analysis of fractional differential equations. In: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)
Lin, W.: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)
Li, C.P., Gong, Z., Qian, D., Chen, Y.: On the bound of the Lyapunov exponents for the fractional differential systems. Chaos 20(1), 013127 (2010). (7 pages)
Li, C.P., Ma, Y.: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71(4), 621–633 (2013)
Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193(1), 27–47 (2011)
Li, K., Peng, J.: Laplace transform and fractional differential equations. Appl. Math. Lett. 24, 2019–2023 (2011)
Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Autom. J. IFAC 45(8), 1965–1969 (2009)
Ma, L., Li, C.: Center manifold of fractional dynamical system. J. Comput. Nonlinear Dynam. 11(2), 021010 (2015). (6 pp)
Matignon, D.: Stability results for fractional differential equations with applications to control processing. Comput Eng. Syst. Appl. 2, 963–968 (1996)
Nieto, J.J.: Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl. Math. Lett. 23, 1248–1251 (2010)
Odibat, Z.M.: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171–1183 (2010)
Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego (1999)
Shilov, G.E.: Linear Algebra. Dover Publications Inc., New York (1977)
This research of the N. D. Cong, T. S. Doan, and H. T. Tuan is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.03-2014.42.
An erratum to this article can be found at http://dx.doi.org/10.1007/s11071-016-3039-z.
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Cong, N.D., Doan, T.S., Siegmund, S. et al. On stable manifolds for fractional differential equations in high-dimensional spaces. Nonlinear Dyn 86, 1885–1894 (2016). https://doi.org/10.1007/s11071-016-3002-z
- Stable manifold
- Fractional differential equation
- Lyapunov stability