Dimension reduction of dynamical system is a significant issue for technical applications, as regards both finite dimensional system and infinite dimensional systems emerging from either science or engineering. Center manifold method is one of the main reduction methods for ordinary differential systems (ODSs). Does there exists a similar method for fractional ODSs (FODSs)? In other words, does there exists a method for reducing the high-dimensional FODS into a lower-dimensional FODS? In this study, we establish a local fractional center manifold for a finite dimensional FODS. Several examples are given to illustrate the theoretical analysis.
Issue Section:
Research Papers
References
1.
Torvik
, P. J.
, and Bagley
, R. L.
, 1984
, “On the Appearance of the Fractional Derivative in the Behavior of Real Materials
,” ASME J. Appl. Mech.
, 51
(2
), pp. 294
–298
.2.
Riewe
, F.
, 1997
, “Mechanics With Fractional Derivatives
,” Phys. Rev. E
, 55
(3
), pp. 3582
–3592
.3.
Atanackovi
, T. M.
, and Stankovic
, B.
, 2004
, “Stability of an Elastic Rod on a Fractional Derivative Type of Foundation
,” J. Sound Vib.
, 277
(1
), pp. 149
–161
.4.
Rossikhin
, Y. A.
, and Shitikova
, M. V.
, 2010
, “Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results
,” ASME Appl. Mech. Rev.
, 63
(1
), p. 010801
.5.
Li
, C. P.
, and Zeng
, F. H.
, 2015
, Numerical Methods for Fractional Calculus
, Chapman and Hall/CRC
, New York
.6.
Monje
, C. A.
, Chen
, Y. Q.
, Vinagre
, B. M.
, Xue
, D. Y.
, and Feliu
, V.
, 2010
, Fractional-Order Systems and Controls: Fundamentals and Applications
, Springer-Verlag
, London
.7.
Margin
, R. L.
, 2006
, Fractional Calculus in Bioengineering
, Begell House
, New York
.8.
Cong
, N. D.
, Doan
, T. S.
, Siegmund
, S.
, and Tuan
, H. T.
, 2014
, “On Stable Manifolds for Planar Fractional Differential Equations
,” Appl. Math. Comput.
, 226
(1
), pp. 157
–168
.9.
Li
, C. P.
, Gong
, Z. Q.
, Qian
, D. L.
, and Chen
, Y. Q.
, 2010
, “On the Bound of the Lyapunov Exponents for the Fractional Differential Systems
,” Chaos
, 20
(1
), p. 013127
.10.
Rega
, G.
, and Troger
, H.
, 2005
, “Dimension Reduction of Dynamical Systems: Methods, Models, Applications
,” Nonlinear Dyn.
, 41
(1–3
), pp. 1
–15
.11.
Chow
, S. N.
, and Hale
, J. K.
, 1982
, Methods of Bifurcation Theory
, Springer-Verlag
, New York
, pp. 312
–322
.12.
Rémi
, V.
, and Andrei
, L. S.
, 1993
, Asymptotic Methods in Mechanics
, American Mathematical Society
, Providence, RI
, pp. 109
–120
.13.
Pliss
, V. A.
, 1964
, “A Reduction Principle in the Theory of Stability of Motion
,” J. Izv. Akad. Nauk SSSR Ser. Mat.
, 28
(6
), pp. 1297
–1324
.14.
Kelley
, A.
, 1967
, “The Stable, Center Stable, Center, Center Unstable and Unstable Manifolds
,” J. Differ. Equations
, 3
(4
), pp. 546
–570
.15.
Carr
, J.
, 1981
, Applications of Centre Manifold Theory
, Springer-Verlag
, New York
, pp. 1
–36
.16.
Guckenheimer
, J.
, and Holmes
, P.
, 1983
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
, Springer-Verlag
, Berlin, Germany
.17.
Vanderbauwhede
, A.
, 1989
, “Center Manifolds, Normal Forms and Elementary Bifurcations
,” Dynamics Reported
, U.
Kirchgraber
and H. O.
Walther
, eds., Vieweg+Teubner Verlag
, Wiesbaden, pp. 89
–169
.18.
Jolly
, M. S.
, and Rosa
, R.
, 2005
, “Computation of Non-Smooth Local Centre Manifolds
,” IMA J. Numer. Anal.
, 25
(4
), pp. 698
–725
.19.
Charles
, L.
, and Wiggins
, S.
, 1997
, Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrödinger Equations
, Springer-Verlag
, New York
, pp. 4
–12
.20.
Gerhard
, D.
, Bernold
, F.
, Klaus
, K.
, and Alexander
, M.
, 1996
, Dynamics of Nonlinear Waves in Dissipative Systems Reduction, Bifurcation and Stability
, Chapman and Hall/CRC
, London
.21.
Haken
, H.
, 2004
, Synergetics: Introduction and Advanced Topics
, Springer
, Berlin, Germany
.22.
Meyer
, C. D.
, 2000
, Matrix Analysis and Applied Linear Algebra
, Society for Industrial and Applied Mathematics
, Philadelphia, PA
.23.
Kilbas
, A. A.
, Srivastava
, H. M.
, and Trujillo
, J. J.
, 2006
, Theory and Applications of Fractional Differential Equations
, Elsevier Science
, Amsterdam, The Netherlands
.24.
Gorenflo
, R.
, Kilbas
, A. A.
, Mainardi
, F.
, and Rogosin
, S. V.
, 2014
, Mittag–Leffler Functions, Related Topics and Applications
, Springer-Verlag
, Berlin, Heidelberg, Germany
.25.
Gorenflo
, R.
, Loutchko
, J.
, and Luchko
, Y.
, 2002
, “Computation of the Mittag-Leffler Function Eα,β and Its Derivative
,” Fractional Calculus Appl. Anal.
, 5
(4
), pp. 491
–518
.26.
Sandor
, J.
, 2005
, “A Note on Certain Inequalities for the Gamma Function
,” J. Inequalities Pure Appl. Math.
, 6
(3
), p. 61
.27.
Samko
, S. G.
, Kilbas
, A.
, and Marichev
, O. I.
, 1993
, Fractional Integrals and Derivatives: Theory and Applications
, Gordon and Breach Science
, Yverdon, Switzerland
.28.
Dixon
, J.
, and Mckee
, S.
, 1986
, “Weakly Singular Discrete Gronwall Inequalities
,” ZAMM-J. Appl. Math. Mech.
, 66
(11
), pp. 535
–544
.29.
Li
, C. P.
, and Ma
, Y. T.
, 2013
, “Fractional Dynamical System and Its Linearization Theorem
,” Nonlinear Dyn.
, 71
(4
), pp. 621
–633
.30.
Kilbas
, A. A.
, and Marzan
, S. A.
, 2005
, “Nonlinear Differential Equations With the Caputo Fractional Derivative in the Space of Continuously Differentiable Functions
,” Diff. Equations
, 41
(1
), pp. 84
–89
.31.
Coddington
, E. A.
, and Levinson
, N.
, 1955
, Theory of Ordinary Differential Equations
, McGraw-Hill
, New York
, pp. 330
–335
.Copyright © 2016 by ASME
You do not currently have access to this content.
