Local stable manifold of Langevin differential equations with two fractional derivatives

Jinrong Wang,D O’Regan,Shan Peng

doi:10.1186/s13662-017-1389-6

Jinrong Wang, D O’Regan + Show 1 more

Open Access

https://doi.org/10.1186/s13662-017-1389-6

Copy DOI

Abstract

In this paper, we investigate the existence of local center stable manifolds of Langevin differential equations with two Caputo fractional derivatives in the two-dimensional case. We adopt the idea of the existence of a local center stable manifold by considering a fixed point of a suitable Lyapunov-Perron operator. A local center stable manifold theorem is given after deriving some necessary integral estimates involving well-known Mittag-Leffler functions.

Highlights

1 Introduction Fractional calculus was introduced by Liouville and Riemann
Fractional calculus is a rapidly growing area with many applications in diverse fields ranging from physical sciences, engineering to biological sciences and economics
In this paper, motivated by [, ], we study local center stable manifolds of fractional Langevin equations of the type:

Summary

Introduction

Fractional calculus was introduced by Liouville and Riemann. The concept of non-integer calculus is a generalization of the traditional integer-order calculus that was mentioned by Leibniz and L’Hospital. In [ ] the authors gave a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations by considering the Lyapunov-Perron operator via the asymptotic behavior of the Mittag-Leffler function. In [ ] a local center manifold result for fractional ordinary differential equations is given. Where cD μ,t and cD ν,t denote the Caputo fractional derivative of order μ, ν ∈ ( , ) with the lower limit zero Where RLDγ ,t denotes the Riemann-Liouville derivative of order γ with the lower limit zero for a function. In Section , we give some fundamental estimates related to Mittag-Leffler functions, and in Section , we present the main result of this paper concerning center stable manifolds.

Proof Note that

If we choose a ι