On the bound of the Lyapunov exponents for the fractional differential systems

Abstract

In recent years, fractional(-order) differential equations have attracted increasing interests due to their applications in modeling anomalous diffusion, time dependent materials and processes with long range dependence, allometric scaling laws, and complex networks. Although an autonomous system cannot define a dynamical system in the sense of semigroup because of the memory property determined by the fractional derivative, we can still use the Lyapunov exponents to discuss its dynamical evolution. In this paper, we first define the Lyapunov exponents for fractional differential systems then estimate the bound of the corresponding Lyapunov exponents. For linear fractional differential system, the bounds of its Lyapunov exponents are conveniently derived which can be regarded as an example for the theoretical results established in this paper. Numerical example is also included which supports the theoretical analysis.

On fractional lyapunov exponent for solutions of linear fractional differential equations

“…This estimate also is used to obtain an upper bound on the Lyapunov exponent in [14]. However, for a sake of completeness we give a short proof of this estimate.…”

“…For linear fractional differential equations, the Lyapunov exponent is first discussed in [14] in which the authors use the asymptotic behavior (in comparison with the exponential function) of the nonsingular eigenvalues of the fundamental matrix to define the Lyapunov spectrum. This notion of the Lyapunov spectrum is used to investigate the chaotic behavior in a class of fractional differential systems, see e.g.…”

Hyperchaotic Chameleon: Fractional Order FPGA Implementation

“…Lyapunov exponents (LEs) are necessary and more convenient for detecting hyperchaos in fractional order hyperchaotic system. A definition of LEs for fractional differential systems was given in [40] based on frequencydomain approximations, but the limitations of frequencydomain approximations are highlighted by Tavazoei and Haeri [41]. Time series based LEs calculation methods like Wolf algorithm [11], Jacobian method [12], and neural network algorithm [13] are popularly known ways of calculating Lyapunov exponents for integer and fractional order systems.…”

Two-particle anomalous diffusion: probability density functions and self-similar stochastic processes

“…Two noteworthy examples of PDFs for anomalous diffusion are obtained, despite the effective microscopic stochastic formulation, by subordination-type integral (2.1) or (2.3) of the Markovian BM, with PDF 5) and variance x 2 = 2t, by selecting as the PDF of the directing process either a unilateral M-Wright function, as introduced in the so-called generalized grey BM (ggBM) [13][14][15], also called Erdélyi-Kober fractional diffusion [16,17], or an extremal Lévy stable density, as considered in Mainardi et al [18][19][20]. In fact, in the former case, for 0 < β ≤ 1 and 0 < α < 2, it holds [16] that…”

“…Fractional systems have been investigated in their dynamic evolution by Lyapunov exponent analysis [5] as well as in their ability to encode memory in nuclear magnetic resonance phenomena [6,7].…”