Synchronization in Tempered Fractional Complex Networks via Auxiliary System Approach

Abstract

In the famous continuous time random walk (CTRW) model, because of the finite lifetime of biological particles, it is sometimes necessary to temper the power law measure such that the waiting time measure has a convergent first moment. The CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. In this article, we introduce the tempered fractional derivative into complex networks to describe the finite life span or bounded physical space of nodes. Some properties of the tempered fractional derivative and tempered fractional systems are discussed. Generalized synchronization in two-layer tempered fractional complex networks via pinning control is addressed based on the auxiliary system approach. The results of the proposed theory are used to derive a sufficient condition for achieving generalized synchronization of tempered fractional networks. Numerical simulations are presented to illustrate the effectiveness of the methods.

Partial topology identification of tempered fractional‐order complex networks via synchronization method

“…Consider the following tempered fractional nonautonomous system: where x (0) ∈ ℝ n , α ∈ (0,1) λ ≥ 0, f ( t , x ) : [0, +∞) × Ω → ℝ n is piecewise continuous in t and locally Lipschitz in x , and domain Ω ∈ ℝ n contains the origin x = 0.Definition If in tempered fractional system , the constant x 0 is an equilibrium point.Lemma (Boyd et al 27 ). If x , y ∈ ℝ n , and P ∈ ℝ n × n is a positive definite matrix, then Lemma (Ma et al 28 ). If x ( t ) ∈ ℝ n is a continuously differentiable vector function, then where t ≥ 0, 0 < α < 1, λ ≥ 0, and Q is a symmetric positive definite matrix.Lemma (Li et al 29 ).…”

Fractional-order delayed Ross–Macdonald model for malaria transmission

“…Fractional calculus, as an extension of traditional calculus, has attracted remarkable attention in recent years since it can better depict memory and hereditary properties of various materials and processes [6]. Various studies have already demonstrated that a fractional-order model better depicted the dynamic behavior of system than the integer-order counterpart [7,8]. Most biological systems have memory and hereditary attributes.…”

Tempered Mittag–Leffler Stability of Tempered Fractional Dynamical Systems

“…Motivated by this, we think it is very necessary and meaningful to study Mittag-Leffler stability of tempered fractional dynamical systems both in theoretical research and practical application. Because tempered fractional operators combine with nonlocal, weak singularity, and exponential factors [31][32][33], it has many differences to fractional case in stability analysis. In this paper, tempered Mittag-Leffler stability is first proposed.…”