Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks

Abstract

We study projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, where projective-anticipating and projective-lag synchronization can be achieved only on two coupled chaotic systems. In this paper, we realize projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In this paper, the dynamics of the nodes of the complex networks are time-delayed chaotic systems without the limitation of the partial linearity. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its existence and sufficient stability conditions. The validity of the proposed method is demonstrated and verified by examining specific examples using Ikeda and Mackey-Glass systems on Erdos-Renyi networks.

Projective synchronization between two different time-delayed chaotic systems using active control approach

“…We use the fourth-order Adam's interpolation method integrating the master system (12) and the response system (14 ) with the length of the integrating Figure 3a shows the time series of the master system (12) x(t) (solid line) and the response system (14 ) y(t) (dash line) for α = 2.5. One can find that the phase angle between the synchronized trajectories is zero.…”

“…In applications to secure communications, this feature can be used to extend binary digital to M-nary digital communication [6] for achieving fast communication. Projective synchronization of identical chaotic systems has been extremely investigated in recent years, including finite dimensional systems [4,[7][8][9], infinite dimensional systems [10][11][12], and complex networks [13,14]. But in practical situations, it is hardly the case that every component can be assumed to be identical.…”

“…Numerical simulations of(12) and(14 ) for α = 2.5: a the time series of the master system x(t) (solid line) and the response system y(t) (dash line); b the synchronization between x and yFig. 4 Numerical simulations of (12) and (14 ) for α = 0.5: a the time series of the master system x(t) (solid line) and the response system y(t) (dash line); b the synchronization between x and y step h = 0.01 in this paper.…”

Projective synchronization in multiple modulated time-delayed systems with adaptive scaling factor

“…While the concept of projective synchronization has been well established in low-dimensional systems [8,9,11,12], only a very few studies have been done in time delay systems [8,9,13,18] in detail. In particular, the mechanism of onset of projective synchronization with an adaptive scaling factor in modulated multiple time-delayed systems has not been yet clearly understood and requires urgent attention.…”

Active backstepping control of combined projective synchronization among different nonlinear systems

“…Synchronization means two or more systems adjust each other to give rise to a common dynamical behaviour. As a key technique of secure communication, chaos synchronization has been extensively studied in recent decades and different notations have been proposed and studied, such as complete synchronization [7][8][9], generalized synchronization [10,11], phase synchronization [12,13], anti-phase synchronization [14][15][16] and projective synchronization [17][18][19][20].…”