Projective synchronization of time-varying delayed neural network with adaptive scaling factors

doi:10.1016/j.chaos.2013.04.007

Chaos, Solitons & Fractals

Volume 53, August 2013, Pages 1-9

https://doi.org/10.1016/j.chaos.2013.04.007 Get rights and content

Highlights

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Projective synchronization in coupled delayed neural chaotic systems with modulated delay time is introduced.
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An adaptive rule for the scaling factors is introduced.
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This scheme is highly applicable in secure communication.

Abstract

In this work, the projective synchronization between two continuous time delayed neural systems with time varying delay is investigated. A sufficient condition for synchronization for the coupled systems with modulated delay is presented analytically with the help of the Krasovskii–Lyapunov approach. The effect of adaptive scaling factors on synchronization are also studied in details. Numerical simulations verify the effectiveness of the analytic results.

Introduction

The synchronization phenomenon has received great attention in recent years, in particular for the potential technological applications in all areas of engineering, science and management. Several theoretical works and laboratory experimentations have also demonstrated the pivotal role of this phenomenon in secure communications [1], [2]. Different types of synchronization techniques have been proposed in the literatures, e.g., complete synchronization (CS) [1], [3], phase synchronization (PS) [4], lag synchronization (LS) [5], generalized synchronization (GS) [6], generalized projective synchronization (GPS) [7], multiplexing synchronization (MS) [8], etc.

Gonzalez-Miranda [9] observed that when chaotic systems exhibit invariance properties under a special type of continuous transformation, amplification and displacement of the attractor is occurred. This degree of amplification or displacement obtained is smoothly dependent on the initial condition. By this definition complete synchronization is not a special case of projective synchronization. Later Mainieri and Rehacek [10] in 1999, this type of synchronization is called as projective synchronization, where they declared that the two partially linear three-dimensional chaotic systems could be synchronized up to a scaling factor α. The scaling factor is a constant transformation between the synchronized variables of the master and slave systems. Projective synchronization is not in the category of GS because the slave system of projective synchronization is not asymptotically stable. The response system attractor possesses the “same topological characteristic (such as Lyapunov exponents and fractal dimensions)” as the slave system attractor [11]. Recently, Hoang and Nakagawa [12] proposed a new type of synchronization in multi-delay feedback systems, which is called projective-anticipating synchronization. This synchronization is a combination of projective and anticipatory synchronization and the states of master and slave are related by ay(t) = bx(t + τ) (τ > 0) where a and b are nonzero real numbers. Projective-lag synchronization are defined as ay(t) = bx(t − τ) (τ > 0).

Projective synchronization is very important for its proportionality between the synchronized dynamical states. There has been a lot of interest on projective synchronization with time delayed systems due to its potential applications in encryption. The inherent delay can generate high order chaos which is effective in terms of security. Also the scaling factor/function can transfer a binary digital to M-nary, which is very useful for faster communications. With the development of research on complex systems, more and more researchers carried out the study about complex dynamical behaviors on networks. While most studies focused on complete and phase synchronization in various networks, little attention has been paid to projective synchronization. Very recently there has been some developments on the generalized analytical conditions [13], [14] on projective synchronization of time delayed systems.

In this paper we have investigated the projective synchronization phenomenon in two neural time delayed systems with delay time modulation. The linear coupling yielding synchronization between two systems can be considered as a combination of a positive-delayed feedback of the drive variables X on Y and of a negative feedback of the driven variables Y onto itself. By exploiting the analogy with electronic circuits this could be thought as meta model of hierarchical signal transfer between neural structures. The idea based on the Lyapunov stability, we theoretically analyze both the existence and sufficient stability conditions of the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. In this study, we extend the study of projective synchronization in infinite-dimensional chaotic system, while most of the studies [9], [10] are done in low dimensional systems. Projective synchronization is usually observed in partially linear system and the scaling factor is dependent on initial conditions but here we propose a projective synchronization scheme without the limitation of partial linearity and initial conditions. Finally we have shown that our numerical calculations well support the analytic results.

The plan of the rest paper is as follows: In Section 2, some chaotic properties is investigated in two time delay neural system. Section 3 represents stability condition for anticipatory, complete and lag synchronization in delay feedback coupled. The stability condition for projective-anticipatory, projective and projective-lag synchronization in coupled neural systems is investigated in Section 4. In Section 5, adaptive scaling factor is considered. Finally conclusions are made in Section 6.

Section snippets

Two-neural system and its chaotic properties

In general n-neuron systems with a single time-delay can be written as ${\dot{x}}_{i} (t) = - u_{i} x_{i} (t) + \sum_{j = 1}^{n} v_{ij} f (x_{j} (t)) + \sum_{j = 1}^{n} w_{ij} f (x_{j} (t - τ)) + I_{i}, i = 1, 2, \dots, n$ where f(·) is the neural activation function, u_i > 0, v_ij and w_ij are real numbers, τ ⩾ 0 is the only time-delay and I_i are external inputs. The initial conditions are x_i(t) = ϕ_i(t), t ∈ [−τ, 0], with some given continuous functions ϕ_i : [−τ, 0] → R. In this paper, without loss of generality, initial conditions are always chosen as constant functions on [−τ, 0].

In this paper,

Coupled system and stability condition for chaos synchronization

We consider drive system as $\dot{x} (t) = - Ux (t) + Vf (x (t)) + Wf (x (t - τ (t)))$ and response system as $\dot{y} (t) = - Uy (t) + Vf (y (t)) + Wf (y (t - τ (t))) + K [x (t - τ_{1}) - y (t)]$ where K = (k₁, k₂)^T is coupling strength vector, τ₁ is coupling delay. For τ₁ < 0, τ₁ = 0 and τ₁ > 0, anticipating, complete and lag synchronization occur between (3), (4) respectively.

Let Δ₁ = x₁(t − τ₁) − y₁, Δ₂ = x₂(t − τ₁) − y₂ be the synchronization errors. Then error dynamics is ${\dot{Δ}}_{1} = - r_{1} Δ_{1} + r_{2} Δ_{2} + s_{1} Δ_{1} (t - τ) + s_{2} Δ_{2} (t - τ)$ ${\dot{Δ}}_{2} = - r_{3} Δ_{1} + r_{4} Δ_{2} + s_{3} Δ_{1} (t - τ) + s_{4} Δ_{2} (t - τ)$ where r₁ = u₁ + k₁ − v₁₁f′(ξ₁), r₂ = v₁₂f

Sufficient condition for projective synchronization

We consider coupled system for projective synchronization as $\dot{x} (t) = - Ux (t) + Vf (x (t)) + Wf (x (t - τ (t)))$ and $\dot{y} (t) = - Uy (t) + Vf (y (t)) + Wf (y (t - τ (t))) + K [α x (t - τ_{1}) - β y (t)]$ where α and β are desired scaling factors for projective synchronization. For α = β = 1, we get the anticipating, complete and lag synchronization as before and for different sign of scaling factors, a new types of anti-phase synchronization arises. We define projective synchronization errors as $Δ_{1} = α x_{1} (t - τ_{1}) - β y_{1}$ $Δ_{2} = α x_{2} (t - τ_{1}) - β y_{2}$ then one have an error

Projective synchronization using adaptive scaling factors

In the previous section, we have chosen the scaling factors α, β arbitrarily. Here we consider a method of projective synchronization to generate the adaptive scaling factors. Corresponding driving and response systems are given by $\dot{x} (t) = - Ux (t) + Vf (x (t)) + Wf (x (t - τ (t)))$ $\dot{y} (t) = - Uy (t) + Vf (y (t)) + Wf (y (t - τ (t))) + K [α (t) x (t - τ_{1}) - β (t) y (t)]$ where the generating equations for α(t) and β(t) can be written as $\dot{α} = γ_{α} [x_{1} (t - τ_{1}) - y_{1} (t)]$ $\dot{β} = γ_{β} [x_{2} (t - τ_{1}) - y_{2} (t)]$ where γ_α and γ_β are the adaptive gains. The synchronization time

Conclusions

In conclusion, we have studied analytically and numerically the projective-lag, projective or projective-anticipatory synchronization in modulated time-delayed coupled systems, related to neural networks. Transition among projective-lag, projective or projective-anticipatory synchronization can be obtained by adjusting the coupling delay. Compared with the previous works [7], [9], [10], [11], [12], the proposed one has the following advantages: (1) The projective synchronization is usually

Acknowledgement

The authors are very thankful to Prof. Roberto Livi and Prof. J.M.G. Miranda for their significant comments and suggestions.

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View full text

Projective synchronization of time-varying delayed neural network with adaptive scaling factors

Highlights

Abstract

Introduction

Section snippets

Two-neural system and its chaotic properties

Coupled system and stability condition for chaos synchronization

Sufficient condition for projective synchronization

Projective synchronization using adaptive scaling factors

Conclusions

Acknowledgement

Chaos

Europhys Lett

Phys Rev Lett

Phys Rev Lett

Phys Rev Lett

Chaos synchronization

Chaos

From Chaos to Order. Methodologies, Perspectives and Applications

Prog Theor Phys

Phys Rev Lett

Phys Rev Lett

Phys Rev E

Phys Rev E

Phys Rev E