Lag synchronization and scaling of chaotic attractor in coupled system

Abstract

We report a design of delay coupling for lag synchronization in two unidirectionally coupled chaotic oscillators. A delay term is introduced in the definition of the coupling to target any desired lag between the driver and the response. The stability of the lag synchronization is ensured by using the Hurwitz matrix stability. We are able to scale up or down the size of a driver attractor at a response system in presence of a lag. This allows compensating the attenuation of the amplitude of a signal during transmission through a delay line. The delay coupling is illustrated with numerical examples of 3D systems, the Hindmarsh-Rose neuron model, the Rössler system, a Sprott system, and a 4D system. We implemented the coupling in electronic circuit to realize any desired lag synchronization in chaotic oscillators and scaling of attractors.

Mixed lag synchronization in chaotic oscillators and experimental observations

“…We use the paradigmatic Rössler model [18] for numerical demonstration and an electronic circuit for experimental evidence. Such a MLS can be engineered [19] in co-rotating chaotic systems by a design of delay coupling, however, to our best knowledge, it is not reported so far in systems under instantaneous diffusive coupling where it is an emergent behavior.…”

Transmission projective synchronization of multi-systems with non-delayed and delayed coupling via impulsive control

“…Sun, Shen, and Zhang Chaos 22, 043107 (2012) ðk 1 þ k 2 þ 1 À 2d à Þðt kþ1 À t k Þ þ lnðha k Þ 0; k ¼ 1; 2; …; (9) where d à is the minimum value of the initial feedback strength d i0 (d i0 d i ; 1 i n), I 1 2 R nÂn denotes the identity matrix, I 2 denotes the suitable identity matrix, and…”

“…For this kind of network, the weights of links are time varying, which results in variations of the network topology and coupling configuration over time. 36 To simulate more realistic networks, the projective synchronization of a class of delayed chaotic systems via impulsive control was proposed, 9,29,37 where the drive-response system can be synchronized to within a desired scaling factor. However, the above chaos synchronization has been paid a lot of attentions by many researchers; so far, the studies of chaos synchronization have mostly been limited to within one drive system and one response system.…”

General hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems

“…Many different types of synchronization phenomenon have been intensively investigated and a lot of theoretical results have been obtained in the past 20 years, such as complete synchronization [1], anti-synchronization [2], generalized synchronization [3,4], phase synchronization [5], anti-phase synchronization [6], lag synchronization [7], partial synchronization [8], projective synchronization [9,10,11,12,13], time scale synchronization [14], combination synchronization [15,16], compound synchronization [17] etc. During the past decades, there exist the following methods to realize chaos synchronization such as OGY method [19], feedback control method [20,21,22], H ∞ control method [23], optimal control method [24], PID control method [25], active control method [26], passive control method [27], backstepping method [28], adaptive control method [29,30], sliding mode control method [31], impulsive control method [32], coupling control method [33,34,35] etc.…”