Time-delay Chaotic Systems Research Articles

An experimental digital communication scheme based on chaotic time-delay system

We develop an experimental system for secure communication with nonlinear mixing of information signal and chaotic signal of a time-delay system. The proposed scheme is based on programmable microcontrollers with digital transmission line. The scheme allows one to transmit and receive speech and musical signals in real time without noticeable distortion. A high quality of extraction of hidden information signal is achieved due to the use of digital elements in the scheme, which ensures identity of the parameters and high stability to noise. We study a possibility of hidden message extraction from a chaotic carrier by a third party in the case of mismatch of the receiver and transmitter parameters.

Hybrid projective synchronization of time-delayed fractional order chaotic systems

Hybrid projective synchronization of fractional order chaotic systems with time-delay is investigated. It is shown that the slave system can be synchronized with the driver up to a scaling matrix. According to the stability theorem of linear fractional order systems with multiple time-delays, a nonlinear controller is proposed for the synchronization. Finally, two different structural time-delayed fractional order chaotic systems are applied to realize hybrid projective synchronization. The corresponding numerical results show the effectiveness and robustness of the controller.

Designing chaotic S-boxes based on time-delay chaotic system

S-box structures used in the encryption architecture are of great importance for constructing powerful block encryption systems, which hold an important place in modern cryptology. The design of S-boxes with sound cryptographic characteristics is of utmost importance for constructing powerful encryption systems. In this study, an S-box design algorithm based on time-delay chaotic systems is proposed. The proposed algorithm is considered relative to other algorithms in the literature as more useful according to such criteria as simplicity and efficient implementation. Theoretical analysis and computer simulations demonstrated that the proposed algorithm meets all the performance requirements for the S-box design criteria, and also verified the efficient and practical structure of the algorithm.

THEORY AND EXPERIMENT OF A FIRST-ORDER CHAOTIC DELAY DYNAMICAL SYSTEM

We report the theory and experiment of a new time-delayed chaotic (hyperchaotic) system with a single scalar time delay and a nonlinearity described by a closed form mathematical function. Detailed stability and bifurcation analyses establish that with the suitable delay and system parameters, the system shows a stable limit cycle through a supercritical Hopf bifurcation. Numerical simulations exemplify that the system depicts mono-scroll and double-scroll chaos and hyperchaos for a range of delay and other system parameters. Nonlinear behavior of the system is characterized by Lyapunov exponents and Kaplan–Yorke dimension. It is established that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate time delay. Finally, the system is implemented in an analogue electronic circuit using off-the-shelf circuit elements. It is shown that the behavior of the time delay chaotic electronic circuit qualitatively agrees well with our analytical and numerical results.

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Chaos pass filter: Linear response of synchronized chaotic systems

The linear response of synchronized time-delayed chaotic systems to small external perturbations, i.e., the phenomenon of chaos pass filter, is investigated for iterated maps. The distribution of distances, i.e., the deviations between two synchronized chaotic units due to external perturbations on the transferred signal, is used as a measure of the linear response. It is calculated numerically and, for some special cases, analytically. Depending on the model parameters this distribution has power law tails in the region of synchronization leading to diverging moments of distances. This is a consequence of multiplicative and additive noise in the corresponding linear equations due to chaos and external perturbations. The linear response can also be quantified by the bit error rate of a transmitted binary message which perturbs the synchronized system. The bit error rate is given by an integral over the distribution of distances and is calculated analytically and numerically. It displays a complex nonmonotonic behavior in the region of synchronization. For special cases the distribution of distances has a fractal structure leading to a devil's staircase for the bit error rate as a function of coupling strength. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. A bidirectionally coupled chain of three units can completely filter out the perturbation. Thus the second moment and the bit error rate become zero.

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Discontinuous attractor dimension at the synchronization transition of time-delayed chaotic systems

The attractor dimension at the transition to complete synchronization in a network of chaotic units with time-delayed couplings is investigated. In particular, we determine the Kaplan-Yorke dimension from the spectrum of Lyapunov exponents for iterated maps and for two coupled semiconductor lasers. We argue that the Kaplan-Yorke dimension must be discontinuous at the transition and compare it to the correlation dimension. For a system of Bernoulli maps, we indeed find a jump in the correlation dimension. The magnitude of the discontinuity in the Kaplan-Yorke dimension is calculated for networks of Bernoulli units as a function of the network size. Furthermore, the scaling of the Kaplan-Yorke dimension as well as of the Kolmogorov entropy with system size and time delay is investigated.

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Projective-anticipating, projective and projective-lag synchronization of chaotic systems with time-varying delays

We investigate different types of projective (projective-anticipating, projective and projective-lag) synchronization in unidirectionally nonlinearly coupled time-delayed chaotic systems with variable time delays. Based on the Krasovskii–Lyapunov approach, we find both the existence and sufficient stability conditions, using a general class of time-delayed chaotic systems related to optical bistable or hybrid optical bistable devices. Our method has the advantage that it requires only one nonlinearly coupled term to achieve different types of projective synchronization in time-delayed chaotic systems with variable time delays. Compared with other existing works, our result provides an easy way to achieve projective-anticipating, projective and projective-lag synchronization. Numerical simulations of the Ikeda system are given to demonstrate the validity of the proposed method.

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Delay-Dependent Exponential Optimal Synchronization for Nonidentical Chaotic Systems via Neural-Network-Based Approach

A novel approach is presented to realize the optimal exponential synchronization of nonidentical multiple time-delay chaotic (MTDC) systems via fuzzy control scheme. A neural-network (NN) model is first constructed for the MTDC system. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of the NN model. Based on this LDI state-space representation, a delay-dependent exponential stability criterion of the error system derived in terms of Lyapunov's direct method is proposed to guarantee that the trajectories of the slave system can approach those of the master system. Subsequently, the stability condition of this criterion is reformulated into a linear matrix inequality (LMI). According to the LMI, a fuzzy controller is synthesized not only to realize the exponential synchronization but also to achieve the optimal performance by minimizing the disturbance attenuation level at the same time. Finally, a numerical example with simulations is given to demonstrate the effectiveness of our approach.

Characteristics of time-delay complex Lorenz chaotic system and its self-synchronization of time delay

Self-synchronization of time delay implies that the synchronization between the time-delay system and the original system keeps the structure and parameters of systems unchanged, thus these various problems produced by time-delay in practice are avoided. Taking a time-delay complex Lorenz system for example, we investigate its dynamic characteristics and the influence of of time lag factor. A nonlinear feedback controller is designed to realize the self-synchronization of time delay of the complex Lorenz system. Numerical simulations verify the effectiveness of the presented controller. The controller adopts some states to realize the synchronization of all states. It is simple in principle and easy to implement in engineering.

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