Ikeda System Research Articles

Design and Implementation of Autonomous and Non-Autonomous Time-Delay Chaotic System Based on Field Programmable Analog Array.

Time-delay chaotic systems can have hyperchaotic attractors with large numbers of positive Lyapunov exponents, and can generate highly stochastic and unpredictable time series with simple structures, which is very suitable as a secured chaotic source in chaotic secure communications. But time-delay chaotic systems are generally designed and implemented by using analog circuit design techniques. Analog implementations require a variety of electronic components and can be difficult and time consuming. At this stage, we can now solve this question by using FPAA (Field-Programmable Analog Array). FPAA is a programmable device for implementing multiple analog functions via dynamic reconfiguration. In this paper, we will introduce two FPAA-based design examples: An autonomous Ikeda system and a non-autonomous Duffing system, to show how a FPAA device is used to design programmable analog time-delay chaotic systems and analyze Shannon entropy and Lyapunov exponents of time series output by circuit and simulation systems.

Enhancing Ikeda Time Delay System by Breaking the Symmetry of Sine Nonlinearity

In the present contribution, an asymmetric central contraction mutation (ACCM) model is proposed to enhance the Ikeda time delay system. The modified Ikeda system model is designed by introducing a superimposed tanh function term into the sine nonlinearity term. Stability and Hopf bifurcation characteristics of the system are analyzed theoretically. Numerical simulations, carried out in terms of bifurcation diagrams, Lyapunov exponents spectrum, phase portraits, and two‐parameter (2D) largest Lyapunov exponent diagrams are employed to highlight the complex dynamical behaviors exhibited by the enhanced system. The results indicate that the modified system has rich dynamical behaviors including limit cycle, multiscroll hyperchaos, chaos, and hyperchaos. Moreover, as a major outcome of this paper, considering the fragile chaos phenomenon, the ACCM‐Ikeda time delay system has better dynamical complexity and larger connected chaotic parameter spaces (connectedness means that there is no stripe corresponding to nonchaotic dynamics embedded in the chaos regions).

A new image encryption scheme based on a simple first-order time-delay system with appropriate nonlinearity

Nonlinear delay differential system (NDDS) is a family of hyperchaotic systems, which has attracted much attention in the last years, especially in optics. The NDDS is often modeled by a time delay that occurs between the signals at the input and the output, and a nonlinearity introduced by a function $$F$$ . This paper introduces the Ikeda system that is considered system of infinite dimension and proposed an image encryption scheme by dynamic block. The encryption algorithm divides the image on dynamic blocks depending on the previous block cipher. The algorithm diffuses the plain image with bitwise XOR between each block and sequence generated by Ikeda system after quantification. Statistical analysis of the proposed encryption scheme demonstrates that this cryptosystem is secure enough to resist the brute-force attack, entropy attack, differential attack, chosen-plaintext attack, known-plaintext attack and statistical attack. In addition, the cryptosystem has high sensitivity of key and large space key.

Adaptive impulsive synchronization of uncertain delayed chaotic system with full unknown parameters via discrete‐time drive signals

This article investigates the adaptive impulsive synchronization of delayed chaotic system with full unknown parameters. Aiming at this problem, we propose a new adaptive strategy, in which both the adaptive–impulsive controller and the parameters adaptive laws are designed via the discrete‐time signals from the drive system. The corresponding theoretical proof is given to guarantee the effectiveness of the proposed strategy. Moreover, the concrete adaptive strategies are achieved for delayed Hopfield neural network, optical Ikeda system and the well‐known delayed Lü chaotic system. As expected, numerical simulations show the effectiveness of the proposed strategy. This method has potential applications in parameters estimation, secure communication, and cryptanalysis when only discrete signals are transmitted in communication channel. © 2014 Wiley Periodicals, Inc. Complexity 21: 43–51, 2016

Exact synchronization bound for coupled time-delay systems

We obtain an exact bound for synchronization in coupled time-delay systems using the generalized Halanay inequality for the general case of time-dependent delay, coupling, and coefficients. Furthermore, we show that the same analysis is applicable to both uni- and bidirectionally coupled time-delay systems with an appropriate evolution equation for their synchronization manifold, which can also be defined for different types of synchronization. The exact synchronization bound assures an exponential stabilization of the synchronization manifold which is crucial for applications. The analytical synchronization bound is independent of the nature of the modulation and can be applied to any time-delay system satisfying a Lipschitz condition. The analytical results are corroborated numerically using the Ikeda system.

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.

Comment on “Generalized projective synchronization in time-delayed systems: Nonlinear observer approach” [Chaos 19, 013102 (2009); 20, 029902 (2010)

In Chaos 19, 013102 (2009), the author proposed generalized projective synchronization for time delay systems using nonlinear observer and obtained sufficient condition to ensure projective synchronization for modulated time varying delay. There are concerns with the obtained conditions as the result was applicable only to trivial case of time varying delay τ[over dot](1)(t)=dτ(1)(t)/dt<1. In this paper, we note the drawbacks of the proposed sufficient condition. The new improved sufficient condition for ensuring the projective synchronization of time varying delayed systems is presented. The proposed new criteria have been verified by adopting the Ikeda system.

Projective Synchronization Between Two Nonidentical Variable Time Delayed Systems

In this paper, we propose a method for the projective synchronization between two different chaotic systems with variable time delays. Using active control approach, the suitable controller is constructed to make the states of two different diverse time delayed systems asymptotically synchronize up to the desired scaling factor. Based on the Lyapunov stability theory, the sufficient condition for the projective synchronization is calculated theoretically. Numerical simulations of the projective synchronization between Mackey—Glass system and Ikeda system with variable time delays are shown to validate the effectiveness of the proposed algorithm.

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

In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.

A constructional method for generalized synchronization of coupled time-delay chaotic systems

A constructional method for detecting the existence and determining the functional relationship of generalized synchronization is introduced in this paper. Based on the stability theory of fixed points of dynamical systems, we show theoretically and numerically that an appropriate coupling scheme allows us to find the synchronization functional relationship between the states of coupled time-delay chaotic systems. On the other hand, given a synchronization function, this approach may help us to design the coupling scheme. To demonstrate the proposed method, Ikeda system is presented as an example.

Prediction of the chaotic time series from parameter-varying systems using artificial neural networks

Prediction of the chaotic time series generated by the complex parameter-varying systems is researched in this paper. The parameter-varying Logistic system is constructed firstly, and the properties of this kind of system are analyzed. These systems, whose parameter values change with time, do not have attractor shape invariable with time evolution because of their continually changing dynamical property. Combining the Takens' embedding theorem and the artificial neural networks (ANN) theory, we interprete the feasible reason that ANN method can be used to predict the chaos systems with the invariable attractor shape, and then discuss the potential problem that will be met when using ANN to predict the parameter-varying system. Experiments of forecasting the chaotic time series from parameter-varying Ikeda system using neural networks have been performed. The previous theoretical analyses are validated by the experiment results. The results also show that if only simply increasing the training data, the neural networks' predicting generalization ability may be reduced, the generalized predicting result on the parameter-varying system is especially seriously affected by the selected training data set. So prediction of the parameter-varying systems must be well resolved before the chaotic time series prediction can be made practical.

ALGORITHM FOR ESTIMATION OF THE STABLE BASIN IN CONTROLLING CHAOTIC DISCRETE DYNAMICS

In this paper, we apply OPCL control to discrete system, and based on relative nonlinear measure, give an algorithm for estimating the radius of stable basin. We rigorously prove that this basin is bound to be of existence for nonlinear discrete system, whose goal dynamics is either periodic orbits or fixed point. We also, in particular, investigate the stable basin in a quadratic polynomial map system, and present that the stable basin is irrelevant to the goal orbits with a negative Jacobian gain matrix. Furthermore, we take the well-known Hénon system and Ikeda system as examples to illustrate the implementation of our theory, and give the corresponding simulations to reinforce our method.

Synchronization of Chaotic Maps by Feedback Control and Application to Secure Communications Using Chaotic Neural Networks

In this paper we propose a method to synchronize chaotic maps by feedback control. This approach is particularly attractive, if it can be made to work on a wide variety of chaotic systems, because it requires very little computational overhead and so is extremely easy to implement in fast hardware. In general the control feedback used here makes use of a diagonal matrix with experimentally determined components. However, for this particular application, synchronization is achieved by using only one element of the feedback matrix. To demonstrate the method we first apply it to the Hénon map and study the local stability properties. We next apply the method to a neural network approximation of the Ikeda system and show that two identical copies of this network approximations of a given chaotic system has potentially interesting applications in secure communications. In the final section we demonstrate how message can be encoded, transmitted and decoded using the Ikeda neural network in combination with the feedback control method.

Synchronized spatiotemporal chaos and spatiotemporal on-off intermittency in a nonlinear ring cavity

We numerically study the transverse Ikeda system describing a ring cavity with a saturable absorber nonlinearity which is driven by a coherent input field. The Gaussian shape of the input beam favors the solutions to have space inversion symmetry. These solutions bifurcate into states of synchronized spatiotemporal chaos where synchronization refers to the identical dynamics of the two beam halves. We demonstrate that the breakdown of this synchronized chaotic state is caused by on-off intermittency. The qualitative and quantitative features of the transition are studied. In particular, the scaling behavior of the transition in the space domain is investigated. The results display a qualitative similarity with the scaling behavior of spatiotemporal intermittency. \textcopyright{} 1996 The American Physical Society.