Continuous-time Chaotic Systems Research Articles

A Deterministic Chaos-Model-Based Gaussian Noise Generator

The abilities of quantitative description of noise are restricted due to its origin, and only statistical and spectral analysis methods can be applied, while an exact time evolution cannot be defined or predicted. This emphasizes the challenges faced in many applications, including communication systems, where noise can play, on the one hand, a vital role in impacting the signal-to-noise ratio, but possesses, on the other hand, unique properties such as an infinite entropy (infinite information capacity), an exponentially decaying correlation function, and so on. Despite the deterministic nature of chaotic systems, the predictability of chaotic signals is limited for a short time window, putting them close to random noise. In this article, we propose and experimentally verify an approach to achieve Gaussian-distributed chaotic signals by processing the outputs of chaotic systems. The mathematical criterion on which the main idea of this study is based on is the central limit theorem, which states that the sum of a large number of independent random variables with similar variances approaches a Gaussian distribution. This study involves more than 40 mostly three-dimensional continuous-time chaotic systems (Chua’s, Lorenz’s, Sprott’s, memristor-based, etc.), whose output signals are analyzed according to criteria that encompass the probability density functions of the chaotic signal itself, its envelope, and its phase and statistical and entropy-based metrics such as skewness, kurtosis, and entropy power. We found that two chaotic signals of Chua’s and Lorenz’s systems exhibited superior performance across the chosen metrics. Furthermore, our focus extended to determining the minimum number of independent chaotic signals necessary to yield a Gaussian-distributed combined signal. Thus, a statistical-characteristic-based algorithm, which includes a series of tests, was developed for a Gaussian-like signal assessment. Following the algorithm, the analytic and experimental results indicate that the sum of at least three non-Gaussian chaotic signals closely approximates a Gaussian distribution. This allows for the generation of reproducible Gaussian-distributed deterministic chaos by modeling simple chaotic systems.

An innovative approach based on optimization for the determination of initial conditions of continuous-time chaotic system as a random number generator

Security has been one of the important problems in the processing, storage and transmission of information. The transfer of information to virtual environments with technological developments has made it necessary to examine the concept of security with different dimensions. Data security can be achieved to a great extent by using randomly generated unpredictable bit sequences in encryption. In this study, approximately 1,000,000 random bits were generated for use in computer science. The Mode2(x) method and the unpredictable nature of chaotic systems are used for random number generation. With this hybrid structure used in the study, it is aimed to gain a different perspective to future studies. However, it is known that the sensitivity of chaotic systems to initial conditions is very high. The most important contribution of the study to the literature is that the initial conditions were determined using optimization algorithms. In this study, Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) were used to calculate the initial conditions. To the authors' knowledge, this is the first study to use optimization to determine initial conditions for continuous-time chaotic systems. NIST SP 800-22 Statistical Test Suite was used to analyze the reliability of the generated random bits. It was seen that the proposed method passed all statistical tests successfully (100 %). Histogram and floating frequency analysis were used for the analysis of random numbers obtained from the generated bits. Histogram plots showed uniform distribution. It is seen that there is a homogeneous distribution in the sliding frequency analysis.

Predictive-based control of complex dynamic networks

This paper addresses the problem of designing a robust controller for a class of complex dynamical networks (CDNs). This class of CDNs is characterized by a scale-free typical structure with Rossler and Lorenz-type nodes. Direct control of every node in a CDN with a large number of nodes might be unnecessary. Based on the concept of pinning control of continuous-time chaotic systems and the matrix measure theory, some simple stability criteria are derived to guarantee the asymptotic stability of the system states such that all the system states of the closed-loop system converge towards the desired unstable equilibrium points. Consequently, the proposed controller developed here for scale-free networks can be employed to handle a broader class of CDNs with different types of nodes. Numerical simulations are provided to show the effectiveness of the proposed control method.

A general method for projective-lag synchronization of heterogeneous chaotic maps with different dimensions

Projective-lag synchronization of complex systems has attracted much attention in the past two decades. However, the majority of previous studies concentrated on continuous-time chaotic systems or discrete-time chaotic systems with the same dimensions. In our present study, a general method for projective-lag synchronization of different discrete-time chaotic systems characterized with different dimensions is first demonstrated. On the basis of stability theory of discrete-time dynamical systems and Lyapunov stability theory, general controllers are designed by using the active control method. The method could achieve projective-lag synchronization in both cases: and . The effectiveness and feasibility of the proposed method is demonstrated by the projective-lag synchronization between two-dimensional Lorenz discrete-time system and three-dimensional Stefanski map, as well as between the three-dimensional generalized Hénon map and the two-dimensional quadratic map, respectively.

Observer-Based H ∞ Fuzzy Synchronization and Output Tracking Control of Time-Varying Delayed Chaotic Systems

Tracking control for the output using an observer-based H ∞ fuzzy synchronization of time-varying delayed discrete- and continuous-time chaotic systems is proposed in this paper. First, from a practical point of view, the chaotic systems here consider the influence of time-varying delays, disturbances, and immeasurable states. Then, to facilitate a uniform control design approach for both discrete- and continuous-time chaotic systems, the dynamic models along with time-varying delays and disturbances are reformulated using the T-S (Takagi–Sugeno) fuzzy representation. For control design considering immeasurable states, a fuzzy observer achieves master-slave synchronization. Third, combining both a fuzzy observer for state estimation and a controller (solved from generalized kinematic constraints) output tracking can be achieved. To make the design more practical, we also consider differences of antecedent variables between the plant, observer, and controller. Finally, using Lyapunov’s stability approach, the results are sufficient conditions represented as LMIs (linear matrix inequalities). The contributions of the method proposed are threefold: (i) systemic and unified problem formulation of master-slave synchronization and tracking control for both discrete and continuous chaotic systems; (ii) practical consideration of time-varying delay, immeasurable state, different antecedent variables (of plant, observer, and controller), and disturbance in the control problem; and (iii) sufficient conditions from Lyapunov’s stability analysis represented as LMIs which are numerically solvable observer and controller gains from LMIs. We carry out numerical simulations on a chaotic three-dimensional discrete-time system and continuous-time Chua’s circuit. Satisfactory numerical results further show the validity of the theoretical derivations.

From Continuous-Time Chaotic Systems to Pseudo Random Number Generators: Analysis and Generalized Methodology.

The use of chaotic systems in electronics, such as Pseudo-Random Number Generators (PRNGs), is very appealing. Among them, continuous-time ones are used less because, in addition to having strong temporal correlations, they require further computations to obtain the discrete solutions. Here, the time step and discretization method selection are first studied by conducting a detailed analysis of their effect on the systems’ statistical and chaotic behavior. We employ an approach based on interpreting the time step as a parameter of the new “maps”. From our analysis, it follows that to use them as PRNGs, two actions should be achieved (i) to keep the chaotic oscillation and (ii) to destroy the inner and temporal correlations. We then propose a simple methodology to achieve chaos-based PRNGs with good statistical characteristics and high throughput, which can be applied to any continuous-time chaotic system. We analyze the generated sequences by means of quantifiers based on information theory (permutation entropy, permutation complexity, and causal entropy × complexity plane). We show that the proposed PRNG generates sequences that successfully pass Marsaglia Diehard and NIST (National Institute of Standards and Technology) tests. Finally, we show that its hardware implementation requires very few resources.

A Novel Random Number Generator Based on Fractional Order Chaotic Chua System

One of the practical applications of chaotic systems is the design of a random number generator. In the literature, generally random number generators are designed using discrete time chaotic systems. The reason for the use of the discrete time chaotic systems in the design architecture is that the latter have a simpler structure than the continuous time chaotic systems. In order to observe chaos in continuous time systems, the system must have at least three degrees. It is shown that for fractional order chaotic systems chaos can be observed even in a lower system degree. The aim of this study is to develop a random number generator using a fractional order chaotic Chua system. The proposed generator is analysed using various randomness tests. The analysis results show that the proposed generator passes the random requirements successfully. On the one hand, this study is important because it demonstrates the practical application of fractional order chaotic systems. On the other hand, it provides an alternative to designs based on discrete time chaotic systems.

Projective–lag synchronization scheme between two different discrete-time chaotic systems

This paper addresses theoretical analysis of projective–lag synchronization between two different discrete-time chaotic systems. We overcome some limitations of previous works where projective–lag synchronization can only be achieved in continuous-time chaotic systems. In this paper, we realize projective–lag synchronization in discrete-time chaotic systems with the same order. Based on the Lyapunov stability theory and a nonlinear control scheme, the controllers are designed by using the relevant variables of master and slave systems, and we find sufficient conditions for the stability of the error dynamics. The correctness of theoretical analysis is verified by 2D and 3D discrete-time chaotic systems, respectively.

Adaptive fuzzy control for the stabilisation of chaotic systems

In this paper, we investigate the stabilisation of unstable periodic orbits of continuous time chaotic systems using adaptive fuzzy controllers. For this aim, we present a control method that can achieve the stabilisation of an unstable periodic orbit (UPO) without any knowledge of the system model. Thus, a fuzzy adaptive linear controller is proposed based on time-delay feedback approach to obtain one that achieves UPO stabilisation. Finally, we use reduced order sliding observer to estimate the necessary state for the controller construction. The efficiency of proposed methods is demonstrated using numerical simulations applied to Chua's system.

Difference synchronization among three chaotic systems with exponential term and its chaos control

Abstract In this article, the difference synchronization and chaos control of chaotic systems with nonlinear exponential terms have been studied by using the feedback control method. The chaotic systems in the presence of an exponential terms behave differently from the polynomial chaotic systems, whose dynamics will also be different. The Routh-Hurwitz condition is used during chaos control and synchronization. The nonlinear ten-ring chaotic system, 3D chaotic system, new 3D chaotic system are considered to simulate the difference synchronization scheme for continuous case, and Wang, 3D Henon map and Rossler systems are considered during simulation of discrete time chaotic systems. The numerical simulations and the graphical results are presented to show the effectiveness and reliability of difference synchronization for continuous and discrete time chaotic systems.

Stabilization of chaotic systems via fuzzy time-delayed controller approac

In this paper, we investigate the stabilization of unstable periodic orbits of continuous time chaotic systems usingfuzzy time-delayed controllers. For this aim, we present a control method that can achieve stabilization of an unstableperiodic orbit (UPO) without any knowledge of the system model. Our proposal is attained progressively. First, wecombine the input-to-state linearizing controller with the fuzzy method to obtain one that achieves UPO stabilization.Then, we use reduced order sliding observer to estimate the necessary state for the controller construction. Finally, theefficiency of the proposed methods is demonstrated using numerical simulations applied to Chua’s system.

On the Discretization of Continuous-Time Chaotic Systems for Digital Implementations

Recently, many continuous-time chaotic systems were synthesized using microcontrollers and FPGAs. This requires applying mathematical discretization to convert integration into recursion. Depending on the approximation algorithms, the speed of the numerical processors, and the number of bits used to represent data, different accuracies and stabilities could be obtained. This article explores the conditions necessary to faithfully generate signals that reflect the true behavior of the chaotic systems, while maintaining the same values for their Lya-punov exponents. This is very important for chaos control and synchronization, especially for applications in secure communication that rely on digital cryptography. The Lorenz system and the Duffing oscillator are investigated to illustrate the effect of having autonomous versus non-autonomous structures. In addition, the Nosé-Hoover dynamical model is investigated to detect the relationship between agility and numerical accuracy. The obtained results prove that identical deterministic chaotic systems can behave differently, for the same set of initial conditions, depending on the discretization algorithm. This added sensitivity necessitates careful design of the mathematical models required for digital implementations of chaotic systems. The article concludes with useful recommendations for best practices in designing, synthesizing, and implementing digital chaotic systems, while commenting on the best compromise between mathematical complexity and numerical accuracy.

Control of Continuous Time Chaotic Systems With Unknown Dynamics and Limitation on State Measurement

This paper has dedicated to study the control of chaos when the system dynamics is unknown and there are some limitations on measuring states. There are many chaotic systems with these features occurring in many biological, economical and mechanical systems. The usual chaos control methods do not have the ability to present a systematic control method for these kinds of systems. To fulfill these strict conditions, we have employed Takens embedding theorem which guarantees the preservation of topological characteristics of the chaotic attractor under an embedding named “Takens transformation.” Takens transformation just needs time series of one of the measurable states. This transformation reconstructs a new chaotic attractor which is topologically similar to the unknown original attractor. After reconstructing a new attractor its governing dynamics has been identified. The measurable state of the original system which is one of the states of the reconstructed system has been controlled by delayed feedback method. Then the controlled measurable state induced a stable response to all of the states of the original system.

On the control of unknown continuous time chaotic systems by applying Takens embedding theory

In this paper, a new approach to control continuous time chaotic systems with an unknown governing equation and limitation on the measurement of states, has been investigated. In many chaotic systems, disability to measure all of the states is a usual limitation, like in some economical, biological and many other engineering systems. Takens showed that a chaotic attractor has an astonishing feature in which it can embed to a mathematically similar attractor by using time series of one of the states. The new embedded attractor saves much information from the original attractor. This phenomenon has been deployed to present a new way to control continuous time chaotic systems, when only one of the states of the system is measurable and the system model is not also available.

Hybrid robust discrete sliding mode control for generalized continuous chaotic systems subject to external disturbances

This paper presents a discrete sliding mode control (DSMC) for the robust chaos suppression of the generalized continuous-time chaotic systems subject to matched/mismatched disturbances. The proposed DSMC first ensures the existence of the sliding manifold. Then, the effect of external disturbances including matched and mismatched cases are discussed after the controlled system is driven into the sliding manifold. The proposed results show that the chaotic behaviour of controlled systems with matched disturbances can be fully suppressed to zero or robustly driven into an estimated bound in the state–space form, which was not explicitly addressed in the literature. Finally, two different chaotic systems are utilized to study the hybrid chaos suppression. The corresponding numerical results show the effectiveness and robustness of the proposed DSMC.

Continuous Time Chaotic Systems for Whale Optimization Algorithm

Discrete time chaotic systems are often used instead of random number arrays in order to improve the convergence properties of optimization algorithms and prevent them to get stuck on ...

Detection of coupling delay: A problem not yet solved.

Nonparametric detection of coupling delay in unidirectionally and bidirectionally coupled nonlinear dynamical systems is examined. Both continuous and discrete-time systems are considered. Two methods of detection are assessed-the method based on conditional mutual information-the CMI method (also known as the transfer entropy method) and the method of convergent cross mapping-the CCM method. Computer simulations show that neither method is generally reliable in the detection of coupling delays. For continuous-time chaotic systems, the CMI method appears to be more sensitive and applicable in a broader range of coupling parameters than the CCM method. In the case of tested discrete-time dynamical systems, the CCM method has been found to be more sensitive, while the CMI method required much stronger coupling strength in order to bring correct results. However, when studied systems contain a strong oscillatory component in their dynamics, results of both methods become ambiguous. The presented study suggests that results of the tested algorithms should be interpreted with utmost care and the nonparametric detection of coupling delay, in general, is a problem not yet solved.

On Inverse Full State Hybrid Function Projective Synchronization For Continuous-time Chaotic Dynamical Systems with Arbitrary Dimensions

Referring to continuous-time chaotic dynamical systems, this paper investigates the inverse full state hybrid function projective synchronization (IFSHFPS) of non-identical systems characterized by different dimensions. By taking a master system of dimension n and a slave system of dimension m, the method enables each master system state to be synchronized with a linear combination of slave system states, where the scaling factor of the linear combination can be any arbitrary differentiable function. The approach, based on the Lyapunov stability theory and stability of linear continuous-time systems, presents some useful features: (i) it enables non-identical chaotic systems with different dimension \(n m\) to be synchronized; (ii) it can be applied to a wide class of chaotic (hyperchaotic) systems for any differentiable scaling function; (iii) it is rigorous, being based on two theorems, one for the case \(n m\). Two different numerical examples are reported. The examples clearly highlight the capability of the conceived approach in effectively achieving synchronized dynamics for any differentiable scaling function.

Dynamic analysis and electronic circuit implementation of a novel 3D autonomous system without linear terms

Mathematical models (ODEs) describing the dynamics of almost all continuous time chaotic nonlinear systems (e.g. Lorenz, Rossler, Chua, or Chen system) involve at least a nonlinear term in addition to linear terms. In this contribution, a novel (and singular) 3D autonomous chaotic system without linear terms is introduced. This system has an especial feature of having two twin strange attractors: one ordinary and one symmetric strange attractor when the time is reversed. The complex behavior of the model is investigated in terms of equilibria and stability, bifurcation diagrams, Lyapunov exponent plots, time series and Poincaré sections. Some interesting phenomena are found including for instance, period-doubling bifurcation, antimonotonicity (i.e. the concurrent creation and annihilation of periodic orbits) and chaos while monitoring the system parameters. Compared to the (unique) case previously reported by Xu and Wang (2014) [31], the system considered in this work displays a more ‘elegant’ mathematical expression and experiences richer dynamical behaviors. A suitable electronic circuit (i.e. the analog simulator) is designed and used for the investigations. Pspice based simulation results show a very good agreement with the theoretical analysis.

Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma

In this paper, a new Lorenz-like chaotic system describing the interaction of three resonantly coupled waves in plasma is studied. Explicit ultimate boundedness and global attraction domain are derived according to stability theory of dynamical systems. The innovation of the paper is that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets for this system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained. Finally, numerical localization of attractor is presented.