Showing papers on "Chaotic published in 2015"
TL;DR: The historical timeline of this topic back to the earliest known paper is established and it is shown that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals.
Abstract: We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
1,139 citations
TL;DR: A new two-dimensional Sine Logistic modulation map (2D-SLMM) which is derived from the Logistic and Sine maps is introduced which has the wider chaotic range, better ergodicity, hyperchaotic property and relatively low implementation cost.
585 citations
TL;DR: This paper introduces a general chaotic framework called the cascade chaotic system (CCS), and introduces a pseudo-random number generator (PRNG) and a data encryption system using a chaotic map generated by CCS.
Abstract: Chaotic maps are widely used in different applications. Motivated by the cascade structure in electronic circuits, this paper introduces a general chaotic framework called the cascade chaotic system (CCS). Using two 1-D chaotic maps as seed maps, CCS is able to generate a huge number of new chaotic maps. Examples and evaluations show the CCS’s robustness. Compared with corresponding seed maps, newly generated chaotic maps are more unpredictable and have better chaotic performance, more parameters, and complex chaotic properties. To investigate applications of CCS, we introduce a pseudo-random number generator (PRNG) and a data encryption system using a chaotic map generated by CCS. Simulation and analysis demonstrate that the proposed PRNG has high quality of randomness and that the data encryption system is able to protect different types of data with a high-security level.
263 citations
TL;DR: The FPGA realization of two multi-scroll chaotic oscillators that are characterized by their maximum Lyapunov exponent (MLE) for generating from 2- to 6-scrolls are shown, and the advantage of realizing those multi- scroll chaotic oscillator is that one can avoid the use of multiplier entities, thus optimizing FPGAs resources and increasing the processing speed, as well as realizing single constant multiplication (SCM) blocks.
174 citations
01 Jan 2015
TL;DR: This research work investigates the global chaos synchronization of Sprott’s jerk chaotic system using backstepping control method, a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict-feedback chaotic systems.
Abstract: This research work investigates the global chaos synchronization of Sprott’s jerk chaotic system using backstepping control method. Sprott’s jerk system (1997) is algebraically the simplest dissipative chaotic system consisting of five terms and a quadratic nonlinearity. Sprott’s chaotic system involves only five terms and one quadratic nonlinearity, while Rossler’s chaotic system (1976) involves seven terms and one quadratic nonlinearity. This work first details the properties of the Sprott’s jerk chaotic system. The phase portraits of the Sprott’s jerk system are described. The Lyapunov exponents of the Sprott’s jerk system are obtained as L 1 = 0.0525, L 2 = 0 and L 3 = −2.0727. The Lyapunov dimension of the Sprott’s jerk system is obtained as D L = 2.0253. Next, an active backstepping controller is designed for the global chaos synchronization of identical Sprott’s jerk systems with known parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict-feedback chaotic systems. Finally, an adaptive backstepping controller is designed for the global chaos synchronization of identical Sprott’s jerk systems with unknown parameters. MATLAB simulations are provided to validate and demonstrate the effectiveness of the proposed active and adaptive chaos synchronization schemes for the Sprott’s jerk systems.
172 citations
TL;DR: A novel system with an exponential nonlinear term, which can exhibit hidden attractors, is proposed in this work, and although new system possesses no equilibrium points, it displays rich dynamical behaviors, like chaos.
Abstract: Studying systems with hidden attractors is new attractive research direction because of its practical and threoretical importance. A novel system with an exponential nonlinear term, which can exhibit hidden attractors, is proposed in this work. Although new system possesses no equilibrium points, it displays rich dynamical behaviors, like chaos. By calculating Lyapunov exponents and bifurcation diagram, the dynamical behaviors of such system are discovered. Moreover, two important features of a chaotic system, the possibility of synchronization and the feasibility of the theoretical model, are also presented by introducing an adaptive synchronization scheme and designing a digital hardware platform-based emulator.
157 citations
TL;DR: In this research work, a novel sliding mode control method is proposed for the global chaos synchronisation of identical chaotic systems and the general result derived is established using Lyapunov stability theory.
Abstract: Synchronisation of chaotic systems is an important research problem in chaos theory. In this research work, a novel sliding mode control method is proposed for the global chaos synchronisation of identical chaotic systems. The general result derived using novel sliding mode control method is established using Lyapunov stability theory. As an application of the general result, the problem of global chaos synchronisation of identical Zhu chaotic systems (2010) is studied and a new sliding mode controller is derived. Numerical simulations have been shown to illustrate the phase portraits of Zhu chaotic system and the sliding mode controller design for the global chaos synchronisation of identical Zhu chaotic systems.
154 citations
TL;DR: A novel approach to fuzzy sampled-data control of chaotic systems is presented by using a time-dependent Lyapunov functional that makes full use of the information on the piecewise constant input and the actual sampling pattern.
Abstract: In this paper, a novel approach to fuzzy sampled-data control of chaotic systems is presented by using a time-dependent Lyapunov functional. The advantage of the new method is that the Lyapunov functional is continuous at sampling times but not necessarily positive definite inside the sampling intervals. Compared with the existing works, the constructed Lyapunov functional makes full use of the information on the piecewise constant input and the actual sampling pattern. In terms of a new parameterized linear matrix inequality (LMI) technique, a less conservative stabilization condition is derived to guarantee the exponential stability for the closed-loop fuzzy sampled-data system. By solving a set of LMIs, the fuzzy sampled-data controller can be easily obtained. Finally, the chaotic Lorenz system and Rossler’s system are employed to illustrate the feasibility and effectiveness of the proposed method.
150 citations
TL;DR: Improved standard FOA is improved by introducing the novel parameter integrated with chaos and overall research findings show that FOA with Chebyshev map show superiority in terms of reliability of global optimality and algorithm success rate.
Abstract: Display Omitted Development of new method named chaotic fruit fly optimization algorithm (CFOA).Fruit fly algorithm (FOA) is integrated with ten different chaos maps.Novel algorithm is tested on ten different well known benchmark problems.CFOA is compared with FOA, FOA with Levy distribution, and similar chaotic methods.Experiments show superiority of CFOA in terms of obtained statistical results. Fruit fly optimization algorithm (FOA) is recently presented metaheuristic technique that is inspired by the behavior of fruit flies. This paper improves the standard FOA by introducing the novel parameter integrated with chaos. The performance of developed chaotic fruit fly algorithm (CFOA) is investigated in details on ten well known benchmark problems using fourteen different chaotic maps. Moreover, we performed comparison studies with basic FOA, FOA with Levy flight distribution, and other recently published chaotic algorithms. Statistical results on every optimization task indicate that the chaotic fruit fly algorithm (CFOA) has a very fast convergence rate. In addition, CFOA is compared with recently developed chaos enhanced algorithms such as chaotic bat algorithm, chaotic accelerated particle swarm optimization, chaotic firefly algorithm, chaotic artificial bee colony algorithm, and chaotic cuckoo search. Overall research findings show that FOA with Chebyshev map show superiority in terms of reliability of global optimality and algorithm success rate.
149 citations
TL;DR: In this paper, a 4D memristive system modified from the 3D chaotic system proposed by Lu and Chen was studied, which has an uncountable infinite number of stable and unstable equilibria.
Abstract: This paper studies a four-dimensional (4D) memristive system modified from the 3D chaotic system proposed by Lu and Chen. The new system keeps the symmetry and dissipativity of the original system and has an uncountable infinite number of stable and unstable equilibria. By varying the strength of the memristor, we find rich complex dynamics, such as limit cycles, torus, chaos, and hyperchaos, which can peacefully coexist with the stable equilibria. To explain such coexistence, we compute the unstable manifolds of the equilibria, find that the manifolds create a safe zone for the hyperchaotic attractor, and also find many heteroclinic orbits. To verify the existence of hyperchaos in the 4D memristive circuit, we carry out a computer-assisted proof via a topological horseshoe with two-directional expansions, as well as a circuit experiment on oscilloscope views.
147 citations
TL;DR: A fast chaos- based image encryption scheme with a dynamic state variables selection mechanism is proposed to enhance the security and promote the efficiency of chaos-based image cryptosystems.
TL;DR: The author employs heterogeneous bit-permutation to reduce computation cost and improve permutation efficiency, then performs expanded XOR operation for R, G, B components of color images, and obtains cipher color images.
TL;DR: In this paper, a new mathematical model of the third-order autonomous deterministic dynamical system with associated chaotic motion has been proposed, whose unique property lies in the existence of circular equilibrium which was not, by referring to the best knowledge of the authors, so far reported.
Abstract: This paper brings a new mathematical model of the third-order autonomous deterministic dynamical system with associated chaotic motion. Its unique property lies in the existence of circular equilibrium which was not, by referring to the best knowledge of the authors, so far reported. Both mathematical analysis and circuitry implementation of the corresponding differential equations are presented. It is shown that discovered system provides a structurally stable strange attractor which fulfills fractal dimensionality and geometrical density and is bounded into a finite state space volume.
01 Jan 2015
TL;DR: A sliding mode controller is derived for the anti-synchronization of the identical Vaidyanathan–Madhavan chaotic systems using sliding mode control and the main result has been proved using Lyapunov stability theory.
Abstract: Anti-synchronization is an important type of synchronization of a pair of chaotic systems called the master and slave systems. The anti-synchronization characterizes the asymptotic vanishing of the sum of the states of the master and slave systems. In other words, anti-synchronization of master and slave system is said to occur when the states of the synchronized systems have the same absolute values but opposite signs. Anti-synchronization has applications in science and engineering. This work derives a general result for the anti-synchronization of identical chaotic systems using sliding mode control. The main result has been proved using Lyapunov stability theory. Sliding mode control (SMC) is well-known as a robust approach and useful for controller design in systems with parameter uncertainties. Next, as an application of the main result, anti-synchronizing controller has been designed for Vaidyanathan–Madhavan chaotic systems (2013). The Lyapunov exponents of the Vaidyanathan–Madhavan chaotic system are found as \(L_1 = 3.2226, L_2 = 0\) and \(L_3 = -30.3406\) and the Lyapunov dimension of the novel chaotic system is found as \(D_L = 2.1095\). The maximal Lyapunov exponent of the Vaidyanathan–Madhavan chaotic system is \(L_1 = 3.2226\). As an application of the general result derived in this work, a sliding mode controller is derived for the anti-synchronization of the identical Vaidyanathan–Madhavan chaotic systems. MATLAB simulations have been provided to illustrate the qualitative properties of the novel 3-D chaotic system and the anti-synchronizer results for the identical novel 3-D chaotic systems.
TL;DR: A seven-term novel 3-D conservative chaotic system with four quadratic nonlinearities with no equilibrium point is announced and an adaptive controller is designed to globally stabilize the novel conservative chaoticSystem with unknown parameters.
Abstract: First, this paper announces a seven-term novel 3-D conservative chaotic system with four quadratic nonlinearities. The conservative chaotic systems are characterized by the important property that they are volume conserving. The phase portraits of the novel conservative chaotic system are displayed and the mathematical properties are discussed. An important property of the proposed novel chaotic system is that it has no equilibrium point. Hence, it displays hidden chaotic attractors. The Lyapunov exponents of the novel conservative chaotic system are obtained as L1 = 0.0395,L2 = 0 and L3 =−0.0395. The Kaplan-Yorke dimension of the novel conservative chaotic system is DKY = 3. Next, an adaptive controller is designed to globally stabilize the novel conservative chaotic system with unknown parameters. Moreover, an adaptive controller is also designed to achieve global chaos synchronization of the identical conservative chaotic systems with unknown parameters. MATLAB simulations have been depicted to illustrate the phase portraits of the novel conservative chaotic system and also the adaptive control results.
TL;DR: The numerical simulations show the effectiveness of the proposed controller, via the improved Adams–Bashforth algorithm, in controlling chaos in a fractional order economic system.
Abstract: In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller.
TL;DR: The simulation experiments and theoretical analyses indicate that the proposed scheme is superior and able to resist exhaustive attack and statistical attack.
TL;DR: This work aims to propose a new approach based on a hybrid model of the Tinkerbell chaotic map, deoxyribonucleic acid (DNA) and cellular automata (CA) to encrypt the plain-image pixels.
TL;DR: This work derives a new result in random matrix theory: the spectral radius of a random connectivity matrix with block-structured variances, and shows how a small group of hyperexcitable neurons within the network can significantly increase the network's computational capacity by bringing it into the chaotic regime.
Abstract: In neural circuits, statistical connectivity rules strongly depend on cell-type identity. We study dynamics of neural networks with cell-type-specific connectivity by extending the dynamic mean-field method and find that these networks exhibit a phase transition between silent and chaotic activity. By analyzing the locus of this transition, we derive a new result in random matrix theory: the spectral radius of a random connectivity matrix with block-structured variances. We apply our results to show how a small group of hyperexcitable neurons within the network can significantly increase the network's computational capacity by bringing it into the chaotic regime.
01 Apr 2015
TL;DR: The effect of augmenting two different chaotic maps along with the uniform random number generator (RNG) in the popular MOO algorithm-the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is explored.
Abstract: Multi-objective optimization-based fractional-order PID controller is designed.NSGA-II algorithm is augmented with chaotic Logistic and Henon map.Load disturbance rejection and controller effort are minimized as two conflicting objectives.FOPID controller outperforms the PID controller in suppressing frequency deviation.Better trade-off is obtained for load-frequency control of power systems with FOPID. Fractional-order proportional-integral-derivative (FOPID) controllers are designed for load-frequency control (LFC) of two interconnected power systems. Conflicting time-domain design objectives are considered in a multi-objective optimization (MOO)-based design framework to design the gains and the fractional differ-integral orders of the FOPID controllers in the two areas. Here, we explore the effect of augmenting two different chaotic maps along with the uniform random number generator (RNG) in the popular MOO algorithm-the Non-dominated Sorting Genetic Algorithm-II (NSGA-II). Different measures of quality for MOO, e.g. hypervolume indicator, moment of inertia-based diversity metric, total Pareto spread, spacing metric, are adopted to select the best set of controller parameters from multiple runs of all the NSGA-II variants (i.e. nominal and chaotic versions). The chaotic versions of the NSGA-II algorithm are compared with the standard NSGA-II in terms of solution quality and computational time. In addition, the Pareto optimal fronts showing the trade-off between the two conflicting time domain design objectives are compared to show the advantage of using the FOPID controller over that with simple PID controller. The nature of fast/slow and high/low noise amplification effects of the FOPID structure or the four quadrant operation in the two inter-connected areas of the power system is also explored. A fuzzy logic-based method has been adopted next to select the best compromise solution from the best Pareto fronts corresponding to each MOO comparison criteria. The time-domain system responses are shown for the fuzzy best compromise solutions under nominal operating conditions. Comparative analysis on the merits and de-merits of each controller structure is reported then. A robustness analysis is also done for the PID and the FOPID controllers.
TL;DR: Results show the good performance of the MLP network model in predicting carbon price, based on phase reconstruction, which possesses good performance in both level and directional measurement.
Abstract: Carbon prices are studied from the chaotic point of view.Carbon prices are found to be chaotic rather than stochastic.The multi-layer perceptron network model is based on reconstructed phase space.The MLP model possesses good performance in both level and directional measurement. Dec14 and Dec15, carbon prices of European Union Emissions Trading Scheme in phase III, are studied from the chaotic point of view. Firstly, chaotic characteristics of carbon price series are identified by three classic indicators: the maximum Lyapunov exponent, the correlation dimension and the Kolmogorov entropy. Both Dec14 and Dec15 have positive maximum Lyapunov exponents, and fractal correlation dimensions and non-zero Kolmogorov entropies, which demonstrates that the fluctuant nature of carbon price can be explained as a chaotic phenomenon. The carbon price dynamic system is recovered by reconstructing the phase space. Based on phase reconstruction, an multi-layer perceptron neural network prediction model is set up for carbon price to characterize its strong nonlinearity. The logic of the MLP are described in detail. K-fold cross-validation method is applied to show the validation of the model. Four measurements in level and directional prediction are used to evaluate the performance of the MLP model. Results show the good performance of the MLP network model in predicting carbon price.
TL;DR: The results of several experimental analyses about randomness, sensitivity and correlation of the cipher-images show that the proposed algorithm has high security level, high sensitivity and high speed which can be adopted for network security and secure communications.
Abstract: In recent years, several algorithms of secure image encryption were studied and developed through chaotic processes. Most of the previous algorithms encrypt color components independently. In this paper, a novel image encryption algorithm based on quantum chaotic map and diffusion–permutation architecture is proposed. First, the new algorithm employs the quantum logistic map to diffuse the relationship of pixels in color components. Next, the keystreams generated by the two-dimensional logistic map are exploited to not only modify the value of diffused pixels, but also spatially permute the pixels of color components at the same time and make the three components affect one another. Finally, the random circular shift operation is applied to the result of the modified and permuted pixels to rearrange bits of each encrypted pixel. In order to achieve the high complexity and the high randomness between these generated keystreams, the two-dimensional logistic map and the quantum chaotic map are independently coupled with nearest-neighboring coupled-map lattices. The results of several experimental analyses about randomness, sensitivity and correlation of the cipher-images show that the proposed algorithm has high security level, high sensitivity and high speed which can be adopted for network security and secure communications.
TL;DR: The stable design of fuzzy logic control systems that deal with a general class of chaotic processes is proposed on the basis of a stability analysis theorem, which employs Lyapunov's direct method and the separate stability analysis of each rule in the fuzzy logic controller (FLC).
Abstract: This paper proposes a new approach to the stable design of fuzzy logic control systems that deal with a general class of chaotic processes. The stable design is carried out on the basis of a stability analysis theorem, which employs Lyapunov's direct method and the separate stability analysis of each rule in the fuzzy logic controller (FLC). The stability analysis theorem offers sufficient conditions for the stability of a general class of chaotic processes controlled by Takagi---Sugeno---Kang FLCs. The approach suggested in this paper is advantageous because inserting a new rule requires the fulfillment of only one of the conditions of the stability analysis theorem. Two case studies concerning the fuzzy logic control of representative chaotic systems that belong to the general class of chaotic systems are included in order to illustrate our stable design approach. A set of simulation results is given to validate the theoretical results.
01 Jan 2015
TL;DR: This research work describes a nine-term 3-D novel chaotic system with four quadratic nonlinearities and describes the adaptive control and synchronization of the identical novel chaotic systems with unknown system parameters.
Abstract: This research work describes a nine-term 3-D novel chaotic system with four quadratic nonlinearities. First, this work describes the dynamic analysis of the novel chaotic system and qualitative properties of the novel chaotic system are derived. The Lyapunov exponents of the nine-term novel chaotic system are obtained as \( L_{1} = 9.45456,\;L_{2} = 0 \) and \( L_{3} = - 30.50532 \). Since the maximal Lyapunov exponent (MLE) of the novel chaotic system is \( L_{1} = 9.45456 \), which is a high value, the novel chaotic system exhibits strong chaotic properties. Next, this work describes the adaptive control of the novel chaotic system with unknown system parameters. Also, this work describes the adaptive synchronization of the identical novel chaotic systems with unknown system parameters. The adaptive control and synchronization results are proved using Lyapunov stability theory. MATLAB simulations are given to demonstrate and validate all the main results derived in this work for the nine-term 3-D novel chaotic system.
01 Jan 2015
TL;DR: A new chaotic-based bat swarm optimisation algorithm, which significantly outperforms conventional BSO, cuckoo search optimisation (CSO), big bang-big crunch algorithm (BBBC), gravitational search algorithm (GSA) and genetic algorithm (GA), and is incorporated into BSO to mitigate premature convergence problem.
Abstract: A new chaotic-based bat swarm optimisation algorithm has been developed.For devising it, different chaotic Bat strategies map functions were tested.In the proposed chaotic BSO, the loudness is updated via multiplying a linearly decreasing function by chaotic map function. Bat swarm optimisation (BSO) is a novel heuristic optimisation algorithm that is being used for solving different global optimisation problems. The paramount problem in BSO is that it severely suffers from premature convergence problem, that is, BSO is easily trapped in local optima. In this paper, chaotic-based strategies are incorporated into BSO to mitigate this problem. Ergodicity and non-repetitious nature of chaotic functions can diversify the bats and mitigate premature convergence problem. Eleven different chaotic map functions along with various chaotic BSO strategies are investigated experimentally and the best one is chosen as the suitable chaotic strategy for BSO. The results of applying the proposed chaotic BSO to different benchmark functions vividly show that premature convergence problem has been mitigated efficiently. Actually, chaotic-based BSO significantly outperforms conventional BSO, cuckoo search optimisation (CSO), big bang-big crunch algorithm (BBBC), gravitational search algorithm (GSA) and genetic algorithm (GA).
TL;DR: In this article, a non-parametric method based on the mapping of a multidimensional time series into a multilayer network is presented, which allows to extract information on a high dimensional dynamical system through the analysis of the structure of the associated multiplex network.
Abstract: Our understanding of a variety of phenomena in physics, biology and economics crucially depends on the analysis of multivariate time series. While a wide range tools and techniques for time series analysis already exist, the increasing availability of massive data structures calls for new approaches for multidimensional signal processing. We present here a non-parametric method to analyse multivariate time series, based on the mapping of a multidimensional time series into a multilayer network, which allows to extract information on a high dimensional dynamical system through the analysis of the structure of the associated multiplex network. The method is simple to implement, general, scalable, does not require ad hoc phase space partitioning, and is thus suitable for the analysis of large, heterogeneous and non-stationary time series. We show that simple structural descriptors of the associated multiplex networks allow to extract and quantify nontrivial properties of coupled chaotic maps, including the transition between different dynamical phases and the onset of various types of synchronization. As a concrete example we then study financial time series, showing that a multiplex network analysis can efficiently discriminate crises from periods of financial stability, where standard methods based on time-series symbolization often fail.
TL;DR: In this paper, the authors proposed a five-term 3D novel conservative chaotic system with a quadratic nonlinearity and a quartic non-linearity, which has the important property that they are volume conserving.
Abstract: This research work proposes a five-term 3-D novel conservative chaotic system with a quadratic nonlinearity and a quartic nonlinearity. The conservative chaotic systems have the important property that they are volume conserving. The Lyapunov exponents of the 3-D novel chaotic system are obtained as L! = 0.0836, L! = 0 and L! = −0.0836. Since the sum of the Lyapunov exponents is zero, the 3-D novel chaotic system is conservative. Thus, the Kaplan-Yorke dimension of the 3-D novel chaotic system is easily seen as 3.0000. The phase portraits of the novel chaotic system simulated using MATLAB depict the chaotic attractor of the novel system. This research work also discusses other qualitative properties of the system. Next, an adaptive controller is designed to achieve Generalized Projective Synchronization (GPS) of two identical novel chaotic systems with unknown system parameters. MATLAB simulations are shown to validate and demonstrate the GPS results derived in this work.
TL;DR: This paper introduces a seven-term novel 3-D chaotic system with three quadratic nonlinearities and derives new results for the adaptive control and synchronization design of the identical novel chaotic systems with unknown parameters.
Abstract: First, this paper introduces a seven-term novel 3-D chaotic system and discusses its qualitative properties. The proposed system is a seven-term novel polynomial chaotic system with three quadratic nonlinearities. The Lyapunov exponents of the novel chaotic system are obtained as L1 = 3.3226, L2 = 0 and L3 = –30.3406. The maximal Lyapunov exponent (MLE) for the novel chaotic system is obtained as L1 = 3.3226 and Lyapunov dimension as DL = 2.1095. Next, we derive new results for the adaptive control design of the novel chaotic system with unknown parameters. Next, we derive new results for the adaptive synchronization design of the identical novel chaotic systems with unknown parameters. The adaptive control and synchronization results have been established using adaptive control theory and Lyapunov stability theory. Numerical simulations with MATLAB have been shown to validate and illustrate all the new results derived in this paper.
TL;DR: The Lyapunov exponents of one and two dimensional fractional logistic maps are calculated and the positive ones are used to distinguish the chaotic areas of the maps.