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Journal articles on the topic 'Fractional chaotic system'

Author: Grafiati

Published: 16 March 2023

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1

Yang, Chunde, Hao Cai, and Ping Zhou. "Compound Generalized Function Projective Synchronization for Fractional-Order Chaotic Systems." Discrete Dynamics in Nature and Society 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/7563416.

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A modified function projective synchronization for fractional-order chaotic system, called compound generalized function projective synchronization (CGFPS), is proposed theoretically in this paper. There are one scaling-drive system, more than one base-drive system, and one response system in the scheme of CGFPS, and the scaling function matrices come from multidrive systems. The proposed CGFPS technique is based on the stability theory of fractional-order system. Moreover, we achieve the CGFPS between three-driver chaotic systems, that is, the fractional-order Arneodo chaotic system, the fractional-order Chen chaotic system, and the fractional-order Lu chaotic system, and one response chaotic system, that is, the fractional-order Lorenz chaotic system. Numerical experiments are demonstrated to verify the effectiveness of the CGFPS scheme.

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2

Hu, Jian-Bing, and Ling-Dong Zhao. "Finite-Time Synchronizing Fractional-Order Chaotic Volta System with Nonidentical Orders." Mathematical Problems in Engineering 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/264136.

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We investigate synchronizing fractional-order Volta chaotic systems with nonidentical orders in finite time. Firstly, the fractional chaotic system with the same structure and different orders is changed to the chaotic systems with identical orders and different structure according to the property of fractional differentiation. Secondly, based on the lemmas of fractional calculus, a controller is designed according to the changed fractional chaotic system to synchronize fractional chaotic with nonidentical order in finite time. Numerical simulations are performed to demonstrate the effectiveness of the method.

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3

Zhou, Ping, and Rui Ding. "An Adaptive Tracking Control of Fractional-Order Chaotic Systems with Uncertain System Parameter." Mathematical Problems in Engineering 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/521549.

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An adaptive tracking control scheme is presented for fractional-order chaotic systems with uncertain parameter. It is theoretically proved that this approach can make the uncertain parameter fractional-order chaotic system track any given reference signal and the uncertain system parameter is estimated through the adaptive tracking control process. Furthermore, the reference signal may belong to other integer-orders chaotic system or belong to different fractional-order chaotic system with different fractional orders. Two examples are presented to demonstrate the effectiveness of the proposed method.

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4

EL-KHAZALI, REYAD, WAJDI AHMAD, and YOUSEF AL-ASSAF. "SLIDING MODE CONTROL OF GENERALIZED FRACTIONAL CHAOTIC SYSTEMS." International Journal of Bifurcation and Chaos 16, no. 10 (October 2006): 3113–25. http://dx.doi.org/10.1142/s0218127406016719.

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A sliding mode control technique is introduced for generalized fractional chaotic systems. These systems are governed by a set of fractional differential equations of incommensurate orders. The proposed design method relies on the fact that the stability region of a fractional system contains the stability region of its underlying integer-order model. A sliding mode controller designed for an equivalent integer-order chaotic system is used to stabilize all its corresponding fractional chaotic systems. The design technique is demonstrated using two generalized fractional chaotic models; a chaotic oscillator and the Chen system. The effect of the total fractional order is investigated with respect to the controller effort and the convergence rate of the system response to the origin. Numerical simulations validate the main results of this work.

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5

Niu, Yujun, Xuming Sun, Cheng Zhang, and Hongjun Liu. "Anticontrol of a Fractional-Order Chaotic System and Its Application in Color Image Encryption." Mathematical Problems in Engineering 2020 (March 12, 2020): 1–12. http://dx.doi.org/10.1155/2020/6795964.

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This paper investigates the anticontrol of the fractional-order chaotic system. The necessary condition of the anticontrol of the fractional-order chaotic system is proposed, and based on this necessary condition, a 3D fractional-order chaotic system is driven to two new 4D fractional-order hyperchaotic systems, respectively, without changing the parameters and fractional order. Hyperchaotic properties of these new fractional dynamic systems are confirmed by Lyapunov exponents and bifurcation diagrams. Furthermore, a color image encryption algorithm is designed based on these fractional hyperchaotic systems. The effectiveness of their application in image encryption is verified.

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6

WANG, XING-YUAN, GUO-BIN ZHAO, and YU-HONG YANG. "DIVERSE STRUCTURE SYNCHRONIZATION OF FRACTIONAL ORDER HYPER-CHAOTIC SYSTEMS." International Journal of Modern Physics B 27, no. 11 (April 25, 2013): 1350034. http://dx.doi.org/10.1142/s0217979213500343.

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This paper studied the dynamic behavior of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system, then numerical analysis of the different fractional orders hyper-chaotic systems are carried out under the predictor–corrector method. We proved the two systems are in hyper-chaos when the maximum and the second largest Lyapunov exponential are calculated. Also the smallest orders of the systems are proved when they are in hyper-chaos. The diverse structure synchronization of the fractional order hyper-chaotic Lorenz system and the fractional order hyper-chaotic Rössler system is realized using active control method. Numerical simulations indicated that the scheme was always effective and efficient.

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7

Jiang, Cuimei, Shutang Liu, and Chao Luo. "A New Fractional-Order Chaotic Complex System and Its Antisynchronization." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/326354.

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We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.

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8

Fang, Jing, and Ruo Xun Zhang. "Synchronization of Incommensurate Fractional-Order Chaotic System Using Adaptive Control." Applied Mechanics and Materials 602-605 (August 2014): 946–49. http://dx.doi.org/10.4028/www.scientific.net/amm.602-605.946.

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This paper investigates the synchronization of incommensurate fractional-order chaotic systems, and proposes a modified adaptive-feedback controller for fractional-order chaos synchronization based on Lyapunov stability theory, fractional order differential inequality and adaptive control theory. This synchronization approach that is simple, global and theoretically rigorous enables synchronization of fractional-order chaotic systems be achieved in a systematic way. Simulation results for a fractional-order chaotic system is provided to illustrate the effectiveness of the proposed scheme.

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9

Zhou, Ping, Rui Ding, and Yu-xia Cao. "Hybrid Projective Synchronization for Two Identical Fractional-Order Chaotic Systems." Discrete Dynamics in Nature and Society 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/768587.

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A hybrid projective synchronization scheme for two identical fractional-order chaotic systems is proposed in this paper. Based on the stability theory of fractional-order systems, a controller for the synchronization of two identical fractional-order chaotic systems is designed. This synchronization scheme needs not to absorb all the nonlinear terms of response system. Hybrid projective synchronization for the fractional-order Chen chaotic system and hybrid projective synchronization for the fractional-order hyperchaotic Lu system are used to demonstrate the validity and feasibility of the proposed scheme.

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10

Cui, Yan, Hongjun He, Guan Sun, and Chenhui Lu. "Analysis and Control of Fractional Order Generalized Lorenz Chaotic System by Using Finite Time Synchronization." Advances in Mathematical Physics 2019 (July 3, 2019): 1–12. http://dx.doi.org/10.1155/2019/3713789.

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In this paper, we present a corresponding fractional order three-dimensional autonomous chaotic system based on a new class of integer order chaotic systems. We found that the fractional order chaotic system belongs to the generalized Lorenz system family by analyzing its linear term and topological structure. We also found that the equilibrium point generated by the fractional order system belongs to the unstable saddle point through the prediction correction method and the fractional order stability theory. The complexity of fractional order chaotic system is given by spectral entropy algorithm andC0algorithm. We concluded that the fractional order chaotic system has a higher complexity. The fractional order system can generate rich dynamic behavior phenomenon with the values of the parameters and the order changed. We applied the finite time stability theory to design the finite time synchronous controller between drive system and corresponding system. The numerical simulations demonstrate that the controller provides fast and efficient method in the synchronization process.

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11

Fu, Xiuwei, Li Fu, and Hashem Imani Marrani. "Synchronization and Anti-Synchronization of a Novel Fractional Order Chaotic System with an exponential term." Electrotehnica, Electronica, Automatica 70, no. 2 (May 15, 2022): 57–65. http://dx.doi.org/10.46904/eea.22.70.2.1108007.

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Today, chaotic systems have become one of the most important tools for encrypting and secure transmission of information. Other applications of these systems in economics, geography, sociology, and the like are not hidden from anyone. Despite the presentation of various chaotic systems, it is necessary to study and present new and more accurate chaotic systems. It is obvious that fractional models are more accurate and yield better results than integer order models. In this paper, the synchronization and anti-synchronization of an innovative fractional order chaotic system is investigated based on the nonlinear control method. In the proposed chaotic system, there is an exponential term that leads to behaviour very different from the integer order chaotic systems. Two different approaches have been proposed to achieve the synchronization and anti-synchronization goals between the proposed new fractional chaotic systems. A backstopping approach has been used to synchronize, and in addition to achieving this goal, it also ensures stability in Lyapunov's concept. Anti-synchronization between the two new fractional systems is also achieved by applying the active control method, and subsequently Lyapunov stability is shown under the proposed method. The simulation results in MATLAB environment show the synchronization and anti-synchronization effectiveness for the proposed innovative fractional order chaotic system.

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12

Odibat, Zaid, Nathalie Corson, M. A. Aziz-Alaoui, and Ahmed Alsaedi. "Chaos in Fractional Order Cubic Chua System and Synchronization." International Journal of Bifurcation and Chaos 27, no. 10 (September 2017): 1750161. http://dx.doi.org/10.1142/s0218127417501619.

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This paper deals with fractional order Chua system with cubic nonlinearity (cubic Chua system), which is a modification of Chua system. The aim is, first, to study the chaotic behavior in fractional order cubic Chua system. We found that chaos indeed exists in the fractional version of this system. The necessary condition for exhibiting chaotic attractors similar to its integer order counterpart is presented. This condition is used to distinguish for which parameters and orders the system generates double scroll chaotic attractors. The synchronization problem between two coupled chaotic fractional order cubic Chua systems is then addressed. An adaptive feedback control scheme for the synchronization with suitable feedback nonlinear controller applied to the response system is presented. Numerical simulations are performed to verify the theoretical analysis.

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13

Zhang, Fan Di. "Anti-Synchronization Control of the Complex Lu System." Applied Mechanics and Materials 721 (December 2014): 269–72. http://dx.doi.org/10.4028/www.scientific.net/amm.721.269.

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This paper propose fractional-order Lu complex system. Moreover, projective synchronization control of the fractional-order hyper-chaotic complex Lu system is studied based on feedback technique and the stability theorem of fractional-order systems, the scheme of anti-synchronization for the fractional-order hyper-chaotic complex Lu system is presented. Numerical simulations on examples are presented to show the effectiveness of the proposed control strategy.

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14

Hu, Xikui, and Ping Zhou. "Coexisting Three-Scroll and Four-Scroll Chaotic Attractors in a Fractional-Order System by a Three-Scroll Integer-Order Memristive Chaotic System and Chaos Control." Complexity 2020 (January 8, 2020): 1–7. http://dx.doi.org/10.1155/2020/5796529.

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Based on the integer-order memristive system that can generate two-scroll, three-scroll, and four-scroll chaotic attractors, in this paper, we found other phenomena that two kinds of three-scroll chaotic attractors coexist in this system with different initial conditions. Furthermore, we proposed a coexisting fractional-order system based on the three-scroll chaotic attractors system, in which the three-scroll or four-scroll chaotic attractors emerged with different fractional-orders q. Meanwhile, with fractional-order q=0.965 and different initial conditions, coexistence of two kinds of three-scroll and four-scroll chaotic attractors is found simultaneously. Finally, we discussed controlling chaos for the fractional-order memristive chaotic system.

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15

Zhou, Ziwei, and Shuo Wang. "Design and implementation of a new fractional-order Hopfield neural network system." Physica Scripta 97, no. 2 (January 28, 2022): 025206. http://dx.doi.org/10.1088/1402-4896/ac4c50.

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Abstract In this work, a novel chaotic system of fractional-order based on the model of Hopfield Neural Network (HNN) is proposed. The numerical solutions of the 4-neurons-based HNN fractional-order chaotic system are obtained by using the Adomain decomposition method. The dynamical performances of the 4-neurons-based HNN fractional-order chaotic system are explored through attractor trajectories, bifurcation diagrams, Lyapunov exponents, SE complexity and chaotic diagram based on SE complexity. In addition, the 4-neurons-based HNN fractional-order chaotic system is implemented based on the Multisim platform. The experimental results indicate that the 4-neurons-based HNN fractional-order chaotic system has rich dynamic behavior, and the influence of different order on the dynamical properties of the system are particularly great. This research will provide theoretical foundation and experimental basis for the hardware implementation and application of the 4-neurons-based HNN fractional-order chaotic system.

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16

Matouk, A. E. "Chaos Synchronization between Two Different Fractional Systems of Lorenz Family." Mathematical Problems in Engineering 2009 (2009): 1–11. http://dx.doi.org/10.1155/2009/572724.

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This work investigates chaos synchronization between two different fractional order chaotic systems of Lorenz family. The fractional order Lü system is controlled to be the fractional order Chen system, and the fractional order Chen system is controlled to be the fractional order Lorenz-like system. The analytical conditions for the synchronization of these pairs of different fractional order chaotic systems are derived by utilizing Laplace transform. Numerical simulations are used to verify the theoretical analysis using different values of the fractional order parameter.

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17

Li, Bo, Xiaobing Zhou, and Yun Wang. "Combination Synchronization of Three Different Fractional-Order Delayed Chaotic Systems." Complexity 2019 (November 7, 2019): 1–9. http://dx.doi.org/10.1155/2019/5184032.

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Time delay is a frequently encountered phenomenon in some practical engineering systems and introducing time delay into a system can enrich its dynamic characteristics. There has been a plenty of interesting results on fractional-order chaotic systems or integer-order delayed chaotic systems, but the problem of synchronization of fractional-order chaotic systems with time delays is in the primary stage. Combination synchronization of three different fractional-order delayed chaotic systems is investigated in this paper. It is an extension of combination synchronization of delayed chaotic systems or combination synchronization of fractional-order chaotic systems. With the help of stability theory of linear fractional-order systems with multiple time delays, we design controllers to achieve combination synchronization of three different fractional-order delayed chaotic systems. In addition, numerical simulations have been performed to demonstrate and verify the theoretical analysis.

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18

Zhou, Ping, Kun Huang, and Chun-de Yang. "A Fractional-Order Chaotic System with an Infinite Number of Equilibrium Points." Discrete Dynamics in Nature and Society 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/910189.

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A new 4D fractional-order chaotic system, which has an infinite number of equilibrium points, is introduced. There is no-chaotic behavior for its corresponded integer-order system. We obtain that the largest Lyapunov exponent of this 4D fractional-order chaotic system is 0.8939 and yield the chaotic attractor. A chaotic synchronization scheme is presented for this 4D fractional-order chaotic system. Numerical simulations is verified the effectiveness of the proposed scheme.

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19

Li, Zhang, and Yang. "Numerical Analysis, Circuit Simulation, and Control Synchronization of Fractional-Order Unified Chaotic System." Mathematics 7, no. 11 (November 8, 2019): 1077. http://dx.doi.org/10.3390/math7111077.

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The traditional method of solving fractional chaotic system has the problem of low precision and is computationally cumbersome. In this paper, different fractional-order calculus solutions, the Adams prediction–correction method, the Adomian decomposition method and the improved Adomian decomposition method, are applied to the numerical analysis of the fractional-order unified chaotic system. The result shows that different methods have higher precision, smaller computational complexity, and shorter running time, in which the improved Adomian decomposition method works best. Then, based on the fractional-order chaotic circuit design theory, the circuit diagram of fractional-order unified chaotic system is designed. The result shows that the circuit simulation diagram of fractional-order unified chaotic system is basically consistent with the phase space diagram obtained from the numerical solution of the system, which verifies the existence of the fractional-order unified chaotic system of 0.9-order. Finally, the active control method is used to control and synchronize in the fractional-order unified chaotic system, and the experiment result shows that the method can achieve synchronization in a shorter time and has a better control performance.

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20

Ozkaynak, Fatih. "A Novel Random Number Generator Based on Fractional Order Chaotic Chua System." Elektronika ir Elektrotechnika 26, no. 1 (February 17, 2020): 52–57. http://dx.doi.org/10.5755/j01.eie.26.1.25310.

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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.

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21

yang, Ningning, Cheng Xu, Chaojun wu, Rong Jia, and Chongxin Liu. "Modeling and Analysis of a Fractional-Order Generalized Memristor-Based Chaotic System and Circuit Implementation." International Journal of Bifurcation and Chaos 27, no. 13 (December 15, 2017): 1750199. http://dx.doi.org/10.1142/s0218127417501991.

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Memristor is a nonlinear “missing circuit element”, that can easily achieve chaotic oscillation. Memristor-based chaotic systems have received more and more attention. Research shows that fractional-order systems are more close to real systems. As an important parameter, the order can increase the flexibility and degree of freedom of the system. In this paper, a fractional-order generalized memristor, which consists of a diode bridge and a parallel circuit with an equivalent unit circuit and a linear resistance, is proposed. Frequency and electrical characteristics of the fractional-order memristor are analyzed. A chain structure circuit is used to implement the fractional-order unit circuit. Then replacing the conventional Chua’s diode by the fractional-order generalized memristor, a fractional-order memristor-based chaotic circuit is proposed. A large amount of research work has been done to investigate the influence of the order on the dynamical behaviors of the fractional-order memristor-based chaotic circuit. Varying with the order, the system enters the chaotic state from the periodic state through the Hopf bifurcation and period-doubling bifurcation. The chaotic state of the system has two types of attractors: single-scroll and double-scroll attractor. The stability theory of fractional-order systems is used to determine the minimum order occurring Hopf bifurcation. And the influence of the initial value on the system is analyzed. Circuit simulations are designed to verify the results of theoretical analysis and numerical simulation.

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Gao, Xin. "Chaotic Dynamics of Fractional-Order Liu System." Applied Mechanics and Materials 55-57 (May 2011): 1327–31. http://dx.doi.org/10.4028/www.scientific.net/amm.55-57.1327.

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In this paper, we numerically investigate the chaotic behaviors of a new fractional-order system. We find that chaotic behaviors exist in the fractional-order system with order less than 3. The lowest order we find to have chaos is 2.4 in such system. In addition, we numerically simulate the continuances of the chaotic behaviors in the fractional-order system with orders from 2.7 to 3. Our investigations are validated through numerical simulations.

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Li, Xiang, Zhijun Li, and Zihao Wen. "One-to-four-wing hyperchaotic fractional-order system and its circuit realization." Circuit World 46, no. 2 (January 10, 2020): 107–15. http://dx.doi.org/10.1108/cw-03-2019-0026.

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Purpose This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems with variable-wing attractors, although some chaotic systems can generate one-to-four-wing attractors, none of them are hyperchaotic attractors, which is incomplete for the dynamic characteristics of chaotic systems. Design/methodology/approach A novel 4D fractional-order hyperchaotic system is proposed based on the classical three-dimensional Lü system. The complex and abundant dynamic behaviors of the fractional-order system are analyzed by phase diagrams, bifurcation diagrams and the corresponding Lyapunov exponents. In addition, SE and C0 algorithms are used to analyze the complexity of the fractional-order system. Then, the influence of order q on the system is also investigated. Finally, the circuit is implemented using physical components. Findings The most particular interest is that the system can generate one-to-four-wing hyperchaotic attractors with only one parameter variation. Then, the hardware circuit experimental results tally with the numerical simulations, which proves the validity and feasibility of the fractional-order hyperchaotic system. Besides, under different initial conditions, coexisting attractors can be obtained by changing the parameter d or the order q. Then, the complexity analysis of the system shows that the fractional-order chaotic system has higher complexity than the corresponding integer-order chaotic system. Originality/value The circuit structure of the fractional-order hyperchaotic system is simple and easy to implement, and one-to-four-wing hyperchaotic attractors can be observed in the circuit. To the best of the knowledge, this unique phenomenon has not been reported in any literature. It is of great reference value to analysis and circuit realization of fractional-order chaotic systems.

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Zhou, Ping, and Rui Ding. "Control and Synchronization of the Fractional-Order Lorenz Chaotic System via Fractional-Order Derivative." Mathematical Problems in Engineering 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/214169.

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The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.

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Zhang, Pei, Renyu Yang, Renhuan Yang, Gong Ren, Xiuzeng Yang, Chuangbiao Xu, Baoguo Xu, Huatao Zhang, Yanning Cai, and Yaosheng Lu. "Parameter estimation for fractional-order chaotic systems by improved bird swarm optimization algorithm." International Journal of Modern Physics C 30, no. 11 (November 2019): 1950086. http://dx.doi.org/10.1142/s0129183119500864.

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The essence of parameter estimation for fractional-order chaotic systems is a multi-dimensional parameter optimization problem, which is of great significance for implementing fractional-order chaos control and synchronization. Aiming at the parameter estimation problem of fractional-order chaotic systems, an improved algorithm based on bird swarm algorithm is proposed. The proposed algorithm further studies the social behavior of the original bird swarm algorithm and optimizes the foraging behavior in the original bird swarm algorithm. This method is applied to parameter estimation of fractional-order chaotic systems. Fractional-order unified chaotic system and fractional-order Lorenz system are selected as two examples for parameter estimation systems. Numerical simulation shows that the algorithm has better convergence accuracy, convergence speed and universality than bird swarm algorithm, artificial bee colony algorithm, particle swarm optimization and genetic algorithm.

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Yuan, Jian, Bao Shi, and Wenqiang Ji. "Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems." Advances in Mathematical Physics 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/576709.

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Recently, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. In this paper we introduce a novel class of fractional chaotic systems in the pseudo state space and propose an adaptive sliding mode control scheme to stabilize the chaotic systems in the presence of uncertainties and external disturbances whose bounds are unknown. To verify the effectiveness of the proposed adaptive sliding mode control technique, numerical simulations of control design of fractional Lorenz's system and Chen's system are presented.

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Haq, Ihtisham Ul, Shabir Ahmad, Sayed Saifullah, Kamsing Nonlaopon, and Ali Akgül. "Analysis of fractal fractional Lorenz type and financial chaotic systems with exponential decay kernels." AIMS Mathematics 7, no. 10 (2022): 18809–23. http://dx.doi.org/10.3934/math.20221035.

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<abstract><p>In this work, we formulate a fractal fractional chaotic system with cubic and quadratic nonlinearities. A fractal fractional chaotic Lorenz type and financial systems are studied using the Caputo Fabrizo (CF) fractal fractional derivative. This study focuses on the characterization of the chaotic nature, and the effects of the fractal fractional-order derivative in the CF sense on the evolution and behavior of each proposed systems. The stability of the equilibrium points for the both systems are investigated using the Routh-Hurwitz criterion. The numerical scheme, which includes the discretization of the CF fractal-fractional derivative, is used to depict the phase portraits of the fractal fractional chaotic Lorenz system and the fractal fractional-order financial system. The simulation results presented in both cases include the two- and three-dimensional phase portraits to evaluate the applications of the proposed operators.</p></abstract>

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Liu, Juan, Xuefeng Cheng, and Ping Zhou. "Circuit Implementation Synchronization between Two Modified Fractional-Order Lorenz Chaotic Systems via a Linear Resistor and Fractional-Order Capacitor in Parallel Coupling." Mathematical Problems in Engineering 2021 (August 19, 2021): 1–8. http://dx.doi.org/10.1155/2021/6771261.

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In this study, a modified fractional-order Lorenz chaotic system is proposed, and the chaotic attractors are obtained. Meanwhile, we construct one electronic circuit to realize the modified fractional-order Lorenz chaotic system. Most importantly, using a linear resistor and a fractional-order capacitor in parallel coupling, we suggested one chaos synchronization scheme for this modified fractional-order Lorenz chaotic system. The electronic circuit of chaos synchronization for modified fractional-order Lorenz chaotic has been given. The simulation results verify that synchronization scheme is viable.

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Zhang, Weiwei, Jinde Cao, Ahmed Alsaedi, and Fuad Eid S. Alsaadi. "Synchronization of Time Delayed Fractional Order Chaotic Financial System." Discrete Dynamics in Nature and Society 2017 (2017): 1–5. http://dx.doi.org/10.1155/2017/1230396.

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The research on a time delayed fractional order financial chaotic system is a hot issue. In this paper, synchronization of time delayed fractional order financial chaotic system is studied. Based on comparison principle of linear fractional equation with delay, by using a fractional order inequality, a sufficient condition is obtained to guarantee the synchronization of master-slave systems. An example is exploited to show the feasibility of the theoretical results.

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Sene, Ndolane. "Theory and applications of new fractional-order chaotic system under Caputo operator." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 12, no. 1 (October 27, 2021): 20–38. http://dx.doi.org/10.11121/ijocta.2022.1108.

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This paper introduces the properties of a fractional-order chaotic system described by the Caputo derivative. The impact of the fractional-order derivative has been focused on. The phase portraits in different orders are obtained with the aids of the proposed numerical discretization, including the discretization of the Riemann-Liouville fractional integral. The stability analysis has been used to help us to delimit the chaotic region. In other words, the region where the order of the Caputo derivative involves and where the presented system in this paper is chaotic. The nature of the chaos has been established using the Lyapunov exponents in the fractional context. The schematic circuit of the proposed fractional-order chaotic system has been presented and simulated in via Mutltisim. The results obtained via Multisim simulation of the chaotic circuit are in good agreement with the results with Matlab simulations. That provided the fractional operators can be applied in real- worlds applications as modeling electrical circuits. The presence of coexisting attractors for particular values of the parameters of the presented fractional-order chaotic model has been studied.

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Wu, Xian Yong, Yi Long Cheng, Kai Liu, Xin Liang Yu, and Xian Qian Wu. "Chaos Synchronization between Fractional-Order Unified Chaotic System and Rossler Chaotic System." Advanced Materials Research 562-564 (August 2012): 2088–91. http://dx.doi.org/10.4028/www.scientific.net/amr.562-564.2088.

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The chaotic dynamics of the unified chaotic system and the Rossler system with different fractional-order are studied in this paper. The research shows that the chaotic attractors can be found in the two systems while the orders of the systems are less than three. Asymptotic synchronization of response and drive systems is realized by active control through designing proper controller when system parameters are known. Theoretical analysis and simulation results demonstrate the effective of this method.

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Fan, J. W., N. Zhao, Y. Gao, and H. L. Lan. "Function Synchronization of the Fractional-Order Chaotic System." Advanced Materials Research 631-632 (January 2013): 1220–25. http://dx.doi.org/10.4028/www.scientific.net/amr.631-632.1220.

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Function synchronization is an important type of chaos synchronization because of enhancing the security of communication. In order to obtain the better conformances of function synchronization, a method of the fractional-order chaotic system is presented, which based on the stability theory of the fractional order system. This method need construct a parameter matrix and a coupled matrix using fractional-order chaotic drive system at first, and then the chaotic response system is set up with these matrixes. The synchronization error function between drive system and response system is satisfied with the asymptotic stability. Function synchronization of the fractional-order Rikitake chaotic system is selected as a typical example. Numerical simulation results demonstrate the validity of the presented method. This method not only has better synchronization conformances, but also can be applied in the chaotic secure communications.

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Feng, Hao, Yang Yang, and Shi Ping Yang. "A New Method for Full State Hybrid Projective Synchronization of Different Fractional Order Chaotic Systems." Applied Mechanics and Materials 385-386 (August 2013): 919–22. http://dx.doi.org/10.4028/www.scientific.net/amm.385-386.919.

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In this paper, the full state hybrid projective synchronization (FSHPS) between two different fractional order chaotic systems is investigated. i.e., the fractional order Chen system and the fractional order Lorenz system. Based on the synchronization error system feedback linearization theory, a new method combining feedback control is proposed for theFSHPSin fractional order chaotic systems. Numerical simulations are presented to verify the effectiveness and the feasibility of the synchronization scheme.

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Xu, Fei. "The Generation of a Series of Multiwing Chaotic Attractors Using Integer and Fractional Order Differential Equation Systems." International Journal of Bifurcation and Chaos 24, no. 10 (October 2014): 1450130. http://dx.doi.org/10.1142/s0218127414501302.

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In this article, we present a systematic approach to design chaos generators using integer order and fractional order differential equation systems. A series of multiwing chaotic attractors and grid multiwing chaotic attractors are obtained using linear integer order differential equation systems with switching controls. The existence of chaotic attractors in the corresponding fractional order differential equation systems is also investigated. We show that, using the nonlinear fractional order differential equation system, or linear fractional order differential equation systems with switching controls, a series of multiwing chaotic attractors can be obtained.

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Wang, Huihai, Shaobo He, and Kehui Sun. "Complex Dynamics of the Fractional-Order Rössler System and Its Tracking Synchronization Control." Complexity 2018 (December 2, 2018): 1–13. http://dx.doi.org/10.1155/2018/4019749.

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Numerical analysis of fractional-order chaotic systems is a hot topic of recent years. The fractional-order Rössler system is solved by a fast discrete iteration which is obtained from the Adomian decomposition method (ADM) and it is implemented on the DSP board. Complex dynamics of the fractional-order chaotic system are analyzed by means of Lyapunov exponent spectra, bifurcation diagrams, and phase diagrams. It shows that the system has rich dynamics with system parameters and the derivative order q. Moreover, tracking synchronization controllers are theoretically designed and numerically investigated. The system can track different signals including chaotic signals from the fractional-order master system and constant signals. It lays a foundation for the application of the fractional-order Rössler system.

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Chen, Liping, Shanbi Wei, Yi Chai, and Ranchao Wu. "Adaptive Projective Synchronization between Two Different Fractional-Order Chaotic Systems with Fully Unknown Parameters." Mathematical Problems in Engineering 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/916140.

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Projective synchronization between two different fractional-order chaotic systems with fully unknown parameters for drive and response systems is investigated. On the basis of the stability theory of fractional-order differential equations, a suitable and effective adaptive control law and a parameter update rule for unknown parameters are designed, such that projective synchronization between the fractional-order chaotic Chen system and the fractional-order chaotic Lü system with unknown parameters is achieved. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed method.

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Abd-Elouahab, Mohammed Salah, Nasr-Eddine Hamri, and Junwei Wang. "Chaos Control of a Fractional-Order Financial System." Mathematical Problems in Engineering 2010 (2010): 1–18. http://dx.doi.org/10.1155/2010/270646.

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Fractional-order financial system introduced by W.-C. Chen (2008) displays chaotic motions at order less than 3. In this paper we have extended the nonlinear feedback control in ODE systems to fractional-order systems, in order to eliminate the chaotic behavior. The results are proved analytically by applying the Lyapunov linearization method and stability condition for fractional system. Moreover numerical simulations are shown to verify the effectiveness of the proposed control scheme.

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Al-Assaf, Yousef, Reyad El-Khazali, and Wajdi Ahmad. "Identification of fractional chaotic system parameters." Chaos, Solitons & Fractals 22, no. 4 (November 2004): 897–905. http://dx.doi.org/10.1016/j.chaos.2004.03.007.

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Deng, Weihua, and Changpin Li. "Synchronization of Chaotic Fractional Chen System." Journal of the Physical Society of Japan 74, no. 6 (June 2005): 1645–48. http://dx.doi.org/10.1143/jpsj.74.1645.

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Zhao, Ling Dong, and Jian Bing Hu. "Synchronizing Fractional Chaotic Genesio-Tesi System via Backstepping Approach." Applied Mechanics and Materials 220-223 (November 2012): 1244–48. http://dx.doi.org/10.4028/www.scientific.net/amm.220-223.1244.

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Abstract. Based on fractional nonlinear stable theorem, backstepping approach for designing controller is extended to fractional order chaotic system. The controller is designed to synchronize fractional order Newton-Leipnik chaotic system via the backstepping approach. Numerical simulation certifies effectiveness of the approach.

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WANG, XING-YUAN, ZUN-WEN HU, and CHAO LUO. "GENERALIZED SYNCHRONIZATION OF NONIDENTICAL FRACTIONAL-ORDER CHAOTIC SYSTEMS." International Journal of Modern Physics B 27, no. 30 (November 7, 2013): 1350195. http://dx.doi.org/10.1142/s0217979213501956.

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In this paper, a chaotic synchronization scheme is proposed to achieve the generalized synchronization between two different fractional-order chaotic systems. Based on the stability theory of fractional-order systems and the pole placement technique, a controller is designed and theoretical proof is given. Two groups of examples are shown to verify the effectiveness of the proposed scheme, the first one is to realize the generalized synchronization between the fractional-order Chen system and the fractional-order Rössler system, the second one is between the fractional-order Lü system and the fractional-order hyperchaotic Lorenz system. The corresponding numerical simulations verify the effectiveness of the proposed scheme.

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Chen, Feng, Long Sheng, Jian Zhang, and Xiao Bin Huang. "Dynamical Analysis of a Fractional Order Multi-Wing Hyper-Chaotic System." Applied Mechanics and Materials 380-384 (August 2013): 1792–95. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1792.

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The dynamic behaviors of fractional-order systems have attracted increasing attentions recently. In this paper, a fractional-order four-wing hyper-chaotic system which has a rich variety of dynamic behaviors is proposed. We numerically study the dynamic behaviors of this fractional-order system with different conditions. Hyper-chaotic behaviors can be found in this system when the order is lower than 3 and four-wing hyper-chaotic attractors similar to integer order system can be generated. The lowest order for Hyper-chaos to exist in this system is 3.6 and the lowest order for chaos to exist in this system is 2.4.

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Gugapriya, G., Prakash Duraisamy, Anitha Karthikeyan, and B. Lakshmi. "Fractional-order chaotic system with hyperbolic function." Advances in Mechanical Engineering 11, no. 8 (August 2019): 168781401987258. http://dx.doi.org/10.1177/1687814019872581.

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In this article, we study bistability, multiscroll, and symmetric properties of fractional-order chaotic system with cubic nonlinearity. The system is configured with hyperbolic function consisting of a parameter “ g.” By varying the parameter “ g,” the dynamical behavior of the system is investigated. Multistability and multiscroll are identified, which makes the system suitable for secure communication applications. When the system is treated as fractional order, for the same parameter values and initial conditions and when the fractional order is varied from 0.96 to 0.99, multiscroll property is obtained. Symmetric property is obtained for the order of 0.99. The fractional system holds only single scroll until 0.965 order and when the order increases to more than 0.99, it is having two-scroll attractor. This property opens a variety of applications for the systems, especially in secure communication. Adaptive synchronization of the system using sliding mode control scheme is presented. For implementing the fractional-order system in field-programmable gate array, Adomian decomposition method is used, and the register-transfer level schematic of the system is presented.

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Wang, Hua, Hang-Feng Liang, Peng Zan, and Zhong-Hua Miao. "A New Scheme on Synchronization of Commensurate Fractional-Order Chaotic Systems Based on Lyapunov Equation." Journal of Control Science and Engineering 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/5975491.

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This paper proposes a new fractional-order approach for synchronization of a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. A simple but practical method to synchronize many familiar fractional-order chaotic systems has been put forward. A new theorem is proposed for a class of cascade fractional-order systems and it is applied in chaos synchronization. Combined with the fact that the states of the fractional chaotic systems are bounded, many coupled items can be taken as zero items. Then, the whole system can be simplified greatly and a simpler controller can be derived. Finally, the validity of the presented scheme is illustrated by numerical simulations of the fractional-order unified system.

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Liu, Jiaxun, Zuoxun Wang, Minglei Shu, Fangfang Zhang, Sen Leng, and Xiaohui Sun. "Secure Communication of Fractional Complex Chaotic Systems Based on Fractional Difference Function Synchronization." Complexity 2019 (August 18, 2019): 1–10. http://dx.doi.org/10.1155/2019/7242791.

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Fractional complex chaotic systems have attracted great interest recently. However, most of scholars adopted integer real chaotic system and fractional real and integer complex chaotic systems to improve the security of communication. In this paper, the advantages of fractional complex chaotic synchronization (FCCS) in secure communication are firstly demonstrated. To begin with, we propose the definition of fractional difference function synchronization (FDFS) according to difference function synchronization (DFS) of integer complex chaotic systems. FDFS makes communication secure based on FCCS possible. Then we design corresponding controller and present a general communication scheme based on FDFS. Finally, we respectively accomplish simulations which transmit analog signal, digital signal, voice signal, and image signal. Especially for image signal, we give a novel image cryptosystem based on FDFS. The results demonstrate the superiority and good performances of FDFS in secure communication.

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Xiaohong, Zhang, and Cheng Peng. "Different-lags Synchronization in Time-delay and Circuit Simulation of Fractional-order Chaotic System Based on Parameter Identification." Open Electrical & Electronic Engineering Journal 9, no. 1 (April 17, 2015): 117–26. http://dx.doi.org/10.2174/1874129001509010117.

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This study constructs a novel four-dimensional fractional-order chaotic system. It verifies chaotic nonlinear dynamic behaviors and physical reliability by numerical simulation and hardware circuit design. For a class of parameter uncertainty fractional different-lags ( τi ) chaotic systems, the authors design a time-delayed (δ) synchronization controllers and parameter adaptive laws. It proves that the drive system and the response system tend to be synchronized and identified parameter when the control parameter matrix K satisfies the condition that K – nE is a positive definite matrix. Simulation results show physical reliability of the fractional-order different-lags chaotic system and verify effectiveness of different-lags synchronization in time-delayed system method design.

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Xu, Fei. "A Class of Integer Order and Fractional Order Hyperchaotic Systems via the Chen System." International Journal of Bifurcation and Chaos 26, no. 06 (June 15, 2016): 1650109. http://dx.doi.org/10.1142/s0218127416501091.

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In this article, we investigate the generation of a class of hyperchaotic systems via the Chen chaotic system using both integer order and fractional order differential equation systems. Based on the Chen chaotic system, we designed a system with four nonlinear ordinary differential equations. For different parameter sets, the trajectory of the system may diverge or display a hyperchaotic attractor with double wings. By linearizing the ordinary differential equation system with divergent trajectory and designing proper switching controls, we obtain a chaotic attractor. Similar phenomenon has also been observed in linearizing the hyperchaotic system. The corresponding fractional order systems are also considered. Our investigation indicates that, switching control can be applied to either linearized chaotic or nonchaotic differential equation systems to create chaotic attractor.

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Huang, Su Hai. "Synchronization of Liu Chaotic System with Fractional-Order." Applied Mechanics and Materials 336-338 (July 2013): 467–70. http://dx.doi.org/10.4028/www.scientific.net/amm.336-338.467.

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This paper deals with chaos synchronization of the Liu chaotic system with fractional-order. Based on the fractional-order stability theory, an adaptive sliding mode controller has been constructed to realize projective synchronization of fractional-order Liu chaotic system with unknown parameter. An illustrative simulation result is given to demonstrate the effectiveness of the proposed sliding mode controller.

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Gao, Xin. "Controlling the Fractional-Order Chaotic System Based on Inverse Optimal Control Approach." Key Engineering Materials 474-476 (April 2011): 108–13. http://dx.doi.org/10.4028/www.scientific.net/kem.474-476.108.

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In this paper, we numerically investigate the chaotic behaviors of a fractional-order system. We find that chaotic behaviors exist in the fractional-order system with an order being less than 3. The lowest order we find to have chaos is 2.4 in such system. In addition, we numerically simulate the continuances of the chaotic behaviors in the fractional-order system with orders ranging from 2.7 to 3. Finally, a simple, but effective, linear state feedback controller is proposed for controlling the fractional-order chaotic system based on an inverse optimal control approach. Numerical simulations show the effectiveness and feasibility of the proposed controller.

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ODIBAT, ZAID M., NATHALIE CORSON, M. A. AZIZ-ALAOUI, and CYRILLE BERTELLE. "SYNCHRONIZATION OF CHAOTIC FRACTIONAL-ORDER SYSTEMS VIA LINEAR CONTROL." International Journal of Bifurcation and Chaos 20, no. 01 (January 2010): 81–97. http://dx.doi.org/10.1142/s0218127410025429.

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The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived analytically to achieve synchronization of the chaotic fractional-order Chen, Rössler and modified Chua systems. Numerical simulations are provided to verify the theoretical analysis.

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