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In this paper, adaptive control of complicated chaotic systems with unknown parameters is discussed via a set of fuzzy modeling-based adaptive strategy. The proposed fuzzy model theory aims to adjust the inner-weighting of each linear sub-equation, simplize the complicated modeling process, and to reveal the similar behaviors of complicated nonlinear dynamic system. Further, based on the modeling concept, a set of fuzzy model-based adaptive control scheme and its creative fuzzy update laws of parameters are proposed to achieve the goal of adaptive synchronization. Two identical complicated dynamic systems, Mathieu-Van der pol system (M-V system) with uncertainties, are designed and illustrated for numerical simulation example to show the effectiveness and feasibility of the proposed novel adaptive control strategy. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, (1) a new fuzzy model is presented to simulate two different chaotic systems with different numbers of nonlinear terms and (2) a new adaptive approach and a new control Lyapunov function are proposed to synchronize these two different fuzzy chaotic systems and speed up the convergence of errors. By using this new model, the numbers of fuzzy rules of chaotic systems can be reduced from to and only 2 subsystems are needed, where N is the number of nonlinear terms. The fuzzy systems become much simpler. In addition, through the new fuzzy model, the new fuzzy systems are much simpler than T–S fuzzy systems (when nonlinear systems are complicated) and can be used to any other kind of application in fuzzy logic control or fuzzy modeling. Mathieu–Van der Pol system (which is called M–V system in this paper) and Quantum cellular neural networks nanosystem (which is called Q-CNN system in this paper) are used for illustrations in numerical simulation results to show the effectiveness and feasibility of our new adaptive approach and new control Lyapunov function. The T–S fuzzy modeling and traditional adaptive control are also given in Appendices B and C for comparison. [Copyright &y& Elsevier]

Abstract: This study demonstrates that synchronization and anti-synchronization can coexist in Chen–Lee chaotic systems by direct linear coupling. Based on Lyapunov’s direct method, a linear controller was designed to assure that two different types of synchronization can simultaneously be achieved. Further, the hybrid projective synchronization of Chen–Lee chaotic systems was studied using a nonlinear control scheme. The nonlinear controller was designed according to the Lyapunov stability theory to guarantee the hybrid projective synchronization, including synchronization, anti-synchronization, and projective synchronization. Finally, numerical examples are presented in order to illustrate the proposed synchronization approach. [Copyright &y& Elsevier]

Abstract: The dynamics of fractional-order systems has attracted increasing attention in recent years. In this paper, the dynamics of the Newton–Leipnik system with fractional order was studied numerically. The system displays many interesting dynamic behaviors, such as fixed points, periodic motions, chaotic motions, and transient chaos. It was found that chaos exists in the fractional-order system with order less than 3. In this study, the lowest order for this system to yield chaos is 2.82. A period-doubling route to chaos in the fractional-order system was also found. [Copyright &y& Elsevier]

Abstract: Vibration phenomena of the fractionally damped systems have attracted increasing attentions in recent years. In this paper, dynamics of the fractionally damped Duffing equation is examined. The fractionally damped Duffing equation is transformed into a set of fractional integral equations solved by a predictor–corrector method. The effect of fractional order of damping on the dynamic behaviors of the motion is the main subject of the study. In this work, bifurcation of the parameter-dependent system is drawn numerically. The time evolutions of the nonlinear dynamical system responses are also described in phase portraits and the Poincaré map technique. In addition, the occurrence and the nature of chaotic attractors are verified by evaluating the largest Lyapunov exponents. Results obtained from this study illustrates that the fractional order of damping has a significant effect on the dynamic behaviors of the motion. The size of the attractor trends to enlarge when fractional order α increases. Regular motions (including period-3 motion) and chaotic motions are examined. Moreover, a period doubling route to chaos is also found. Many period-3 windows are also observed in bifurcation diagram. [Copyright &y& Elsevier]

Abstract: The dynamics of fractional-order systems have attracted a great deal of attentions in recent years. With fractional order, the dynamics of a system which includes comprehensive dynamical behaviors, such as fixed point, periodic motion, chaotic motion, and transient chaos is studied numerically in this paper. It is known that chaos exists in the fractional-order system with order less than 3. In this study, the lowest order found for this system to yield chaos is 2.43. The results are validated by the existence of a positive Lyapunov exponent. Period doubling routes to chaos in the fractional-order system are also obtained. Moreover, generation of a four-scroll chaotic attractor by the system is observed. [Copyright &y& Elsevier]

Abstract: This work presents chaos synchronization between two different chaotic systems by using active control. This technique is applied to achieve chaos synchronization for a new system and each of the dynamical systems Lorenz, Chen and Lü. Numerical simulations are also shown to verify the results. [Copyright &y& Elsevier]

Abstract: The dynamic behaviors of a SMIB power system are studied in this paper. A single modal equation is used to analyze the qualitative behaviors of the system. The famous equation of motion is called “swing equation”. The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system. The bifurcation of the parameter dependent system is studied numerically. Besides, the phase portraits, the Poincaré maps, and the Lyapunov exponents are presented to observe periodic and chaotic motions. Further, the addition of periodic force and the feedback control are used to control chaos effectively. Finally, the chaotification problem of the SMIB power system is also issued. [Copyright &y& Elsevier]

Abstract: The dynamic behaviors of a dissipative gyroscope mounted on a vibrating base are investigated qualitatively and numerically. It is shown that the nonlinear system can exhibit regular and chaotic motions. The qualitative behaviors of the system are studied by the center manifold theorem and the normal form theorem. The co-dimension one bifurcation analysis for the Hopf bifurcation is carried out. The pitchfork, Hopf, and saddle connection bifurcations for co-dimension two bifurcation are also found in this study. Regular and chaotic motions are shown to be possible in the parameter space. Numerical methods are used to obtain the time histories, the Poincaré maps, the Liapunov exponents, and the Liapunov dimensions. The effect of the spin speed of the gyroscope on its dynamic behavior is also studied by numerical simulation in conjunction with the Liapunov exponents, and it has been found that the higher spin speed of the gyroscope can quench the chaotic motion. [Copyright &y& Elsevier]

Abstract: Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V(e)=1/2eTe. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach. [Copyright &y& Elsevier]

Anti-control of chaos for a rigid body has been studied in the paper. For certain feedback gains, a rigid body can easily generate chaotic motion. Basic dynamical behaviors, such as symmetry, invariance, dissipativity and existence of attractor, are also discussed. The transient behaviors of the chaotic system have also been presented as the feedback gain changed. Of particular interesting is the fact that the chaotic system can generate a complex multi-scroll chaotic attractor under the appropriate feedback gains. Finally, it was shown that the system could be related to the famous Lorenz equations and Chen system. In other words, the system can easily display all the dynamical behaviors of the famous Lorenz equations and Chen system. [Copyright &y& Elsevier]

Abstract: The chaotic Chen–Lee system was developed with a formalism based on the Euler equations for the motion of a rigid body. It was proved that this system is the governing set of equations for gyro motion with feedback control. Recently, studies were conducted to explore the dynamic behavior of this system, including fractional order behavior, the generation of hyperchaos and perturbation analysis, control and anti-control of chaos, synchronization, etc. In this study, we further explore (1) the stability of the equilibrium points and (2) the implementation of an electronic circuit using the Chen–Lee system. It is shown that not only is this system related to gyro motion but can also be applied to electronic circuits for encryption purposes. [Copyright &y& Elsevier]

Abstract: Since Lorenz’s work, many autonomous systems for investigating chaotic thermal convection in pure fluid layers or in fluid-saturated porous media have been presented. From considering thermal convection in Oldroydian fluid-saturated porous media using thermal non-equilibrium model, a novel, unified system with six-order dynamics of chaotic convection was derived in this study. The six-order system can be used to investigate thermal convection in both pure Newtonian/viscoelastic fluid layers and in Newtonian/viscoelastic fluid-saturated porous media. Most of the chaotic convection systems previously reported in the literature are shown to be special cases of the system presented in this study. [Copyright &y& Elsevier]

Abstract: A system with more than one positive Lyapunov exponent can be classified as a hyperchaotic system. In this study, a sinusoidal perturbation was designed for generating hyperchaos from the Chen–Lee chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents and bifurcation diagrams. The system is hyperchaotic in several different regions of the parameters c, ε, and ω. It was found that this method not only can enhance or suppress chaotic behavior, but also induces chaos in non-chaotic parameter ranges. In addition, two interesting dynamical behaviors, Hopf bifurcation and intermittency, were also found in this study. [Copyright &y& Elsevier]

Abstract: Buoyancy-induced convection in a viscoelastic fluid-saturated porous medium was analyzed using an Oldroydian-type constitutive relation. An autonomous system with four differential equations was deduced by applying the truncated Galerkin expansion to the momentum and heat transfer equations. The four-dimensional system can be reduced to many systems provided in the literature such as the Lorenz system, Vadasz system, Khayat system, and Akhatov system. Depending on the flow parameters, the asymptotic behavior can be stationary, periodic, or chaotic. Generation of a four-scroll, or two-“butterfly”, chaotic attractor was observed. Results also show that stress relaxation tends to precipitate the onset of chaos. [Copyright &y& Elsevier]