Abstract
This research work describes an eight-term 3-D novel polynomial chaotic system consisting of three quadratic nonlinearities. First, this work presents the 3-D dynamics of the novel chaotic system and depicts the phase portraits of the system. Next, the qualitative properties of the novel chaotic system are discussed in detail. The novel chaotic system has four equilibrium points. We show that two equilibrium points are saddle points and the other equilibrium points are saddle-foci. The Lyapunov exponents of the novel chaotic system are obtained as \(L_1 = 0.4715, L_2 = 0\) and \(L_3 = -2.4728\). The Lyapunov dimension of the novel chaotic system is obtained as \(D_{L} = 2.1907\). Next, we present the design of adaptive feedback controller for globally stabilizing the trajectories of the novel chaotic system with unknown parameters. Furthermore, we present the design of adaptive feedback controller for achieving complete synchronization of the identical novel chaotic systems with unknown parameters. The main adaptive control results are proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this research work for eight-term 3-D novel chaotic system.
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1 Introduction
Chaotic systems are defined as nonlinear dynamical systems which are sensitive to initial conditions, topologically mixing and with dense periodic orbits [10, 11, 13].
Sensitivity to initial conditions of chaotic systems is popularly known as the butterfly effect. Small changes in an initial state will make a very large difference in the behavior of the system at future states.
Poincaré [15] suspected chaotic behaviour in the study of three bodies problem at the end of the 19th century, but chaos was experimentally established by Lorenz [41] only a few decades ago in the study of 3-D weather models.
The Lyapunov exponent is a measure of the divergence of phase points that are initially very close and can be used to quantify chaotic systems. It is common to refer to the largest Lyapunov exponent as the Maximal Lyapunov Exponent (MLE). A positive maximal Lyapunov exponent and phase space compactness are usually taken as defining conditions for a chaotic system.
In the last five decades, there is significant interest in the literature in discovering new chaotic systems [73]. Some popular chaotic systems are Lorenz system [41], Rössler system [63], Arneodo system [2], Henon-Heiles system [27], Genesio-Tesi system [25], Sprott systems [72], Chen system [19], Lü system [42], Rikitake dynamo system [62], Liu system [40], Shimizu system [71], etc.
In the recent years, many new chaotic systems have been found such as Pandey system [46], Qi system [54], Li system [35], Wei-Yang system [171], Zhou system [178], Zhu system [179], Sundarapandian systems [76, 81], Dadras system [21], Tacha system [84], Vaidyanathan systems [92, 93, 95,96,97,98, 101, 112, 126,127,128,129,130,131,132,133, 135,136,137, 139, 148, 150, 159, 161, 163, 165,166,167], Vaidyanathan-Azar systems [142, 143, 145,146,147], Pehlivan system [48], Sampath system [64], Akgul system [1], Pham system [49, 51,52,53], etc.
Chaos theory and control systems have many important applications in science and engineering [3, 10,11,12, 14, 180]. Some commonly known applications are oscillators [115, 119, 121,122,123,124, 134], lasers [37, 174], chemical reactions [102, 103, 107,108,109, 111, 113, 114, 117, 118, 120], biology [22, 33, 100, 104,105,106, 110, 116], ecology [26, 74], encryption [34, 177], cryptosystems [61, 85], mechanical systems [5,6,7,8,9], secure communications [23, 44, 175], robotics [43, 45, 169], cardiology [55, 172], intelligent control [4, 38], neural networks [28, 31, 39], memristors [50, 170], etc.
Synchronization of chaotic systems is a phenomenon that occurs when two or more chaotic systems are coupled or when a chaotic system drives another chaotic system. Because of the butterfly effect which causes exponential divergence of the trajectories of two identical chaotic systems started with nearly the same initial conditions, the synchronization of chaotic systems is a challenging research problem in the chaos literature.
Major works on synchronization of chaotic systems deal with the complete synchronization of a pair of chaotic systems called the master and slave systems. The design goal of the complete synchronization is to apply the output of the master system to control the slave system so that the output of the slave system tracks the output of the master system asymptotically with time. Active feedback control is used when the system parameters are available for measurement. Adaptive feedback control is used when the system parameters are unknown.
Pecora and Carroll pioneered the research on synchronization of chaotic systems with their seminal papers [18, 47]. The active control method [30, 65, 66, 75, 80, 86, 90, 151, 152, 155] is typically used when the system parameters are available for measurement.
Adaptive control method [67,68,69, 77,78,79, 88, 94, 125, 138, 144, 149, 153, 154, 160, 164, 168] is typically used when some or all the system parameters are not available for measurement and estimates for the uncertain parameters of the systems. Adaptive control method has more relevant for many practical situations for systems with unknown parameters. In the literature, adaptive control method is preferred over active control method due to the wide applicability of the adaptive control method.
Intelligent control methods like fuzzy control method [16, 17] are also used for the synchronization of chaotic systems. Intelligent control methods have advantages like robustness, insensitive to small variations in the parameters, etc.
Sampled-data feedback control method [24, 36, 173, 176] and time-delay feedback control method [20, 29, 70] are also used for synchronization of chaotic systems. Backstepping control method [56,57,58,59,60, 83, 156, 162] is also used for the synchronization of chaotic systems, which is a recursive method for stabilizing the origin of a control system in strict-feedback form.
Another popular method for the synchronization of chaotic systems is the sliding mode control method [82, 87, 89, 91, 99, 140, 141, 157, 158], which is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to “slide” along a cross-section of the system’s normal behavior.
In this research work, we describe an eight-term 3-D novel polynomial chaotic system with three quadratic nonlinearities. Section 2 describes the 3-D dynamical model and phase portraits of the novel chaotic system.
Section 3 describes the dynamic analysis of the novel chaotic system. We show that the novel chaotic system has four equilibrium points of which two equilibrium points are saddle points and the other two equilibrium points are saddle-foci.
The Lyapunov exponents of the eight-term novel chaotic system are obtained as \(L_1 = 0.4715\), \(L_2 = 0\) and \(L_3 = -2.4728\). Since the sum of the Lyapunov exponents of the novel chaotic system is negative, this chaotic system is dissipative. Also, the Lyapunov dimension of the novel chaotic system is obtained as \(D_{L} = 2.1907\).
Section 4 describes the adaptive feedback control of the novel chaotic system with unknown parameters. Section 5 describes the adaptive feedback synchronization of the identical novel chaotic systems with unknown parameters. The adaptive feedback control and synchronization results are proved using Lyapunov stability theory [32].
MATLAB simulations are depicted to illustrate all the main results for the 3-D novel chaotic system. Section 6 concludes this work with a summary of the main results.
2 A Novel 3-D Chaotic System
In this research work, we announce an eight-term 3-D chaotic system described by
where \(x_1, x_2, x_3\) are the states and a, b, c, p are constant, positive parameters.
The 3-D system (1) is chaotic when the parameter values are taken as
For numerical simulations, we take the initial state of the chaotic system (1) as
The Lyapunov exponents of the novel chaotic system (1) for the parameter values (2) and the initial values (3) are numerically determined as
The Lyapunov dimension of the novel chaotic system (1) is calculated as
which is fractional.
The presence of a positive Lyapunov exponent in (4) shows that the 3-D novel system (1) is chaotic (Fig. 1).
The novel 3-D chaotic system (1) exhibits a strange chaotic attractor. It is interesting to note that the strange chaotic attractor looks like a trumpet. Hence, the novel chaotic system (1) can be also called as a trumpet attractor.
Figure 2 describes the 2-D projection of the strange chaotic attractor of the novel chaotic system (1) on \((x_1, x_2)\)-plane.
Figure 3 describes the 2-D projection of the strange chaotic attractor of the novel chaotic system (1) on \((x_2, x_3)\)-plane.
Figure 4 describes the 2-D projection of the strange chaotic attractor of the novel chaotic system (1) on \((x_1, x_3)\)-plane.
3 Analysis of the 3-D Novel Chaotic System
This section gives the qualitative properties of the novel chaotic system (1).
3.1 Dissipativity
In vector notation, the system (1) can be expressed as
where
We take the parameter values as
The divergence of the vector field f on is obtained as
where
Let \(\varOmega \) be any region in with a smooth boundary. Let \(\varOmega (t) = \varPhi _t(\varOmega )\), where \(\varPhi _t\) is the flow of the vector field f. Let V(t) denote the volume of \(\varOmega (t)\).
By Liouville’s theorem, it follows that
Substituting the value of \(\text{ div } f\) in (11) leads to
Integrating the linear differential equation (12), V(t) is obtained as
From Eq. (13), it follows that the volume V(t) shrinks to zero exponentially as \(t \rightarrow \infty \).
Thus, the novel chaotic system (1) is dissipative. Hence, any asymptotic motion of the system (1) settles onto a set of measure zero, i.e. a strange attractor.
3.2 Invariance
It is easily seen that the \(x_3\)-axis is invariant for the flow of the novel chaotic system (1). The invariant motion along the \(x_3\)-axis is characterized by the scalar dynamics
which is globally exponentially stable.
3.3 Equilibria
The equilibrium points of the novel chaotic system (1) are obtained by solving the nonlinear equations
We take the parameter values as in the chaotic case, viz.
Solving the nonlinear system (15) with the parameter values (16), we obtain four equilibrium points of the novel chaotic system (1), viz.
The Jacobian matrix of the novel chaotic system (1) at \((x_1^\star , x_2^\star , x_3^\star )\) is obtained as
The matrix \(J_0 = J(E_0)\) has the eigenvalues
This shows that the equilibrium point \(E_0\) is a saddle-point, which is unstable.
The matrix \(J_1 = J(E_1)\) has the eigenvalues
This shows that the equilibrium point \(E_1\) is a saddle point, which is unstable.
The matrix \(J_2 = J(E_2)\) has the eigenvalues
This shows that the equilibrium point \(E_2\) is a saddle-focus, which is unstable.
The matrix \(J_3 = J(E_3)\) has the eigenvalues
This shows that the equilibrium point \(E_3\) is a saddle-focus, which is unstable.
Hence, \(E_0, E_1, E_2, E_3\) are all unstable equilibrium points of the 3-D novel chaotic system (1), where \(E_0, E_1\) are saddle points and \(E_3, E_4\) are saddle-foci.
3.4 Lyapunov Exponents and Lyapunov Dimension
We take the initial values of the novel chaotic system (1) as in (3) and the parameter values of the novel chaotic system (1) as in (2).
Then the Lyapunov exponents of the novel chaotic system (1) are numerically obtained as
Since \(L_1 + L_2 + L_3 = - 2.0013 < 0\), the system (1) is dissipative.
Also, the Lyapunov dimension of the system (1) is obtained as
which is fractional.
Figure 5 depicts the Lyapunov exponents of the novel chaotic system (1). From this figure, it is seen that the Maximal Lyapunov Exponent (MLE) of the novel chaotic system (1) is \(L_1 = 0.4715\).
4 Adaptive Feedback Control of the 3-D Novel Chaotic System
This section derives new results for adaptive feedback controller design in order to stabilize the unstable novel chaotic system with unknown parameters for all initial conditions.
The controlled novel 3-D chaotic system is given by
where \(x_1, x_2, x_3\) are state variables, a, b, c, p are constant, unknown, parameters of the system and \(u_1, u_2, u_3\) are adaptive feedback controls to be designed.
An adaptive feedback control law is taken as
In (26), \(\hat{a}(t), \hat{b}(t), \hat{c}(t), \hat{p}(t)\) are estimates for the unknown parameters a, b, c, p, respectively, and \(k_1, k_2, k_3\) are positive gain constants.
The closed-loop control system is obtained by substituting (26) into (25) as
To simplify (27), we define the parameter estimation error as
Using (28), the closed-loop system (27) can be simplified as
Differentiating the parameter estimation error (28) with respect to t, we get
Next, we find an update law for parameter estimates using Lyapunov stability theory.
Consider the quadratic Lyapunov function defined by
which is positive definite on .
Differentiating V along the trajectories of (29) and (30), we get
In view of Eq. (32), an update law for the parameter estimates is taken as
Theorem 1
The novel chaotic system (25) with unknown system parameters is globally and exponentially stabilized for all initial conditions by the adaptive control law (26) and the parameter update law (33), where \(k_i, (i = 1, 2, 3)\) are positive constants.
Proof
The result is proved using Lyapunov stability theory [32]. We consider the quadratic Lyapunov function V defined by (31), which is positive definite on .
Substitution of the parameter update law (33) into (32) yields
which is a negative semi-definite function on .
Therefore, it can be concluded that the state vector x(t) and the parameter estimation error are globally bounded, i.e.
Define
Then it follows from (34) that
Integrating the inequality (37) from 0 to t, we get
From (38), it follows that \(\mathbf {x}(t) \in \mathbf{L}_2\).
Using (29), it can be deduced that \(\dot{x}(t) \in \mathbf{L}_\infty \).
Hence, using Barbalat’s lemma, we can conclude that \(\mathbf {x}(t) \rightarrow 0\) exponentially as \(t \rightarrow \infty \) for all initial conditions .
This completes the proof. \(\qquad \square \)
For numerical simulations, the parameter values of the novel system (25) are taken as in the chaotic case, viz.
The gain constants are taken as \(k_i = 6, (i = 1, 2, 3)\).
The initial values of the parameter estimates are taken as
The initial values of the novel system (25) are taken as
Figure 6 shows the time-history of the controlled states \(x_1(t), x_2(t), x_3(t)\).
Figure 6 depicts the exponential convergence of the controlled states and the efficiency of the adaptive controller defined by (26).
5 Adaptive Synchronization of the Identical 3-D Novel Chaotic Systems
This section derives new results for the adaptive synchronization of the identical novel chaotic systems with unknown parameters.
The master system is given by the novel chaotic system
where \(x_1, x_2, x_3\) are state variables and a, b, c, p are constant, unknown, parameters of the system.
The slave system is given by the controlled novel chaotic system
where \(y_1, y_2, y_3\) are state variables and \(u_1, u_2, u_3\) are adaptive controls to be designed.
The synchronization error is defined as
The error dynamics is easily obtained as
An adaptive control law is taken as
where \(\hat{a}(t), \hat{b}(t), \hat{c}(t), \hat{p}(t)\) are estimates for the unknown parameters a, b, c, p, respectively, and \(k_1, k_2, k_3\) are positive gain constants.
The closed-loop control system is obtained by substituting (46) into (45) as
To simplify (47), we define the parameter estimation error as
Using (48), the closed-loop system (47) can be simplified as
Differentiating the parameter estimation error (48) with respect to t, we get
Next, we find an update law for parameter estimates using Lyapunov stability theory.
Consider the quadratic Lyapunov function defined by
which is positive definite on .
Differentiating V along the trajectories of (49) and (50), we get
In view of Eq. (52), an update law for the parameter estimates is taken as
Theorem 2
The identical novel chaotic systems (42) and (43) with unknown system parameters are globally and exponentially synchronized for all initial conditions by the adaptive control law (46) and the parameter update law (53), where \(k_i, (i = 1, 2, 3)\) are positive constants.
Proof
The result is proved using Lyapunov stability theory [32].
We consider the quadratic Lyapunov function V defined by (51), which is positive definite on .
Substitution of the parameter update law (53) into (52) yields
which is a negative semi-definite function on .
Therefore, it can be concluded that the synchronization error vector e(t) and the parameter estimation error are globally bounded, i.e.
Define
Then it follows from (54) that
Integrating the inequality (57) from 0 to t, we get
From (58), it follows that \(\mathbf {e}(t) \in \mathbf {L}_2\).
Using (49), it can be deduced that \(\dot{\mathbf {e}}(t) \in \mathbf {L}_\infty \).
Hence, using Barbalat’s lemma, we can conclude that \(\mathbf {e}(t) \rightarrow 0\) exponentially as \(t \rightarrow \infty \) for all initial conditions .
This completes the proof. \(\qquad \square \)
For numerical simulations, the parameter values of the novel systems (42) and (43) are taken as in the chaotic case, viz.
The gain constants are taken as \(k_i = 6\) for \(i = 1, 2, 3\).
The initial values of the parameter estimates are taken as
The initial values of the master system (42) are taken as
The initial values of the slave system (43) are taken as
Figures 7-9 show the complete synchronization of the identical chaotic systems (42) and (43).
Figure 7 shows that the states \(x_1(t)\) and \(y_1(t)\) are synchronized in one second (MATLAB).
Figure 8 shows that the states \(x_2(t)\) and \(y_2(t)\) are synchronized in one second (MATLAB).
Figure 9 shows that the states \(x_3(t)\) and \(y_3(t)\) are synchronized in one second (MATLAB).
Figure 10 shows the time-history of the synchronization errors \(e_1(t), e_2(t), e_3(t)\). From Fig. 10, it is seen that the errors \(e_1(t), e_2(t)\) and \(e_3(t)\) are stabilized in one second (MATLAB).
6 Conclusions
In this work, we described an eight-term 3-D novel polynomial chaotic system consisting of three quadratic nonlinearities. The qualitative properties of the novel chaotic system have been discussed in detail. We showed that the novel chaotic system has four equilibrium points of which two equilibrium points are saddle points and the other equilibrium points are saddle-foci. The Lyapunov exponents of the novel chaotic system were derived as \(L_1 = 0.4715, L_2 = 0\) and \(L_3 = -2.4728\). The Lyapunov dimension of the novel chaotic system was obtained as \(D_{L} = 2.1907\). Next, we worked on the design of adaptive feedback controller for globally stabilizing the trajectories of the novel chaotic system with unknown parameters. Furthermore, we derived new results for the design of adaptive feedback controller for achieving complete synchronization of the identical novel chaotic systems with unknown parameters. The main adaptive control results were proved using Lyapunov stability theory. MATLAB simulations were displayed to illustrate all the main results presented in this research work.
References
Akgul, A., Moroz, I., Pehlivan, I., & Vaidyanathan, S. (2016). A new four-scroll chaotic attractor and its engineering applications. Optik, 127, 5491–5499.
Arneodo, A., Coullet, P., & Tresser, C. (1981). Possible new strange attractors with spiral structure. Communications in Mathematical Physics, 79, 573–579.
Azar, A. T. (2010). Fuzzy systems. Vienna, Austria: IN-TECH.
Azar, A. T. (2012). Overview of type-2 fuzzy logic systems. International Journal of Fuzzy System Applications, 2(4), 1–28.
Azar, A. T., & Serrano, F. E. (2014). Robust IMC-PID tuning for cascade control systems with gain and phase margin specifications. Neural Computing and Applications, 25(5), 983–995.
Azar, A. T., & Serrano, F. E. (2015). Adaptive sliding mode control of the Furuta pendulum. In A. T. Azar & Q. Zhu (Eds.), Advances and applications in sliding mode control systems. Studies in computational intelligence (Vol. 576, pp. 1–42). Germany: Springer.
Azar, A. T., & Serrano, F. E. (2015). Deadbeat control for multivariable systems with time varying delays. In A. T. Azar & S. Vaidyanathan (Eds.), Chaos modeling and control systems design. Studies in computational intelligence (Vol. 581, pp. 97–132). Germany: Springer.
Azar, A. T., & Serrano, F. E. (2015). Design and modeling of anti wind up PID controllers. In Q. Zhu & A. T. Azar (Eds.), Complex system modelling and control through intelligent soft computations. Studies in fuzziness and soft computing (Vol. 319, pp. 1–44). Germany: Springer.
Azar, A. T., & Serrano, F. E. (2015). Stabilizatoin and control of mechanical systems with backlash. In A. T. Azar & S. Vaidyanathan (Eds.), Handbook of research on advanced intelligent control engineering and automation. Advances in computational intelligence and robotics (ACIR) (pp. 1–60). USA: IGI-Global.
Azar, A. T., & Vaidyanathan, S. (2015). Chaos modeling and control systems design. Studies in computational intelligence (Vol. 581). Germany: Springer.
Azar, A. T., & Vaidyanathan, S. (2015). Computational intelligence applications in modeling and control. Studies in computational intelligence (Vol. 575). Germany: Springer.
Azar, A. T., & Vaidyanathan, S. (2015). Handbook of research on advanced intelligent control engineering and automation. Advances in computational intelligence and robotics (ACIR). USA: IGI-Global.
Azar, A. T., & Vaidyanathan, S. (2016). Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337). Germany: Springer.
Azar, A. T., & Zhu, Q. (2015). Advances and applications in sliding mode control systems. Studies in computational intelligence (Vol. 576). Germany: Springer.
Barrow-Green, J. (1997). Poincaré and the three body problem. American Mathematical Society.
Boulkroune, A., Bouzeriba, A., Bouden, T., & Azar, A. T. (2016a). Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 681–697). Germany: Springer.
Boulkroune, A., Hamel, S., Azar, A. T., & Vaidyanathan, S. (2016b). Fuzzy control-based function synchronization of unknown chaotic systems with dead-zone input. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 699–718). Germany: Springer.
Carroll, T. L., & Pecora, L. M. (1991). Synchronizing chaotic circuits. IEEE Transactions on Circuits and Systems, 38(4), 453–456.
Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and Chaos, 9(7), 1465–1466.
Chen, W. H., Wei, D., & Lu, X. (2014). Global exponential synchronization of nonlinear time-delay Lur’e systems via delayed impulsive control. Communications in Nonlinear Science and Numerical Simulation, 19(9), 3298–3312.
Dadras, S., & Momeni, H. R. (2009). A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors. Physics Letters A, 373, 3637–3642.
Das, S., Goswami, D., Chatterjee, S., & Mukherjee, S. (2014). Stability and chaos analysis of a novel swarm dynamics with applications to multi-agent systems. Engineering Applications of Artificial Intelligence, 30, 189–198.
Feki, M. (2003). An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons and Fractals, 18(1), 141–148.
Gan, Q., & Liang, Y. (2012). Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control. Journal of the Franklin Institute, 349(6), 1955–1971.
Genesio, R., & Tesi, A. (1992). Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica, 28(3), 531–548.
Gibson, W. T., & Wilson, W. G. (2013). Individual-based chaos: Extensions of the discrete logistic model. Journal of Theoretical Biology, 339, 84–92.
Henon, M., & Heiles, C. (1964). The applicability of the third integral of motion: Some numerical experiments. The Astrophysical Journal, 69, 73–79.
Huang, X., Zhao, Z., Wang, Z., & Li, Y. (2012). Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing, 94, 13–21.
Jiang, G. P., Zheng, W. X., & Chen, G. (2004). Global chaos synchronization with channel time-delay. Chaos, Solitons & Fractals, 20(2), 267–275.
Karthikeyan, R., & Sundarapandian, V. (2014). Hybrid chaos synchronization of four-scroll systems via active control. Journal of Electrical Engineering, 65(2), 97–103.
Kaslik, E., & Sivasundaram, S. (2012). Nonlinear dynamics and chaos in fractional-order neural networks. Neural Networks, 32, 245–256.
Khalil, H. K. (2001). Nonlinear systems. New Jersey, USA: Prentice Hall.
Kyriazis, M. (1991). Applications of chaos theory to the molecular biology of aging. Experimental Gerontology, 26(6), 569–572.
Lang, J. (2015). Color image encryption based on color blend and chaos permutation in the reality-preserving multiple-parameter fractional Fourier transform domain. Optics Communications, 338, 181–192.
Li, D. (2008). A three-scroll chaotic attractor. Physics Letters A, 372(4), 387–393.
Li, N., Zhang, Y., & Nie, Z. (2011). Synchronization for general complex dynamical networks with sampled-data. Neurocomputing, 74(5), 805–811.
Li, N., Pan, W., Yan, L., Luo, B., & Zou, X. (2014). Enhanced chaos synchronization and communication in cascade-coupled semiconductor ring lasers. Communications in Nonlinear Science and Numerical Simulation, 19(6), 1874–1883.
Li, Z., & Chen, G. (2006). Integration of fuzzy logic and chaos theory. Studies in fuzziness and soft computing (Vol. 187). Germany: Springer.
Lian, S., & Chen, X. (2011). Traceable content protection based on chaos and neural networks. Applied Soft Computing, 11(7), 4293–4301.
Liu, C., Liu, T., Liu, L., & Liu, K. (2004). A new chaotic attractor. Chaos, Solitions and Fractals, 22(5), 1031–1038.
Lorenz, E. N. (1963). Deterministic periodic flow. Journal of the Atmospheric Sciences, 20(2), 130–141.
Lü, J., & Chen, G. (2002). A new chaotic attractor coined. International Journal of Bifurcation and Chaos, 12(3), 659–661.
Mondal, S., & Mahanta, C. (2014). Adaptive second order terminal sliding mode controller for robotic manipulators. Journal of the Franklin Institute, 351(4), 2356–2377.
Murali, K., & Lakshmanan, M. (1998). Secure communication using a compound signal from generalized chaotic systems. Physics Letters A, 241(6), 303–310.
Nehmzow, U., & Walker, K. (2005). Quantitative description of robot-environment interaction using chaos theory. Robotics and Autonomous Systems, 53(3–4), 177–193.
Pandey, A., Baghel, R. K., & Singh, R. P. (2012). Synchronization analysis of a new autonomous chaotic system with its application in signal masking. IOSR Journal of Electronics and Communication Engineering, 1(5), 16–22.
Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64(8), 821–824.
Pehlivan, I., Moroz, I. M., & Vaidyanathan, S. (2014). Analysis, synchronization and circuit design of a novel butterfly attractor. Journal of Sound and Vibration, 333(20), 5077–5096.
Pham, V. T., Vaidyanathan, S., Volos, C. K., & Jafari, S. (2015). Hidden attractors in a chaotic system with an exponential nonlinear term. European Physical Journal Special Topics, 224(8), 1507–1517.
Pham, V. T., Volos, C. K., Vaidyanathan, S., Le, T. P., & Vu, V. Y. (2015). A memristor-based hyperchaotic system with hidden attractors: Dynamics, synchronization and circuital emulating. Journal of Engineering Science and Technology Review, 8(2), 205–214.
Pham, V. T., Jafari, S., Vaidyanathan, S., Volos, C., & Wang, X. (2016). A novel memristive neural network with hidden attractors and its circuitry implementation. Science China Technological Sciences, 59(3), 358–363.
Pham, V. T., Vaidyanathan, S., Volos, C., Jafari, S., & Kingni, S. T. (2016). A no-equilibrium hyperchaotic system with a cubic nonlinear term. Optik, 127(6), 3259–3265.
Pham, V. T., Vaidyanathan, S., Volos, C. K., Jafari, S., Kuznetsov, N. V., & Hoang, T. M. (2016). A novel memristive time-delay chaotic system without equilibrium points. European Physical Journal Special Topics, 225(1), 127–136.
Qi, G., & Chen, G. (2006). Analysis and circuit implementation of a new 4D chaotic system. Physics Letters A, 352, 386–397.
Qu, Z. (2011). Chaos in the genesis and maintenance of cardiac arrhythmias. Progress in Biophysics and Molecular Biology, 105(3), 247–257.
Rasappan, S., & Vaidyanathan, S. (2012). Global chaos synchronization of WINDMI and Coullet chaotic systems by backstepping control. Far East Journal of Mathematical Sciences, 67(2), 265–287.
Rasappan, S., & Vaidyanathan, S. (2012). Hybrid synchronization of n-scroll Chua and Lur’e chaotic systems via backstepping control with novel feedback. Archives of Control Sciences, 22(3), 343–365.
Rasappan, S., & Vaidyanathan, S. (2012). Synchronization of hyperchaotic Liu system via backstepping control with recursive feedback. Communications in Computer and Information Science, 305, 212–221.
Rasappan, S., & Vaidyanathan, S. (2013). Hybrid synchronization of \(n\)-scroll chaotic Chua circuits using adaptive backstepping control design with recursive feedback. Malaysian Journal of Mathematical Sciences, 7(2), 219–246.
Rasappan, S., & Vaidyanathan, S. (2014). Global chaos synchronization of WINDMI and Coullet chaotic systems using adaptive backstepping control design. Kyungpook Mathematical Journal, 54(1), 293–320.
Rhouma, R., & Belghith, S. (2011). Cryptoanalysis of a chaos based cryptosystem on DSP. Communications in Nonlinear Science and Numerical Simulation, 16(2), 876–884.
Rikitake, T. (1958). Oscillations of a system of disk dynamos. Mathematical Proceedings of the Cambridge Philosophical Society, 54(1), 89–105.
Rössler, O. E. (1976). An equation for continuous chaos. Physics Letters A, 57(5), 397–398.
Sampath, S., Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). An eight-term novel four-scroll chaotic system with cubic nonlinearity and its circuit simulation. Journal of Engineering Science and Technology Review, 8(2), 1–6.
Sarasu, P., & Sundarapandian, V. (2011). Active controller design for the generalized projective synchronization of four-scroll chaotic systems. International Journal of Systems Signal Control and Engineering Application, 4(2), 26–33.
Sarasu, P., & Sundarapandian, V. (2011). The generalized projective synchronization of hyperchaotic Lorenz and hyperchaotic Qi systems via active control. International Journal of Soft Computing, 6(5), 216–223.
Sarasu, P., & Sundarapandian, V. (2012). Adaptive controller design for the generalized projective synchronization of 4-scroll systems. International Journal of Systems Signal Control and Engineering Application, 5(2), 21–30.
Sarasu, P., & Sundarapandian, V. (2012). Generalized projective synchronization of three-scroll chaotic systems via adaptive control. European Journal of Scientific Research, 72(4), 504–522.
Sarasu, P., & Sundarapandian, V. (2012). Generalized projective synchronization of two-scroll systems via adaptive control. International Journal of Soft Computing, 7(4), 146–156.
Shahverdiev, E. M., & Shore, K. A. (2009). Impact of modulated multiple optical feedback time delays on laser diode chaos synchronization. Optics Communications, 282(17), 3568–2572.
Shimizu, T., & Morioka, N. (1980). On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Physics Letters A, 76(3–4), 201–204.
Sprott, J. C. (1994). Some simple chaotic flows. Physical Review E, 50(2), 647–650.
Sprott, J. C. (2010). Elegant chaos. World Scientific.
Suérez, I. (1999). Mastering chaos in ecology. Ecological Modelling, 117(2–3), 305–314.
Sundarapandian, V. (2010). Output regulation of the Lorenz attractor. Far East Journal of Mathematical Sciences, 42(2), 289–299.
Sundarapandian, V. (2013). Analysis and anti-synchronization of a novel chaotic system via active and adaptive controllers. Journal of Engineering Science and Technology Review, 6(4), 45–52.
Sundarapandian, V., & Karthikeyan, R. (2011). Anti-synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems by adaptive control. International Journal of Systmes Signal Control and Engineering Application, 4(2), 18–25.
Sundarapandian, V., & Karthikeyan, R. (2011). Anti-synchronization of Lü and Pan chaotic systems by adaptive nonlinear control. European Journal of Scientific Research, 64(1), 94–106.
Sundarapandian, V., & Karthikeyan, R. (2012). Adaptive anti-synchronization of uncertain Tigan and Li systems. Journal of Engineering and Applied Sciences, 7(1), 45–52.
Sundarapandian, V., & Karthikeyan, R. (2012). Hybrid synchronization of hyperchaotic Lorenz and hyperchaotic Chen systems via active control. Journal of Engineering and Applied Sciences, 7(3), 254–264.
Sundarapandian, V., & Pehlivan, I. (2012). Analysis, control, synchronization, and circuit design of a novel chaotic system. Mathematical and Computer Modelling, 55(7–8), 1904–1915.
Sundarapandian, V., & Sivaperumal, S. (2011). Sliding controller design of hybrid synchronization of four-wing chaotic systems. International Journal of Soft Computing, 6(5), 224–231.
Suresh, R., & Sundarapandian, V. (2013). Global chaos synchronization of a family of \(n\)-scroll hyperchaotic Chua circuits using backstepping control with recursive feedback. Far East Journal of Mathematical Sciences, 73(1), 73–95.
Tacha, O. I., Volos, C. K., Kyprianidis, I. M., Stouboulos, I. N., Vaidyanathan, S., & Pham, V. T. (2016). Analysis, adaptive control and circuit simulation of a novel nonlinear finance system. Applied Mathematics and Computation, 276, 200–217.
Usama, M., Khan, M. K., Alghatbar, K., & Lee, C. (2010). Chaos-based secure satellite imagery cryptosystem. Computers and Mathematics with Applications, 60(2), 326–337.
Vaidyanathan, S. (2011). Hybrid chaos synchronization of Liu and Lu systems by active nonlinear control. Communications in Computer and Information Science, 204, 1–10.
Vaidyanathan, S. (2012). Analysis and synchronization of the hyperchaotic Yujun systems via sliding mode control. Advances in Intelligent Systems and Computing, 176, 329–337.
Vaidyanathan, S. (2012). Anti-synchronization of Sprott-L and Sprott-M chaotic systems via adaptive control. International Journal of Control Theory and Applications, 5(1), 41–59.
Vaidyanathan, S. (2012). Global chaos control of hyperchaotic Liu system via sliding control method. International Journal of Control Theory and Applications, 5(2), 117–123.
Vaidyanathan, S. (2012). Output regulation of the Liu chaotic system. Applied Mechanics and Materials, 110–116, 3982–3989.
Vaidyanathan, S. (2012). Sliding mode control based global chaos control of Liu-Liu-Liu-Su chaotic system. International Journal of Control Theory and Applications, 5(1), 15–20.
Vaidyanathan, S. (2013). A new six-term 3-D chaotic system with an exponential nonlinearity. Far East Journal of Mathematical Sciences, 79(1), 135–143.
Vaidyanathan, S. (2013). Analysis and adaptive synchronization of two novel chaotic systems with hyperbolic sinusoidal and cosinusoidal nonlinearity and unknown parameters. Journal of Engineering Science and Technology Review, 6(4), 53–65.
Vaidyanathan, S. (2013). Analysis, control and synchronization of hyperchaotic Zhou system via adaptive control. Advances in Intelligent Systems and Computing, 177, 1–10.
Vaidyanathan, S. (2014). A new eight-term 3-D polynomial chaotic system with three quadratic nonlinearities. Far East Journal of Mathematical Sciences, 84(2), 219–226.
Vaidyanathan, S. (2014). Analysis and adaptive synchronization of eight-term 3-D polynomial chaotic systems with three quadratic nonlinearities. European Physical Journal Special Topics, 223(8), 1519–1529.
Vaidyanathan, S. (2014). Analysis, control and synchronisation of a six-term novel chaotic system with three quadratic nonlinearities. International Journal of Modelling, Identification and Control, 22(1), 41–53.
Vaidyanathan, S. (2014). Generalized projective synchronisation of novel 3-D chaotic systems with an exponential non-linearity via active and adaptive control. International Journal of Modelling, Identification and Control, 22(3), 207–217.
Vaidyanathan, S. (2014). Global chaos synchronization of identical Li-Wu chaotic systems via sliding mode control. International Journal of Modelling, Identification and Control, 22(2), 170–177.
Vaidyanathan, S. (2015). 3-cells cellular neural network (CNN) attractor and its adaptive biological control. International Journal of PharmTech Research, 8(4), 632–640.
Vaidyanathan, S. (2015). A 3-D novel highly chaotic system with four quadratic nonlinearities, its adaptive control and anti-synchronization with unknown parameters. Journal of Engineering Science and Technology Review, 8(2), 106–115.
Vaidyanathan, S. (2015). A novel chemical chaotic reactor system and its adaptive control. International Journal of ChemTech Research, 8(7), 146–158.
Vaidyanathan, S. (2015). A novel chemical chaotic reactor system and its output regulation via integral sliding mode control. International Journal of ChemTech Research, 8(11), 669–683.
Vaidyanathan, S. (2015). Adaptive backstepping control of enzymes-substrates system with ferroelectric behaviour in brain waves. International Journal of PharmTech Research, 8(2), 256–261.
Vaidyanathan, S. (2015). Adaptive biological control of generalized Lotka-Volterra three-species biological system. International Journal of PharmTech Research, 8(4), 622–631.
Vaidyanathan, S. (2015). Adaptive chaotic synchronization of enzymes-substrates system with ferroelectric behaviour in brain waves. International Journal of PharmTech Research, 8(5), 964–973.
Vaidyanathan, S. (2015). Adaptive control design for the anti-synchronization of novel 3-D chemical chaotic reactor systems. International Journal of ChemTech Research, 8(11), 654–668.
Vaidyanathan, S. (2015). Adaptive control of a chemical chaotic reactor. International Journal of PharmTech Research, 8(3), 377–382.
Vaidyanathan, S. (2015). Adaptive synchronization of chemical chaotic reactors. International Journal of ChemTech Research, 8(2), 612–621.
Vaidyanathan, S. (2015). Adaptive synchronization of generalized Lotka-Volterra three-species biological systems. International Journal of PharmTech Research, 8(5), 928–937.
Vaidyanathan, S. (2015). Adaptive synchronization of novel 3-D chemical chaotic reactor systems. International Journal of ChemTech Research, 8(7), 159–171.
Vaidyanathan, S. (2015). Analysis, properties and control of an eight-term 3-D chaotic system with an exponential nonlinearity. International Journal of Modelling, Identification and Control, 23(2), 164–172.
Vaidyanathan, S. (2015). Anti-synchronization of Brusselator chemical reaction systems via adaptive control. International Journal of ChemTech Research, 8(6), 759–768.
Vaidyanathan, S. (2015). Anti-synchronization of chemical chaotic reactors via adaptive control method. International Journal of ChemTech Research, 8(8), 73–85.
Vaidyanathan, S. (2015). Anti-synchronization of Mathieu-Van der Pol chaotic systems via adaptive control method. International Journal of ChemTech Research, 8(11), 638–653.
Vaidyanathan, S. (2015). Chaos in neurons and adaptive control of Birkhoff-Shaw strange chaotic attractor. International Journal of PharmTech Research, 8(5), 956–963.
Vaidyanathan, S. (2015). Dynamics and control of Brusselator chemical reaction. International Journal of ChemTech Research, 8(6), 740–749.
Vaidyanathan, S. (2015). Dynamics and control of Tokamak system with symmetric and magnetically confined plasma. International Journal of ChemTech Research, 8(6), 795–803.
Vaidyanathan, S. (2015). Global chaos control of Mathieu-Van der pol system via adaptive control method. International Journal of ChemTech Research, 8(9), 406–417.
Vaidyanathan, S. (2015). Global chaos synchronization of chemical chaotic reactors via novel sliding mode control method. International Journal of ChemTech Research, 8(7), 209–221.
Vaidyanathan, S. (2015). Global chaos synchronization of Duffing double-well chaotic oscillators via integral sliding mode control. International Journal of ChemTech Research, 8(11), 141–151.
Vaidyanathan, S. (2015). Global chaos synchronization of Mathieu-Van der Pol chaotic systems via adaptive control method. International Journal of ChemTech Research, 8(10), 148–162.
Vaidyanathan, S. (2015). Global chaos synchronization of novel coupled Van der Pol conservative chaotic systems via adaptive control method. International Journal of ChemTech Research, 8(8), 95–111.
Vaidyanathan, S. (2015). Global chaos synchronization of the forced Van der Pol chaotic oscillators via adaptive control method. International Journal of PharmTech Research, 8(6), 156–166.
Vaidyanathan, S. (2015). Hyperchaos, qualitative analysis, control and synchronisation of a ten-term 4-D hyperchaotic system with an exponential nonlinearity and three quadratic nonlinearities. International Journal of Modelling, Identification and Control, 23(4), 380–392.
Vaidyanathan, S. (2016). A novel 2-D chaotic enzymes-substrates reaction system and its adaptive backstepping control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 507–528). Germany: Springer.
Vaidyanathan, S. (2016). A novel 3-D conservative jerk chaotic system with two quadratic nonlinearities and its adaptive control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 349–376). Germany: Springer.
Vaidyanathan, S. (2016). A novel 3-D jerk chaotic system with three quadratic nonlinearities and its adaptive control. Archives of Control Sciences, 26(1), 19–47.
Vaidyanathan, S. (2016). A novel 4-D hyperchaotic thermal convection system and its adaptive control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 75–100). Germany: Springer.
Vaidyanathan, S. (2016). A novel double convecton system, its analysis, adaptive control and synchronization. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 553–579). Germany: Springer.
Vaidyanathan, S. (2016). A seven-term novel 3-D jerk chaotic system with two quadratic nonlinearities and its adaptive backstepping control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 581–607). Germany: Springer.
Vaidyanathan, S. (2016). Analysis, adaptive control and synchronization of a novel 3-D chaotic system with a quartic nonlinearity and two quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 429–453). Germany: Springer.
Vaidyanathan, S. (2016). Analysis, control and synchronization of a novel 4-D highly hyperchaotic system with hidden attractors. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 529–552). Germany: Springer.
Vaidyanathan, S. (2016). Anti-synchronization of duffing double-well chaotic oscillators via integral sliding mode control. International Journal of ChemTech Research, 9(2), 297–304.
Vaidyanathan, S. (2016). Dynamic analysis, adaptive control and synchronization of a novel highly chaotic system with four quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 405–428). Germany: Springer.
Vaidyanathan, S. (2016). Global chaos synchronization of a novel 3-D chaotic system with two quadratic nonlinearities via active and adaptive control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 481–506). Germany: Springer.
Vaidyanathan, S. (2016). Qualitative analysis and properties of a novel 4-D hyperchaotic system with two quadratic nonlinearities and its adaptive control. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 455–480). Germany: Springer.
Vaidyanathan, S., & Azar, A. T. (2015). Analysis and control of a 4-D novel hyperchaotic system. In A. T. Azar & S. Vaidyanathan (Eds.), Chaos modeling and control systems design. Studies in computational intelligence (Vol. 581, pp. 19–38). Germany: Springer.
Vaidyanathan, S., & Azar, A. T. (2015). Analysis, control and synchronization of a nine-term 3-D novel chaotic system. In A. T. Azar & S. Vaidyanathan (Eds.), Chaos modelling and control systems design. Studies in computational intelligence (Vol. 581, pp. 19–38). Germany: Springer.
Vaidyanathan, S., & Azar, A. T. (2015). Anti-synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan-Madhavan chaotic systems. Studies in Computational Intelligence, 576, 527–547.
Vaidyanathan, S., & Azar, A. T. (2015). Hybrid synchronization of identical chaotic systems using sliding mode control and an application to Vaidhyanathan chaotic systems. Studies in Computational Intelligence, 576, 549–569.
Vaidyanathan, S., & Azar, A. T. (2016). A novel 4-D four-wing chaotic system with four quadratic nonlinearities and its synchronization via adaptive control method. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 203–224). Germany: Springer.
Vaidyanathan, S., & Azar, A. T. (2016). Adaptive backstepping control and synchronization of a novel 3-D jerk system with an exponential nonlinearity. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 249–274). Germany: Springer.
Vaidyanathan, S., & Azar, A. T. (2016). Adaptive control and synchronization of Halvorsen circulant chaotic systems. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 225–247). Germany: Springer.
Vaidyanathan, S., & Azar, A. T. (2016). Dynamic analysis, adaptive feedback control and synchronization of an eight-term 3-D novel chaotic system with three quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 155–178). Germany: Springer.
Vaidyanathan, S., & Azar, A. T. (2016). Generalized projective synchronization of a novel hyperchaotic four-wing system via adaptive control method. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 275–296). Germany: Springer.
Vaidyanathan, S., & Azar, A. T. (2016). Qualitative study and adaptive control of a novel 4-D hyperchaotic system with three quadratic nonlinearities. In A. T. Azar & S. Vaidyanathan (Eds.), Advances in chaos theory and intelligent control. Studies in fuzziness and soft computing (Vol. 337, pp. 179–202). Germany: Springer.
Vaidyanathan, S., & Madhavan, K. (2013). Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system. International Journal of Control Theory and Applications, 6(2), 121–137.
Vaidyanathan, S., & Pakiriswamy, S. (2013). Generalized projective synchronization of six-term Sundarapandian chaotic systems by adaptive control. International Journal of Control Theory and Applications, 6(2), 153–163.
Vaidyanathan, S., & Pakiriswamy, S. (2015). A 3-D novel conservative chaotic system and its generalized projective synchronization via adaptive control. Journal of Engineering Science and Technology Review, 8(2), 52–60.
Vaidyanathan, S., & Rajagopal, K. (2011a). Anti-synchronization of Li and T chaotic systems by active nonlinear control. Communications in Computer and Information Science, 198, 175–184.
Vaidyanathan, S., & Rajagopal, K. (2011b). Global chaos synchronization of hyperchaotic Pang and Wang systems by active nonlinear control. Communications in Computer and Information Science, 204, 84–93.
Vaidyanathan, S., & Rajagopal, K. (2011c). Global chaos synchronization of Lü and Pan systems by adaptive nonlinear control. Communications in Computer and Information Science, 205, 193–202.
Vaidyanathan, S., & Rajagopal, K. (2012). Global chaos synchronization of hyperchaotic Pang and hyperchaotic Wang systems via adaptive control. International Journal of Soft Computing, 7(1), 28–37.
Vaidyanathan, S., & Rasappan, S. (2011). Global chaos synchronization of hyperchaotic Bao and Xu systems by active nonlinear control. Communications in Computer and Information Science, 198, 10–17.
Vaidyanathan, S., & Rasappan, S. (2014). Global chaos synchronization of \(n\)-scroll Chua circuit and Lur’e system using backstepping control design with recursive feedback. Arabian Journal for Science and Engineering, 39(4), 3351–3364.
Vaidyanathan, S., & Sampath, S. (2011). Global chaos synchronization of hyperchaotic Lorenz systems by sliding mode control. Communications in Computer and Information Science, 205, 156–164.
Vaidyanathan, S., & Sampath, S. (2012). Anti-synchronization of four-wing chaotic systems via sliding mode control. International Journal of Automation and Computing, 9(3), 274–279.
Vaidyanathan, S., & Volos, C. (2015). Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system. Archives of Control Sciences, 25(3), 333–353.
Vaidyanathan, S., Volos, C., & Pham, V. T. (2014). Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation. Archives of Control Sciences, 24(4), 409–446.
Vaidyanathan, S., Volos, C., Pham, V. T., Madhavan, K., & Idowu, B. A. (2014). Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities. Archives of Control Sciences, 24(3), 375–403.
Vaidyanathan, S., Idowu, B. A., & Azar, A. T. (2015). Backstepping controller design for the global chaos synchronization of Sprott’s jerk systems. Studies in Computational Intelligence, 581, 39–58.
Vaidyanathan, S., Rajagopal, K., Volos, C. K., Kyprianidis, I. M., & Stouboulos, I. N. (2015). Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system with three quadratic nonlinearities and its digital implementation in LabVIEW. Journal of Engineering Science and Technology Review, 8(2), 130–141.
Vaidyanathan, S., Volos, C., Pham, V. T., & Madhavan, K. (2015). Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation. Archives of Control Sciences, 25(1), 5–28.
Vaidyanathan, S., Volos, C. K., Kyprianidis, I. M., Stouboulos, I. N., & Pham, V. T. (2015). Analysis, adaptive control and anti-synchronization of a six-term novel jerk chaotic system with two exponential nonlinearities and its circuit simulation. Journal of Engineering Science and Technology Review, 8(2), 24–36.
Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). Analysis, adaptive control and adaptive synchronization of a nine-term novel 3-D chaotic system with four quadratic nonlinearities and its circuit simulation. Journal of Engineering Science and Technology Review, 8(2), 174–184.
Vaidyanathan, S., Volos, C. K., & Pham, V. T. (2015). Global chaos control of a novel nine-term chaotic system via sliding mode control. In A. T. Azar & Q. Zhu (Eds.), Advances and applications in sliding mode control systems. Studies in computational intelligence (Vol. 576, pp. 571–590). Germany: Springer.
Vaidyanathan, S., Pham, V. T., & Volos, C. K. (2015). A 5-D hyperchaotic Rikitake dynamo system with hidden attractors. European Physical Journal Special Topics, 224(8), 1575–1592.
Volos, C. K., Kyprianidis, I. M., & Stouboulos, I. N. (2013). Experimental investigation on coverage performance of a chaotic autonomous mobile robot. Robotics and Autonomous Systems, 61(12), 1314–1322.
Volos, C. K., Kyprianidis, I. M., Stouboulos, I. N., Tlelo-Cuautle, E., & Vaidyanathan, S. (2015). Memristor: A new concept in synchronization of coupled neuromorphic circuits. Journal of Engineering Science and Technology Review, 8(2), 157–173.
Wei, Z., & Yang, Q. (2010). Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci. Applied Mathematics and Computation, 217(1), 422–429.
Witte, C. L., & Witte, M. H. (1991). Chaos and predicting varix hemorrhage. Medical Hypotheses, 36(4), 312–317.
Xiao, X., Zhou, L., & Zhang, Z. (2014). Synchronization of chaotic Lur’e systems with quantized sampled-data controller. Communications in Nonlinear Science and Numerical Simulation, 19(6), 2039–2047.
Yuan, G., Zhang, X., & Wang, Z. (2014). Generation and synchronization of feedback-induced chaos in semiconductor ring lasers by injection-locking. Optik - International Journal for Light and Electron Optics, 125(8), 1950–1953.
Zaher, A. A., & Abu-Rezq, A. (2011). On the design of chaos-based secure communication systems. Communications in Nonlinear Systems and Numerical Simulation, 16(9), 3721–3727.
Zhang, H., & Zhou, J. (2012). Synchronization of sampled-data coupled harmonic oscillators with control inputs missing. Systems & Control Letters, 61(12), 1277–1285.
Zhang, X., Zhao, Z., & Wang, J. (2014). Chaotic image encryption based on circular substitution box and key stream buffer. Signal Processing: Image Communication, 29(8), 902–913.
Zhou, W., Xu, Y., Lu, H., & Pan, L. (2008). On dynamics analysis of a new chaotic attractor. Physics Letters A, 372(36), 5773–5777.
Zhu, C., Liu, Y., & Guo, Y. (2010). Theoretic and numerical study of a new chaotic system. Intelligent Information Management, 2, 104–109.
Zhu, Q., & Azar, A. T. (2015). Complex system modelling and control through intelligent soft computations. Studies in fuzzines and soft computing (Vol. 319). Germany: Springer.
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Vaidyanathan, S., Azar, A.T., Ouannas, A. (2017). An Eight-Term 3-D Novel Chaotic System with Three Quadratic Nonlinearities, Its Adaptive Feedback Control and Synchronization. In: Azar, A., Vaidyanathan, S., Ouannas, A. (eds) Fractional Order Control and Synchronization of Chaotic Systems. Studies in Computational Intelligence, vol 688. Springer, Cham. https://doi.org/10.1007/978-3-319-50249-6_25
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