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A novel hyperchaos system only with one equilibrium
Physics Letters A 360 (2007) 696–701 www.elsevier.com/locate/pla
A novel hyperchaos system only with one equilibrium Zengqiang Chen a,∗ , Yong Yang a , Guoyuan Qi b , Zhuzhi Yuan a a Department of Automation, Nankai University, Tianjin 300071, PR China b Department of Electronic Information and Automation, Tianjin University of Science and Technology, Tianjin 300222, PR China
Received 7 March 2006; received in revised form 23 July 2006; accepted 31 August 2006 Available online 25 September 2006 Communicated by A.P. Fordy
Abstract This Letter presents a new hyperchaotic system by introducing an additional state feedback into a three-dimensional quadratic chaotic system. The system only has one equilibrium, but it can evolve into periodic, quasi-periodic, chaotic and hyperchaotic dynamical behaviors. Basic bifurcation analysis of the new system is investigated by means of Lyapunov exponent spectrum and bifurcation diagrams. We find that the new hyperchaotic system possesses two big positive Lyapunov exponents within a large range of parameters. Therefore, the new hyperchaotic system may have good application prospects. © 2006 Elsevier B.V. All rights reserved. PACS: 05.45.Pq; 05.45.-a Keywords: Chaos; Hyperchaos; Four-dimensional chaotic system; Lyapunov exponent; Equilibrium
1. Introduction In 1963, Lorenz found the first chaotic attractor in a threedimensional autonomous system when he studied atmospheric convection [1]. Since then, the Lorenz system has been extensively studied in the field of chaos theory and dynamical systems. In 1999, Chen constructed a 3D autonomous chaotic system based upon Lorenz system from an engineering feedback control approach [2]. A hyperchaotic attractor is characterized as a chaotic attractor with more than one positive Lyapunov exponents, and indicates that the dynamics of the system is expanded in more than one direction. For a chaotic system, there is just one positive Lyapunov exponent. Messages masked by such simple chaotic systems are not always safe [3]. It is suggested that this problem can be overcome by using higher-dimensional hyperchaotic systems, which have increased randomness and higher unpredictability [4]. Due to its higher unpredictability than chaotic system, the hyperchaos may be more useful in some fields
such as communication, encryption etc. Hyperchaos was first reported by Rössler in 1979 [5]. In the last years, the generation of hyperchaos have been studied with increasing interest [6,7]. Hyperchaotic Rössler system [5] and hyperchaotic Chua’s circuit [8] are two well-known hyperchaos examples. For hyperchaos, some basic properties are described as follows: (i) Hyperchaos exists only in higher-dimensional systems, i.e., not less than four-dimensional (4D) autonomous system for the continuous time cases. (ii) It was suggested that the number of terms in the coupled equations that give rise to instability should be at least two, in which one should be a nonlinear function [5]. However, for most existing hyperchaotic systems, the second biggest Lyapunov exponent is relatively small. Recently, based on the Lorenz system, a new chaotic system was reported by Qi et al. [9]. The system is described by x˙ = a(y − x) + yz,
* Corresponding author.
E-mail address: [email protected] (Z. Chen). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.085
y˙ = cx − xz − y, z˙ = xy − bz.
(1)
Z. Chen et al. / Physics Letters A 360 (2007) 696–701
The chaotic system (1) has complex nonlinear dynamical characters. This system can generate complex dynamics within wide parameters ranges, including chaos, Hopf bifurcation, period-doubling bifurcation, periodic orbit, sink and source, and so on [9]. In this Letter, a new four-dimensional hyperchaotic system is designed by introducing state feedback control and constant multipliers to the two quadratic terms in system (1). The new system reported in this Letter only has one equilibrium, but it has bigger positive Lyapunov exponents than those already known hyperchaotic systems. So the new system may be more useful in some fields such as encryption, communication. The new system is not only demonstrated by numerical computing but also verified with bifurcation analysis. 2. Introducing a hyperchaotic system Based on the above chaotic system (1), a simple nonlinear state feedback controller is introduced to the second equation. At the same time, in the first and the second equation, a constant multiplier is added to the cross-product item respectively, and the sign of item y in the second equation is also changed. As a result, the following four-order system can be obtained ⎧ x˙ = a(y − x) + eyz, ⎪ ⎨ y˙ = cx − dxz + y + u, (2) ⎪ ⎩ z˙ = xy − bz, u˙ = −ky. For the new 4D system described by (2), a, b, c, d, e, k are constants to be tuned and x, y, z, u are state variables. The new system (2) has the following basic properties:
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3. Bifurcation analysis In the two sections below, some properties of the new fourdimensional system are further discussed with k and b varying. And the simulation results are obtained by using 4-order Runge–Kutta method with the step length taken as 0.001. 3.1. Fixing a = 35, b = 4.9, c = 25, d = 5, e = 35 and varying k When k varies, the corresponding three Lyapunov exponents [10,11] of system (2) are shown in Fig. 1. The bifurcation diagram of state variable y with increasing k is given in Fig. 2. It can be observed that the bifurcation diagram well coincides with the spectrum of Lyapunov exponents. Fig. 1 shows that system (2) is hyperchaotic for a very wide range of k, and the system can also evolve into chaotic orbits, periodic orbits, and quasi-periodic orbits. From Figs. 1 and 2, the dynamical behaviors of system (2) can be clearly observed. When k ∈ (331, 340), (386, 536.5), (556, 570), (577, 583), the largest Lyapunov exponent almost equals zeros, which means that system (2) is periodic. Especially there are also some periodic windows in the chaotic or quasi-periodic regions. Some periodic phase portraits for special values of k are depicted in Fig. 3. When k ∈ (570, 577), (583, 843), the first and the second largest Lyapunov exponents are almost zero, and the others are negative, which means the system (2) is quasi-periodic. Some phase portraits of special k are shown in Fig. 4. When k belongs to (171, 241), (297, 331), (536.5, 556), there are one positive, one zero, and two negative Lyapunov ex-
(1) It is symmetric with respect to the z-axis, which is invariant for the coordinate transformation (x, y, z, u) → (−x, −y, z, −u). (2) It is dissipative when (a + b − 1) > 0, since ∇V =
∂ x˙ ∂ y˙ ∂z ∂u + + + = −(a + b − 1). ∂x ∂y ∂z ∂u
(3) It only has zero equilibrium. By linearizing at zero, the following Jacobian matrix is obtained ⎡ ⎤ −a a 0 0 1 0 1⎥ ⎢ c J =⎣ ⎦. 0 0 −b 0 0 −k 0 0
Fig. 1. The Lyapunov exponents vs. k with a = 35, b = 4.9, c = 25, d = 5, e = 35.
The eigenvalues of matrix J are given by the roots of the characteristic equation
(λ + b) λ3 + (a − 1)λ2 + (k − a − ac)λ + ak = 0. For example, when choose a = 35, b = 4.9, c = 25, d = 5, e = 35, k = 100, the four eigenvalues of matrix J are 10.8806, 6.2867, −4.9, −51.1673 respectively. In this case, zero is a saddle point.
Fig. 2. Bifurcation diagram for k with a = 35, b = 4.9, c = 25, d = 5, e = 35.
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Z. Chen et al. / Physics Letters A 360 (2007) 696–701
(a)
(b)
(c) Fig. 3. Periodic phase portraits of the system (2) in the x–z plane for different k. (a) k = 264.5; (b) k = 367; (c) k = 535.
(a)
(b)
(c) Fig. 4. Quasi-periodic phase portraits of the system (2) in the y–u plane for different k. (a) k = 390; (b) k = 573; (c) k = 700.
Z. Chen et al. / Physics Letters A 360 (2007) 696–701
(a)
699
(b)
(c) Fig. 5. Chaotic phase portraits of the system (2) in the x–y plane for different k. (a) k = 190; (b) k = 303; (c) k = 538.
(a)
(b)
(c) Fig. 6. Hyperchaotic phase portraits of the system (2) with k = 22: (a) x–y plane; (b) x–z plane; (c) x–u plane.
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Z. Chen et al. / Physics Letters A 360 (2007) 696–701
ponents, which means system (2) is in chaotic mode. The phase portraits for some special values of k are depicted in Fig. 5. When k ∈ (0, 171), (241, 297), system (2) has two positive, one zero, and one negative Lyapunov exponents. In these cases, the system is hyperchaotic. Note that, there are some periodic windows in the hyperchaotic region. For example, when k = 41.5, system (2) is periodic. When k = 22, system (2) is hyperchaotic and the corresponding phase portraits are shown in Fig. 6. Some typical hyperchaotic systems with different k and their Lyapunov exponents are listed in Table 1.
It is shown that, when k varies continuously within a very wide range, system (2) is always hyperchaotic. Further more,
Table 1 Some typical parameter values of k to generate hyperchaos k
λ1
λ2
λ3
λ4
10 22 67 109 126
4.409 4.031 3.069 2.411 2.158
0.131 0.252 0.669 0.792 0.859
0 0 0 0 0
−43.440 −43.182 −42.638 −42.103 −41.917
Fig. 7. The Lyapunov exponents vs. b with a = 35, c = 25, d = 5, e = 35, k = 100.
Table 2 Some typical parameter values of b to generate hyperchaos b
λ1
λ2
λ3
λ4
3.8 5.5 7.6 9.9 11
2.251 2.496 2.761 2.562 2.247
0.680 0.751 0.760 0.743 0.705
0 0 0 0 0
−40.734 −42.747 −45.118 −47.203 −47.951
Fig. 8. Bifurcation diagram for b with a = 35, c = 25, d = 5, e = 35, k = 100.
(a)
(b)
(c)
(d)
Fig. 9. Phase portraits of the system (2) in the x–y plane for increasing b, with a = 35, c = 25, d = 5, e = 35, k = 100: (a) hyperchaotic; (b) chaotic; (c) quasi-periodic; (d) periodic attractor. (a) b = 5.2; (b) b = 15.5; (c) b = 20; (d) b = 39.7.
Z. Chen et al. / Physics Letters A 360 (2007) 696–701
the two positive Lyapunov exponents are all big, which have not yet been reported in other known hyperchaotic systems. 3.2. Fixing a = 35, c = 25, d = 5, e = 35, k = 100 and varying b
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different parameters, the hyperchaotic orbits, chaotic orbits, periodic orbits and quasi-periodic orbits have also been observed. The hyperchaotic system has been confirmed in various numerical computing and verified with detailed bifurcation analysis. Acknowledgements
For the new hyperchaotic system, Fig. 7 shows the three Lyapunov exponents of system (2) with respect to parameter b. And the corresponding bifurcation diagram is shown in Fig. 8. With b increasing, the system (2) can evolve into hyperchaotic, chaotic, periodic, and quasi-periodic orbits. The Lyapunov exponent spectrum and bifurcation diagram demonstrate that system (2) has complex behaviors as b varies. Some special attractors are briefly depicted in Fig. 9. Fig. 7 clearly shows that when b varies in a very wide range, the system is still hyperchaotic. For some typical values of b, the corresponding Lyapunov exponents are tabulate in Table 2. From Table 2, it is found that, when b varies within the proper range, system (2) is also hyperchaotic as shown in Table 1. 4. Conclusions In this Letter, we have presented an autonomous hyperchaotic system based on a three-dimensional quadratic chaotic system. The new system has some good properties. For example, it only has one equilibrium, but it has two relatively big Lyapunov exponents, and it is still hyperchaotic when the parameter k or b varies in a very wide range. So, the new hyperchaotic system may have good application prospects. With
This work was partially supported by the National Nature Science Foundation of China (Grant Nos. 60374037 and 60574036), the Specialized Research Fund for the Doctoral Program of China (Grant No. 20050055013), the program for New Excellent Talents in University of China (NCET), and Science and Technology Development Foundation of Tianjin Colleges, PR China (20051528). References [1] C. Sparrow, The Lorenz Equations: Bifurcations Chaos and Strange Attractors, Springer, New York, 1982. [2] G. Chen, T. Ueta, Int. J. Bifur. Chaos 9 (1999) 1465. [3] G. Perez, H.A. Cerdeira, Phys. Rev. Lett. 74 (1995) 1970. [4] L. Pecora, Phys. World 9 (1996) 17. [5] O.E. Rossler, Phys. Lett. A 71 (1979) 155. [6] Y. Li, G. Chen, L. Wallace, K.S. Tang, IEEE Trans. CAS-II 52 (2005) 204. [7] Y. Li, Wallace, K.S. Tang, G. Chen, Int. J. Circuits Theory Appl. 33 (2005) 235. [8] D. Cafagna, G. Grassi, Int. J. Bifur. Chaos 13 (2003) 2889. [9] G. Qi, G. Chen, S. Du, Z. Chen, Z. Yuan, Physica A: Stat. Mech. Appl. 352 (2005) 295. [10] A. Wolf, J. Swift, H. Swinney, J. Vastano, Physica D 16 (1985) 285. [11] J.P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57 (1985) 617.
A novel hyperchaos system only with one equilibrium Zengqiang Chen a,∗ , Yong Yang a , Guoyuan Qi b , Zhuzhi Yuan a a Department of Automation, Nankai University, Tianjin 300071, PR China b Department of Electronic Information and Automation, Tianjin University of Science and Technology, Tianjin 300222, PR China
Received 7 March 2006; received in revised form 23 July 2006; accepted 31 August 2006 Available online 25 September 2006 Communicated by A.P. Fordy
Abstract This Letter presents a new hyperchaotic system by introducing an additional state feedback into a three-dimensional quadratic chaotic system. The system only has one equilibrium, but it can evolve into periodic, quasi-periodic, chaotic and hyperchaotic dynamical behaviors. Basic bifurcation analysis of the new system is investigated by means of Lyapunov exponent spectrum and bifurcation diagrams. We find that the new hyperchaotic system possesses two big positive Lyapunov exponents within a large range of parameters. Therefore, the new hyperchaotic system may have good application prospects. © 2006 Elsevier B.V. All rights reserved. PACS: 05.45.Pq; 05.45.-a Keywords: Chaos; Hyperchaos; Four-dimensional chaotic system; Lyapunov exponent; Equilibrium
1. Introduction In 1963, Lorenz found the first chaotic attractor in a threedimensional autonomous system when he studied atmospheric convection [1]. Since then, the Lorenz system has been extensively studied in the field of chaos theory and dynamical systems. In 1999, Chen constructed a 3D autonomous chaotic system based upon Lorenz system from an engineering feedback control approach [2]. A hyperchaotic attractor is characterized as a chaotic attractor with more than one positive Lyapunov exponents, and indicates that the dynamics of the system is expanded in more than one direction. For a chaotic system, there is just one positive Lyapunov exponent. Messages masked by such simple chaotic systems are not always safe [3]. It is suggested that this problem can be overcome by using higher-dimensional hyperchaotic systems, which have increased randomness and higher unpredictability [4]. Due to its higher unpredictability than chaotic system, the hyperchaos may be more useful in some fields
such as communication, encryption etc. Hyperchaos was first reported by Rössler in 1979 [5]. In the last years, the generation of hyperchaos have been studied with increasing interest [6,7]. Hyperchaotic Rössler system [5] and hyperchaotic Chua’s circuit [8] are two well-known hyperchaos examples. For hyperchaos, some basic properties are described as follows: (i) Hyperchaos exists only in higher-dimensional systems, i.e., not less than four-dimensional (4D) autonomous system for the continuous time cases. (ii) It was suggested that the number of terms in the coupled equations that give rise to instability should be at least two, in which one should be a nonlinear function [5]. However, for most existing hyperchaotic systems, the second biggest Lyapunov exponent is relatively small. Recently, based on the Lorenz system, a new chaotic system was reported by Qi et al. [9]. The system is described by x˙ = a(y − x) + yz,
* Corresponding author.
E-mail address: [email protected] (Z. Chen). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.085
y˙ = cx − xz − y, z˙ = xy − bz.
(1)
Z. Chen et al. / Physics Letters A 360 (2007) 696–701
The chaotic system (1) has complex nonlinear dynamical characters. This system can generate complex dynamics within wide parameters ranges, including chaos, Hopf bifurcation, period-doubling bifurcation, periodic orbit, sink and source, and so on [9]. In this Letter, a new four-dimensional hyperchaotic system is designed by introducing state feedback control and constant multipliers to the two quadratic terms in system (1). The new system reported in this Letter only has one equilibrium, but it has bigger positive Lyapunov exponents than those already known hyperchaotic systems. So the new system may be more useful in some fields such as encryption, communication. The new system is not only demonstrated by numerical computing but also verified with bifurcation analysis. 2. Introducing a hyperchaotic system Based on the above chaotic system (1), a simple nonlinear state feedback controller is introduced to the second equation. At the same time, in the first and the second equation, a constant multiplier is added to the cross-product item respectively, and the sign of item y in the second equation is also changed. As a result, the following four-order system can be obtained ⎧ x˙ = a(y − x) + eyz, ⎪ ⎨ y˙ = cx − dxz + y + u, (2) ⎪ ⎩ z˙ = xy − bz, u˙ = −ky. For the new 4D system described by (2), a, b, c, d, e, k are constants to be tuned and x, y, z, u are state variables. The new system (2) has the following basic properties:
697
3. Bifurcation analysis In the two sections below, some properties of the new fourdimensional system are further discussed with k and b varying. And the simulation results are obtained by using 4-order Runge–Kutta method with the step length taken as 0.001. 3.1. Fixing a = 35, b = 4.9, c = 25, d = 5, e = 35 and varying k When k varies, the corresponding three Lyapunov exponents [10,11] of system (2) are shown in Fig. 1. The bifurcation diagram of state variable y with increasing k is given in Fig. 2. It can be observed that the bifurcation diagram well coincides with the spectrum of Lyapunov exponents. Fig. 1 shows that system (2) is hyperchaotic for a very wide range of k, and the system can also evolve into chaotic orbits, periodic orbits, and quasi-periodic orbits. From Figs. 1 and 2, the dynamical behaviors of system (2) can be clearly observed. When k ∈ (331, 340), (386, 536.5), (556, 570), (577, 583), the largest Lyapunov exponent almost equals zeros, which means that system (2) is periodic. Especially there are also some periodic windows in the chaotic or quasi-periodic regions. Some periodic phase portraits for special values of k are depicted in Fig. 3. When k ∈ (570, 577), (583, 843), the first and the second largest Lyapunov exponents are almost zero, and the others are negative, which means the system (2) is quasi-periodic. Some phase portraits of special k are shown in Fig. 4. When k belongs to (171, 241), (297, 331), (536.5, 556), there are one positive, one zero, and two negative Lyapunov ex-
(1) It is symmetric with respect to the z-axis, which is invariant for the coordinate transformation (x, y, z, u) → (−x, −y, z, −u). (2) It is dissipative when (a + b − 1) > 0, since ∇V =
∂ x˙ ∂ y˙ ∂z ∂u + + + = −(a + b − 1). ∂x ∂y ∂z ∂u
(3) It only has zero equilibrium. By linearizing at zero, the following Jacobian matrix is obtained ⎡ ⎤ −a a 0 0 1 0 1⎥ ⎢ c J =⎣ ⎦. 0 0 −b 0 0 −k 0 0
Fig. 1. The Lyapunov exponents vs. k with a = 35, b = 4.9, c = 25, d = 5, e = 35.
The eigenvalues of matrix J are given by the roots of the characteristic equation
(λ + b) λ3 + (a − 1)λ2 + (k − a − ac)λ + ak = 0. For example, when choose a = 35, b = 4.9, c = 25, d = 5, e = 35, k = 100, the four eigenvalues of matrix J are 10.8806, 6.2867, −4.9, −51.1673 respectively. In this case, zero is a saddle point.
Fig. 2. Bifurcation diagram for k with a = 35, b = 4.9, c = 25, d = 5, e = 35.
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Z. Chen et al. / Physics Letters A 360 (2007) 696–701
(a)
(b)
(c) Fig. 3. Periodic phase portraits of the system (2) in the x–z plane for different k. (a) k = 264.5; (b) k = 367; (c) k = 535.
(a)
(b)
(c) Fig. 4. Quasi-periodic phase portraits of the system (2) in the y–u plane for different k. (a) k = 390; (b) k = 573; (c) k = 700.
Z. Chen et al. / Physics Letters A 360 (2007) 696–701
(a)
699
(b)
(c) Fig. 5. Chaotic phase portraits of the system (2) in the x–y plane for different k. (a) k = 190; (b) k = 303; (c) k = 538.
(a)
(b)
(c) Fig. 6. Hyperchaotic phase portraits of the system (2) with k = 22: (a) x–y plane; (b) x–z plane; (c) x–u plane.
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Z. Chen et al. / Physics Letters A 360 (2007) 696–701
ponents, which means system (2) is in chaotic mode. The phase portraits for some special values of k are depicted in Fig. 5. When k ∈ (0, 171), (241, 297), system (2) has two positive, one zero, and one negative Lyapunov exponents. In these cases, the system is hyperchaotic. Note that, there are some periodic windows in the hyperchaotic region. For example, when k = 41.5, system (2) is periodic. When k = 22, system (2) is hyperchaotic and the corresponding phase portraits are shown in Fig. 6. Some typical hyperchaotic systems with different k and their Lyapunov exponents are listed in Table 1.
It is shown that, when k varies continuously within a very wide range, system (2) is always hyperchaotic. Further more,
Table 1 Some typical parameter values of k to generate hyperchaos k
λ1
λ2
λ3
λ4
10 22 67 109 126
4.409 4.031 3.069 2.411 2.158
0.131 0.252 0.669 0.792 0.859
0 0 0 0 0
−43.440 −43.182 −42.638 −42.103 −41.917
Fig. 7. The Lyapunov exponents vs. b with a = 35, c = 25, d = 5, e = 35, k = 100.
Table 2 Some typical parameter values of b to generate hyperchaos b
λ1
λ2
λ3
λ4
3.8 5.5 7.6 9.9 11
2.251 2.496 2.761 2.562 2.247
0.680 0.751 0.760 0.743 0.705
0 0 0 0 0
−40.734 −42.747 −45.118 −47.203 −47.951
Fig. 8. Bifurcation diagram for b with a = 35, c = 25, d = 5, e = 35, k = 100.
(a)
(b)
(c)
(d)
Fig. 9. Phase portraits of the system (2) in the x–y plane for increasing b, with a = 35, c = 25, d = 5, e = 35, k = 100: (a) hyperchaotic; (b) chaotic; (c) quasi-periodic; (d) periodic attractor. (a) b = 5.2; (b) b = 15.5; (c) b = 20; (d) b = 39.7.
Z. Chen et al. / Physics Letters A 360 (2007) 696–701
the two positive Lyapunov exponents are all big, which have not yet been reported in other known hyperchaotic systems. 3.2. Fixing a = 35, c = 25, d = 5, e = 35, k = 100 and varying b
701
different parameters, the hyperchaotic orbits, chaotic orbits, periodic orbits and quasi-periodic orbits have also been observed. The hyperchaotic system has been confirmed in various numerical computing and verified with detailed bifurcation analysis. Acknowledgements
For the new hyperchaotic system, Fig. 7 shows the three Lyapunov exponents of system (2) with respect to parameter b. And the corresponding bifurcation diagram is shown in Fig. 8. With b increasing, the system (2) can evolve into hyperchaotic, chaotic, periodic, and quasi-periodic orbits. The Lyapunov exponent spectrum and bifurcation diagram demonstrate that system (2) has complex behaviors as b varies. Some special attractors are briefly depicted in Fig. 9. Fig. 7 clearly shows that when b varies in a very wide range, the system is still hyperchaotic. For some typical values of b, the corresponding Lyapunov exponents are tabulate in Table 2. From Table 2, it is found that, when b varies within the proper range, system (2) is also hyperchaotic as shown in Table 1. 4. Conclusions In this Letter, we have presented an autonomous hyperchaotic system based on a three-dimensional quadratic chaotic system. The new system has some good properties. For example, it only has one equilibrium, but it has two relatively big Lyapunov exponents, and it is still hyperchaotic when the parameter k or b varies in a very wide range. So, the new hyperchaotic system may have good application prospects. With
This work was partially supported by the National Nature Science Foundation of China (Grant Nos. 60374037 and 60574036), the Specialized Research Fund for the Doctoral Program of China (Grant No. 20050055013), the program for New Excellent Talents in University of China (NCET), and Science and Technology Development Foundation of Tianjin Colleges, PR China (20051528). References [1] C. Sparrow, The Lorenz Equations: Bifurcations Chaos and Strange Attractors, Springer, New York, 1982. [2] G. Chen, T. Ueta, Int. J. Bifur. Chaos 9 (1999) 1465. [3] G. Perez, H.A. Cerdeira, Phys. Rev. Lett. 74 (1995) 1970. [4] L. Pecora, Phys. World 9 (1996) 17. [5] O.E. Rossler, Phys. Lett. A 71 (1979) 155. [6] Y. Li, G. Chen, L. Wallace, K.S. Tang, IEEE Trans. CAS-II 52 (2005) 204. [7] Y. Li, Wallace, K.S. Tang, G. Chen, Int. J. Circuits Theory Appl. 33 (2005) 235. [8] D. Cafagna, G. Grassi, Int. J. Bifur. Chaos 13 (2003) 2889. [9] G. Qi, G. Chen, S. Du, Z. Chen, Z. Yuan, Physica A: Stat. Mech. Appl. 352 (2005) 295. [10] A. Wolf, J. Swift, H. Swinney, J. Vastano, Physica D 16 (1985) 285. [11] J.P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57 (1985) 617.