- Email: [email protected]
Dynamic Analysis and Synchronization for a Generalized Class of Nonlinear Systems
2012 IFAC Conference on Analysis and Control of Chaotic Systems The International Federation of Automatic Control June 20-22, 2012. Cancún, México
Dynamic Analysis and Synchronization for a Generalized Class of Nonlinear Systems D. Becker-Bessudo ∗ A. I. Klip-Kahan ∗ S. Leboreiro-Velez ∗ S. Carrillo-Moreno ∗ J. J. Flores-Godoy ∗ G. Fernandez-Anaya ∗ ∗
Departamento de F´ısica y Matem´ aticas, Universidad Iberoamericana, Prolongaci´ on Paseo de la Reforma 880, 01219 M´exico, D.F., M´exico. Tel.: +52 55 59504071; fax: +52 55 59504284. (e-mails: [email protected], [email protected], lebo [email protected], [email protected], [email protected], [email protected])
Abstract: This paper presents a generalized differential equation structure which gives rise to new nonlinear, chaotic systems and also encompasses many well known systems, i.e. Lorenz, Chen, L¨ u, R¨ossler, Sprott and others. Throughout the paper we shall analyze several properties from a few of the systems derived from this general structure. Some of the systems described in the article, to the extent of the authors knowledge, have not been published. The analytical and numerical results derived from these analysis have shown evidence of chaotic behavior. We will also address the possibility of quasi-simultaneous synchronization for some members of this class of chaotic systems. Keywords: Chaotic-Nonlinear Systems, Control, Observers, Stabilization, and Synchronization 1. INTRODUCTION Ever since the discovery of physical chaotic systems, Lorenz (Lorenz, 1963), Chen (Chen and Ueta, 1999), Chua (Matsumoto, 1984), L¨ u (L¨ u and Chen, 2002), etc., research interests have been stirred to find new and ever more complex systems exhibiting these qualities. Applications for these systems, such as signal and imaging encryption Bu and Wang (2004), could benefit from finding new and more diverse chaotic systems. New systems exhibiting particular properties could also make them more suitable to specific applications. This paper discusses a generalized structure which describes a class of dynamical system under which the authors were able to identify several new systems which exhibited complex dynamics. Numerical and theoretical analysis of these systems confirmed the existence of chaotic dynamics. The interrelation between these systems given by the general structure led the authors to pursue the possibility of synchronizing these new attractors using a nonlinear feedback law Bai and Lonngren (2000) with a generalized linear feedback matrix. The success of this generalized error stabilizing controller can be attested in the numerical simulations of the error dynamics presented in the paper. As a measure for comparison we used the more conventional LQR method to find stabilizing feedback matrices for each particular system and compared the results with the generalized linear feedback. The motivation for this paper is due in part to previous research presented in Pan et al. (2010). Both articles are 978-3-902823-02-1/12/$20.00 © 2012 IFAC
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interested in analyzing the dynamical properties of new chaotic systems through the use of numerical and analytical tools. However, this article goes a step further by introducing the possibility of quasi-simultaneous synchronization through a generalized control law using a common linear feedback matrix. 2. DETERMINING THE STRUCTURE FOR THE FAMILY OF CHAOTIC DYNAMICAL SYSTEMS The structure for our generalized nonlinear dynamical system class is determined as follows: ˙ = f (X) = A0 + M X + N R + Q S, X
(1)
where A0 , M, N, Q are ! a1 A0 = a2 , M = a3
parameter matrices defined by ! m11 m12 m13 m21 m22 m23 , m31 m32 m33 (2) ! ! n11 n12 n13 q11 q12 q13 N = n21 n22 n23 , Q = q21 q22 q23 , n31 n32 n33 q31 q32 q33 where X is the state variable vector and R, S are vector functions of X 2 ! ! x x yz (3) X = y , R = xz , S = y 2 . z xy z2
Through this generalized structure it is possible to identify several dynamical systems that present chaotic behavior. In the following section we analyze several properties of
10.3182/20120620-3-MX-3012.00036
CHAOS'12 June 20-22, 2012. Cancún, México
eight select cases which present particularly interesting dynamics.
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This structure is similar to the one presented by J. C. Sprott in Sprott (1994). However Sprott was interested in finding algebraically simple ODE’s to contend the idea of R¨ossler’s system being the simplest chaotic flow. This article is focused on finding and analyzing new systems and thus we have implemented a broader spectrum of R parameters as well as newer tools from MATLAB to solve and analyze them.
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In this section we present the parameter values that give rise to the nonlinear systems in question. Given the general structure mentioned above, consider any parameter not mentioned in a particular system to be exactly zero.
Systems 1 through 5, to the authors knowledge, have not been published before. This list includes previously published systems 6 and 7, Pan et al. (2010), as well as the R¨ossler system R¨ossler (1976). The reason we include them here is because they fit into the general structure we proposed. 4. SIMULATIONS AND PROPERTIES OF THE SYSTEMS Due to space constraints we show simulations and graphical information related to the systems 1, 3 and 5 due to their interesting dynamical behavior. 4.1 Trajectories, Poincar´e Maps and Bifurcation Diagrams System 1 shown in Fig. 1 looks like a relatively common systems very similar to some of the attractors found by Sprott. System 3 shown in Fig. 2 shows trajectories which the authors have dubbed “m¨obius vortex”. 155
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(1) System 1: m12 = −22.5, m13 = −45, m21 = 27.5, m22 = 7.5, m23 = −50, m32 = −2.5, m33 = −45, n23 = −1, n31 = 1, n33 = 1, q13 = −1, q22 = −1, q31 = 1, q32 = −1. (2) System 2: m11 = 15, m13 = 22.5, m21 = −42.5, m23 = 2.5, m31 = −1, m32 = 27.5, m33 = −10, n11 = 1, n12 = −1, n23 = −1, q32 = 1. (3) System 3: m12 = −40, m13 = −35, m21 = 5, m31 = −12.5, m32 = 1, m33 = −12.5, n32 = 1, q13 = −1, q32 = −1. (4) System 4: m11 = −50, m12 = 32.5, m21 = −10, m22 = 27.5, m31 = 35, m33 = −2.5, n13 = −1, n22 = 1, q11 = −1, q32 = −1. (5) System 5: m12 = −35, m13 = 12.5, m21 = 5, m32 = −10, m33 = −25, n11 = −1, n22 = 1. (6) System 6*: m11 = −10, m12 = 10, m21 = 40, m33 = −2.5, n22 = −1, q31 = 4. (7) System 7*: m11 = −10, m12 = 10, m21 = 40, m22 = −2, m33 = −2.5, n22 = −1, q31 = 4. (8) R¨ ossler*: a3 = 0.1, m12 = −1, m13 = −1, m21 = 1, m22 = 0.1, m33 = −14, n32 = 1.
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Fig. 2. Phase space and Poincar´e maps for system 3 System 5 which can be seen en Fig. 3 bears certain similarity with the Lorenz, Chen and L¨ u systems in the sense that it appears to be a double scroll attractor with a spiral vortex on the transition boundary between the scrolls. The Poincar´e maps where elaborated by placing 3 sampling planes, one parallel to each axis, on the average value for the solutions of each state variable. The bifurcation diagrams were constructed by changing the value of the parameter m12 , by trial an error we found that the chaotic behavior of the dynamical systems is more sensible to this parameter than for some others, Figs. 4, 5 and 6, show the regions where chaotic orbits are present along a value interval for the m12 parameter. The results for systems 1 and 3 are consistent with what is typically observed for chaotic systems with period doubling bifurcations. System 5 however shows a different behavior with windows of periodic orbits followed by chaotic regions.
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Figs. (7.a), (8.a) and (9.a) show the absolute value of the state errors between the trajectories of the master/slave systems using the quasi-simultaneous stabilizing feedback matrix. Figs. (7.b), (8.b) and (9.b) show the same system errors using the individual feedback matrices derived from the LQR method. It can be easily appreciated that both controllers effectively synchronize both master/slave pairs. However it should be noted that synchronization is not achieved at the same rate by both controllers.
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Fig. 7. Absolute error |E| = |X − Y| between the trajectories of the master/slave system 1. Controls (K, KLQR1 , corresponding to graphs a) and b) respectively) are active from t = 4 onwards.
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Just as it was seen in (6) the error dynamics are reduced to a set of linear equations of the form ˙ i = Ai Ei + gi (X) − gi (Y) − Ui E (9) = Ai Ei + Li (X, Y) − KLQRi Ei − Li (X, Y) = (Ai − KLQRi )Ei where Ai − KLQRi are Hurwitz matrices for all i.
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The second method employed to achieve synchronization for these systems involved the use of a linear quadratic state-feedback regulator to find a stabilizing feedback matrix for each individual system and using the same feedback law. The state feedback matrix is determined by the solution, Pi , of the Riccati equation −1 ⊤ A⊤ B +T =0 (8) i Pi + Pi Ai + Pi BR where R and T positive definite and a positive semidefinite weighting matrices, respectively; yielding KLQRi = R−1 BPi .
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The constant feedback matrix found by the previous heuristic procedure for the 3 master/slave pairs formed by systems 1, 3 and 5 was
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The employed feedback matrix K was chosen to be a diagonal matrix and the task of finding the appropriate values of its components was carried out by finding the stability regions of the closed loop system running the values of K over a predefined interval. Next the results of this search were sifted in order to find the intersection of all these regions. Finally the triad of values with the smallest Euclidean norm of these intersections was selected and its values increased by a relatively significant amount to ensure stability across all systems.
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tion of the error dynamics for the systems in question. The notion of quasi-simultaneous stabilizations consist on using a fixed controller to stabilize several different error systems. ˙ i = Ai Ei + gi (Xi ) − gi (Yi ) − Ui E = Ai Ei + Li (Xi , Yi ) − KEi − Li (Xi , Yi ) (6) = (Ai − K)Ei this renders the error dynamics equation a linear system; as such, simultaneous stability for the error systems may be assured as long as the set A1 − K, A2 − K, . . . , An − K be comprised entirely of Hurwitz matrices.
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Fig. 9. Absolute error |E| = |X − Y| between the trajectories of the master/slave system 5. Controls (K, KLQR5 , corresponding to graphs a) and b) respectively) are active from t = 4 onwards. To the authors knowledge the concept of quasi-simultaneous synchronization has not been addressed or reported in previous publications. 6. CONCLUSIONS In this paper we have proposed a generalized structure for dynamical systems (1) which under different arrays of parameters gives rise to several new chaotic systems. The numerical and analytical data corroborate the existence of chaotic dynamics for the particular examples exhibited in this article. As a first approach towards stabilization of the dynamics of these systems, numerical data supports the proposition of quasi-simultaneous synchronization through a common linear feedback matrix. The results derived from this paper coincide with those found in Pan et al. (2010)
CHAOS'12 June 20-22, 2012. Cancún, México
reinforcing the analysis methodology used to characterize the systems, despite the fact that the methods the were employed to find the new systems on each article are different. It should be noted, however that this paper has introduced a general class under which the aforementioned article’s system is a particular case. REFERENCES Alligood, K.T., Sauer, T., and Yorke, J.A. (1996). Chaos: an introduction to dynamical systems. Springer. Bai, E.W. and Lonngren, K.E. (2000). Sequential synchronization of two lorenz systems using active control. Chaos, Solitons & Fractals, 11(7), 1041 – 1044. Bu, S. and Wang, B.H. (2004). Improving the security of chaotic encryption by using a simple modulating method. Chaos, Solitons & Fractals, 19(4), 919 – 924. Chen, G. and Ueta, T. (1999). Yet another chaotic attractor. Int. J. of Bifur. Chaos, 9, 1465–1466. Doyle, J., Francis, B., and Tannenbaum, A. (1990). Feedback Control Theory. Macmillan, New York. Lorenz, E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci., 20(2), 130–141. L¨ u, J.H. and Chen, G. (2002). A new chaotic attractor coined. Int. J. of Bifur Chaos, 12(3), 659–661. Matsumoto, T. (1984). A chaotic attractor from Chua’s circuit. IEEE Trans. on Circuits & Systems, CAS31(12), 1055–1058. Nijmeijer, H. (2001). A dynamical control view on synchronization. Physica D: Nonlinear Phenomena, 154, 219–228. Pan, L., Zhou, W., and Fang, J. (2010). Dynamics analysis of a new simple chaotic attractor. Int. J. Control Autom., 8(2), 468–472. R¨ossler, O.E. (1976). An equation for continuous chaos. Phys. Lett. A, 57(5), 397–398. Sprott, J.C. (1994). Some simple chaotic flow. Physical Review E, 50(2), R647–R650.
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Dynamic Analysis and Synchronization for a Generalized Class of Nonlinear Systems D. Becker-Bessudo ∗ A. I. Klip-Kahan ∗ S. Leboreiro-Velez ∗ S. Carrillo-Moreno ∗ J. J. Flores-Godoy ∗ G. Fernandez-Anaya ∗ ∗
Departamento de F´ısica y Matem´ aticas, Universidad Iberoamericana, Prolongaci´ on Paseo de la Reforma 880, 01219 M´exico, D.F., M´exico. Tel.: +52 55 59504071; fax: +52 55 59504284. (e-mails: [email protected], [email protected], lebo [email protected], [email protected], [email protected], [email protected])
Abstract: This paper presents a generalized differential equation structure which gives rise to new nonlinear, chaotic systems and also encompasses many well known systems, i.e. Lorenz, Chen, L¨ u, R¨ossler, Sprott and others. Throughout the paper we shall analyze several properties from a few of the systems derived from this general structure. Some of the systems described in the article, to the extent of the authors knowledge, have not been published. The analytical and numerical results derived from these analysis have shown evidence of chaotic behavior. We will also address the possibility of quasi-simultaneous synchronization for some members of this class of chaotic systems. Keywords: Chaotic-Nonlinear Systems, Control, Observers, Stabilization, and Synchronization 1. INTRODUCTION Ever since the discovery of physical chaotic systems, Lorenz (Lorenz, 1963), Chen (Chen and Ueta, 1999), Chua (Matsumoto, 1984), L¨ u (L¨ u and Chen, 2002), etc., research interests have been stirred to find new and ever more complex systems exhibiting these qualities. Applications for these systems, such as signal and imaging encryption Bu and Wang (2004), could benefit from finding new and more diverse chaotic systems. New systems exhibiting particular properties could also make them more suitable to specific applications. This paper discusses a generalized structure which describes a class of dynamical system under which the authors were able to identify several new systems which exhibited complex dynamics. Numerical and theoretical analysis of these systems confirmed the existence of chaotic dynamics. The interrelation between these systems given by the general structure led the authors to pursue the possibility of synchronizing these new attractors using a nonlinear feedback law Bai and Lonngren (2000) with a generalized linear feedback matrix. The success of this generalized error stabilizing controller can be attested in the numerical simulations of the error dynamics presented in the paper. As a measure for comparison we used the more conventional LQR method to find stabilizing feedback matrices for each particular system and compared the results with the generalized linear feedback. The motivation for this paper is due in part to previous research presented in Pan et al. (2010). Both articles are 978-3-902823-02-1/12/$20.00 © 2012 IFAC
154
interested in analyzing the dynamical properties of new chaotic systems through the use of numerical and analytical tools. However, this article goes a step further by introducing the possibility of quasi-simultaneous synchronization through a generalized control law using a common linear feedback matrix. 2. DETERMINING THE STRUCTURE FOR THE FAMILY OF CHAOTIC DYNAMICAL SYSTEMS The structure for our generalized nonlinear dynamical system class is determined as follows: ˙ = f (X) = A0 + M X + N R + Q S, X
(1)
where A0 , M, N, Q are ! a1 A0 = a2 , M = a3
parameter matrices defined by ! m11 m12 m13 m21 m22 m23 , m31 m32 m33 (2) ! ! n11 n12 n13 q11 q12 q13 N = n21 n22 n23 , Q = q21 q22 q23 , n31 n32 n33 q31 q32 q33 where X is the state variable vector and R, S are vector functions of X 2 ! ! x x yz (3) X = y , R = xz , S = y 2 . z xy z2
Through this generalized structure it is possible to identify several dynamical systems that present chaotic behavior. In the following section we analyze several properties of
10.3182/20120620-3-MX-3012.00036
CHAOS'12 June 20-22, 2012. Cancún, México
eight select cases which present particularly interesting dynamics.
10
5 z
This structure is similar to the one presented by J. C. Sprott in Sprott (1994). However Sprott was interested in finding algebraically simple ODE’s to contend the idea of R¨ossler’s system being the simplest chaotic flow. This article is focused on finding and analyzing new systems and thus we have implemented a broader spectrum of R parameters as well as newer tools from MATLAB to solve and analyze them.
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In this section we present the parameter values that give rise to the nonlinear systems in question. Given the general structure mentioned above, consider any parameter not mentioned in a particular system to be exactly zero.
Systems 1 through 5, to the authors knowledge, have not been published before. This list includes previously published systems 6 and 7, Pan et al. (2010), as well as the R¨ossler system R¨ossler (1976). The reason we include them here is because they fit into the general structure we proposed. 4. SIMULATIONS AND PROPERTIES OF THE SYSTEMS Due to space constraints we show simulations and graphical information related to the systems 1, 3 and 5 due to their interesting dynamical behavior. 4.1 Trajectories, Poincar´e Maps and Bifurcation Diagrams System 1 shown in Fig. 1 looks like a relatively common systems very similar to some of the attractors found by Sprott. System 3 shown in Fig. 2 shows trajectories which the authors have dubbed “m¨obius vortex”. 155
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(1) System 1: m12 = −22.5, m13 = −45, m21 = 27.5, m22 = 7.5, m23 = −50, m32 = −2.5, m33 = −45, n23 = −1, n31 = 1, n33 = 1, q13 = −1, q22 = −1, q31 = 1, q32 = −1. (2) System 2: m11 = 15, m13 = 22.5, m21 = −42.5, m23 = 2.5, m31 = −1, m32 = 27.5, m33 = −10, n11 = 1, n12 = −1, n23 = −1, q32 = 1. (3) System 3: m12 = −40, m13 = −35, m21 = 5, m31 = −12.5, m32 = 1, m33 = −12.5, n32 = 1, q13 = −1, q32 = −1. (4) System 4: m11 = −50, m12 = 32.5, m21 = −10, m22 = 27.5, m31 = 35, m33 = −2.5, n13 = −1, n22 = 1, q11 = −1, q32 = −1. (5) System 5: m12 = −35, m13 = 12.5, m21 = 5, m32 = −10, m33 = −25, n11 = −1, n22 = 1. (6) System 6*: m11 = −10, m12 = 10, m21 = 40, m33 = −2.5, n22 = −1, q31 = 4. (7) System 7*: m11 = −10, m12 = 10, m21 = 40, m22 = −2, m33 = −2.5, n22 = −1, q31 = 4. (8) R¨ ossler*: a3 = 0.1, m12 = −1, m13 = −1, m21 = 1, m22 = 0.1, m33 = −14, n32 = 1.
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Fig. 2. Phase space and Poincar´e maps for system 3 System 5 which can be seen en Fig. 3 bears certain similarity with the Lorenz, Chen and L¨ u systems in the sense that it appears to be a double scroll attractor with a spiral vortex on the transition boundary between the scrolls. The Poincar´e maps where elaborated by placing 3 sampling planes, one parallel to each axis, on the average value for the solutions of each state variable. The bifurcation diagrams were constructed by changing the value of the parameter m12 , by trial an error we found that the chaotic behavior of the dynamical systems is more sensible to this parameter than for some others, Figs. 4, 5 and 6, show the regions where chaotic orbits are present along a value interval for the m12 parameter. The results for systems 1 and 3 are consistent with what is typically observed for chaotic systems with period doubling bifurcations. System 5 however shows a different behavior with windows of periodic orbits followed by chaotic regions.
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Figs. (7.a), (8.a) and (9.a) show the absolute value of the state errors between the trajectories of the master/slave systems using the quasi-simultaneous stabilizing feedback matrix. Figs. (7.b), (8.b) and (9.b) show the same system errors using the individual feedback matrices derived from the LQR method. It can be easily appreciated that both controllers effectively synchronize both master/slave pairs. However it should be noted that synchronization is not achieved at the same rate by both controllers.
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Fig. 7. Absolute error |E| = |X − Y| between the trajectories of the master/slave system 1. Controls (K, KLQR1 , corresponding to graphs a) and b) respectively) are active from t = 4 onwards.
10
Just as it was seen in (6) the error dynamics are reduced to a set of linear equations of the form ˙ i = Ai Ei + gi (X) − gi (Y) − Ui E (9) = Ai Ei + Li (X, Y) − KLQRi Ei − Li (X, Y) = (Ai − KLQRi )Ei where Ai − KLQRi are Hurwitz matrices for all i.
5 t
a)
10
−4
−4
10 abs(X−Y)
10 abs(X−Y)
The second method employed to achieve synchronization for these systems involved the use of a linear quadratic state-feedback regulator to find a stabilizing feedback matrix for each individual system and using the same feedback law. The state feedback matrix is determined by the solution, Pi , of the Riccati equation −1 ⊤ A⊤ B +T =0 (8) i Pi + Pi Ai + Pi BR where R and T positive definite and a positive semidefinite weighting matrices, respectively; yielding KLQRi = R−1 BPi .
3
−18
10
The constant feedback matrix found by the previous heuristic procedure for the 3 master/slave pairs formed by systems 1, 3 and 5 was
−8
10
−10
10
abs(X−Y)
The employed feedback matrix K was chosen to be a diagonal matrix and the task of finding the appropriate values of its components was carried out by finding the stability regions of the closed loop system running the values of K over a predefined interval. Next the results of this search were sifted in order to find the intersection of all these regions. Finally the triad of values with the smallest Euclidean norm of these intersections was selected and its values increased by a relatively significant amount to ensure stability across all systems.
a)
abs(X−Y)
tion of the error dynamics for the systems in question. The notion of quasi-simultaneous stabilizations consist on using a fixed controller to stabilize several different error systems. ˙ i = Ai Ei + gi (Xi ) − gi (Yi ) − Ui E = Ai Ei + Li (Xi , Yi ) − KEi − Li (Xi , Yi ) (6) = (Ai − K)Ei this renders the error dynamics equation a linear system; as such, simultaneous stability for the error systems may be assured as long as the set A1 − K, A2 − K, . . . , An − K be comprised entirely of Hurwitz matrices.
−6
10
−8
−6
10
−8
10
10
−10
−10
10
10
−12
−12
10
10 abs(x1−y1)
−14
2
abs(x1−y1)
−14
abs(x −y )
10
abs(x −y )
10
2
2
abs(x −y ) 3
2
abs(x −y )
3
3
−16
10
3
−16
0
2
4 t
6
10
0
10
20
30
t
Fig. 9. Absolute error |E| = |X − Y| between the trajectories of the master/slave system 5. Controls (K, KLQR5 , corresponding to graphs a) and b) respectively) are active from t = 4 onwards. To the authors knowledge the concept of quasi-simultaneous synchronization has not been addressed or reported in previous publications. 6. CONCLUSIONS In this paper we have proposed a generalized structure for dynamical systems (1) which under different arrays of parameters gives rise to several new chaotic systems. The numerical and analytical data corroborate the existence of chaotic dynamics for the particular examples exhibited in this article. As a first approach towards stabilization of the dynamics of these systems, numerical data supports the proposition of quasi-simultaneous synchronization through a common linear feedback matrix. The results derived from this paper coincide with those found in Pan et al. (2010)
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reinforcing the analysis methodology used to characterize the systems, despite the fact that the methods the were employed to find the new systems on each article are different. It should be noted, however that this paper has introduced a general class under which the aforementioned article’s system is a particular case. REFERENCES Alligood, K.T., Sauer, T., and Yorke, J.A. (1996). Chaos: an introduction to dynamical systems. Springer. Bai, E.W. and Lonngren, K.E. (2000). Sequential synchronization of two lorenz systems using active control. Chaos, Solitons & Fractals, 11(7), 1041 – 1044. Bu, S. and Wang, B.H. (2004). Improving the security of chaotic encryption by using a simple modulating method. Chaos, Solitons & Fractals, 19(4), 919 – 924. Chen, G. and Ueta, T. (1999). Yet another chaotic attractor. Int. J. of Bifur. Chaos, 9, 1465–1466. Doyle, J., Francis, B., and Tannenbaum, A. (1990). Feedback Control Theory. Macmillan, New York. Lorenz, E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci., 20(2), 130–141. L¨ u, J.H. and Chen, G. (2002). A new chaotic attractor coined. Int. J. of Bifur Chaos, 12(3), 659–661. Matsumoto, T. (1984). A chaotic attractor from Chua’s circuit. IEEE Trans. on Circuits & Systems, CAS31(12), 1055–1058. Nijmeijer, H. (2001). A dynamical control view on synchronization. Physica D: Nonlinear Phenomena, 154, 219–228. Pan, L., Zhou, W., and Fang, J. (2010). Dynamics analysis of a new simple chaotic attractor. Int. J. Control Autom., 8(2), 468–472. R¨ossler, O.E. (1976). An equation for continuous chaos. Phys. Lett. A, 57(5), 397–398. Sprott, J.C. (1994). Some simple chaotic flow. Physical Review E, 50(2), R647–R650.
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