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Complex Dynamic Behaviors of the Complex Lorenz System

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Complex Dynamic Behaviors of the Complex Lorenz System View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Scientia Iranica D (2012) 19 (3), 733–738 Sharif University of Technology Scientia Iranica Transactions D: Computer Science & Engineering and Electrical Engineering www.sciencedirect.com Complex dynamic behaviors of the complex Lorenz system M. Moghtadaei, M.R. Hashemi Golpayegani ∗ Complex Systems and Cybernetic Control Lab., Faculty of Biomedical Engineering, Amirkabir University of Technology, Tehran, P.O. Box 1591634311, Iran Received 4 August 2010; revised 11 September 2010; accepted 29 November 2010 KEYWORDS Abstract This study compares the dynamic behaviors of the Lorenz system with complex variables to that Lorenz system; of the standard Lorenz system involving real variables. Different methodologies, including the Lyapunov Complex variables; Exponents spectrum, the bifurcation diagram, the first return map to the Poincaré section and topological Chaos; entropy, were used to investigate and compare the behaviors of these two systems. The results show that Hyper-chaos. expressing the Lorenz system in terms of complex variables leads to more distinguished behaviors, which could not be achieved in the Lorenz system with real variables, such as quasi-periodic and hyper-chaotic behaviors. ' 2012 Sharif University of Technology. Production and hosting by Elsevier B.V. Open access under CC BY-NC-ND license. 1. Introduction cannot be observed in 3-D systems. So, in order to achieve hyper-chaos, one should always add at least one extra variable In 1963, the concept of chaos was introduced by Edward to the three variable Lorenz equations [6,8,14–18]. Lorenz via his 3D autonomous system [1]. Over time, interest in Choosing different feedback controllers to the Lorenz the complex dynamic behaviors of nonlinear systems increased, equations can lead to different chaotic and hyper-chaotic due to their potential applications in different fields, such as systems. Many researchers [6,8,9,14–21] have introduced their detecting changes of biological signals (mostly EEG) in different hyper-chaotic systems by adding extra variables to the Lorenz abnormalities [2], data and image encryption [3], studying equations. In contrast, in this paper, we show that, indeed, sunspot cycles [4], and lasers [5]. Consequently, a large number simply changing Lorenz variables to complex forms not only of novel systems have been developed based on the original keeps the chaotic dynamics of the Lorenz equations but will Lorenz system [6–9]. also change the system into a higher dimensional system, A more complex form of dynamic behavior, hyper-chaos, where it has the potential to be hyper-chaotic for some ranges was introduced by Rössler in 1979, as a higher form of chaos, of parameter value. There are numerous ways to evaluate with at least two directions of hyperbolic instability on the chaotic behavior in systems, such as estimation of the fractal attractor [10]. In contrast to chaotic attractors, the hyper- dimension [22–24], correlation dimension [24], Lyapunov chaotic attractor has attracted more attention owing to its Exponent spectrum [24,25], and bifurcation diagram [24,26] higher complexity, which makes it harder to predict. Since etc. Here, we investigate the level of chaos in our system messages masked by simple chaotic systems are not always through numerical simulations by means of computing the safe, the use of higher dimensional hyper-chaotic systems Lyapunov Exponents spectrum, bifurcation diagram, first return to secure communication, and data and image encryption map to the Poincaré section, topological entropy of the first has been suggested because of their higher unpredictability return map, and the phase diagram. [11–13]. By definition [10], complex hyper-chaotic behavior In the following sections, first, the two Lorenz system forms that are based on real and complex variables are described, and ∗ then the requirements of chaotic and hyper-chaotic behaviors, Corresponding author. Tel.: +98 21 64542389. E-mail addresses: [email protected] (M. Moghtadaei), and our methods for measuring them are briefly explained. [email protected] (M.R. Hashemi Golpayegani). Subsequently, the results of our analysis on the behavior of Peer review under responsibility of Sharif University of Technology. these two forms of Lorenz system are reported and finally in the conclusion and discussion section, the main distinguished dynamical and behavioral features of the complex form of the Lorenz system are compared with those of its standard real form. 1026-3098 ' 2012 Sharif University of Technology. Production and hosting by Elsevier B.V. Open access under CC BY-NC-ND license. doi:10.1016/j.scient.2010.11.001 734 M. Moghtadaei, M.R. Hashemi Golpayegani / Scientia Iranica, Transactions D: Computer Science & Engineering and Electrical Engineering 19 (2012) 733–738 2. Complex and real Lorenz systems, and methods to 3. The number of terms giving rise to instability should be more quantify their behaviors than one, which means the system should expand in at least two directions, at least one of which should have a nonlinear The Lorenz system is described as [1]: function. (xP D a.y − x/ As mentioned above, Requirements 1 and 2 are already f V yP D cx − y − xz (1) satisfied for the Lorenz system involving complex variables. zP D xy − bz: To satisfy the third requirement, the existence of at least two positive Lyapunov Exponents is necessary [14]. Therefore, in Some important basic features of this system are: order to prove hyper-chaotic behavior in the complex form of 1. It is autonomous, which means that time does not explicitly the Lorenz system, we evaluated the presence of these two appear on the right hand side. positive Lyapunov Exponents. Using the method reported by Wolf et al. [25], the corre- 2. The equations involve only first order time derivatives, so sponding Lyapunov exponent spectrums of the real and com- the evolution depends only on the values of x; y, and z at the plex Lorenz systems were computed by fixing two parameters, time. a D 10 and b D 8=3, and varying the third parameter, c, as the 3. Due to the terms of xz and xy in the second and third equa- control parameter. The initial conditions in all these simulations tions, the system is non-linear. are the same and are equal to x ; x ; y ; y ; z ; z D 4. The system is dissipative when the following inequality 10 20 10 20 10 20 T0:1; 0:2; 0:3; 0:4; 1; 2U. holds: In addition, the bifurcation diagram was plotted for our @xP @yP @zP rf D C C D −a − 1 − b < 0: systems, which is the most useful graphical representation @x @y @z of the sequence of bifurcations that take place in the system Since parameters a and b, denoting the physical characteris- when the control parameter changes [26]. Also, the first return tics of the air flow, are positive, the inequality always holds map to the Poincaré section and the topological entropy of the and, thus, solutions are bounded. first return map were computed to check the complexity in dynamics for both complex and real based Lorenz systems. 5. The system is symmetric, with respect to the z axis, which The first return map to the Poincaré section is a common tool means it is invariant for the coordinate transformation: for analyzing the existence and stability of periodic trajectories .x; y; z/ ! .−x; −y; z/ of dynamical systems. It is defined on a hyper-surface formed by a Poincaré section, which is transverse to the trajectory The dual form of the Lorenz system expressed in terms of of the system [31]. In this study, these maps were formed complex variables is described by a similar formula [7]: by plotting the local maxima of the variable, z (in the Lorenz system with real variables), and the real part of the variable, (P D − x a.y x/ z (in the Lorenz system with complex variables), against their P D − − ∗ y cx y x z (2) next local maxima (at c D 30). P D − z xy bz The topological entropy quantifies the degree of complexity of a trajectory and it is the most important quantity related to where x; y, and z are complex variables defined as x D x1 C the orbit growth [32]. We used the symbolic dynamics of the ix2; y D y1 C iy2; z D z1 C iz2. The basic features of this complex form of Lorenz system first return maps at c D 30 to compute topological entropies by are similar to the aforementioned features. That means it is visualizing the complexity of the first return maps in the space autonomous, non-linear, dissipative with bounded solutions, of symbol sequences [33]. Comparing the topological entropy of symmetric with respect to the z axis, and involves only first both systems provides a finer distinction between the states of order time derivatives. Besides, this complex form is equal to their complexity. the third order Lorenz when the imaginary part of all variables is equal to zero. 3. Dynamical analysis of the Lorenz system involving Chaos is described as the irregular, unpredictable behavior of complex variables deterministic, non-linear dynamical systems. In order to have chaotic behavior, simultaneous stretching and folding in the The corresponding Lyapunov Exponent spectrums of the dynamics of the system are essential [27,28]. Stretching ensures System (1) (real form of Lorenz equations), and System (2) the sensitivity to initial conditions and folding guarantees (complex form of Lorenz equations), are shown in Figures 1 that the attractor is bounded [29]. Stretching and folding and 2, respectively. Also, the bifurcation diagrams of the state are equivalent to positive and negative Lyapunov Exponents, variables, z, in System (1), and z1 in Systems (2) are shown respectively.
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