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Simple Environment for Developing Methods of Controlling Chaos in Spatially Distributed Systems

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Simple Environment for Developing Methods of Controlling Chaos in Spatially Distributed Systems Int. J. Appl. Math. Comput. Sci., 2011, Vol. 21, No. 1, 149–159 DOI: 10.2478/v10006-011-0011-4 SIMPLE ENVIRONMENT FOR DEVELOPING METHODS OF CONTROLLING CHAOS IN SPATIALLY DISTRIBUTED SYSTEMS ŁUKASZ KORUS Institute of Informatics, Automatics and Robotics Wrocław University of Technology, ul. Janiszewskiego 11/17, 50–370 Wrocław, Poland e-mail: [email protected] The paper presents a simple mathematical model called a coupled map lattice (CML). For some range of its parameters, this model generates complex, spatiotemporal behavior which seems to be chaotic. The main purpose of the paper is to provide results of stability analysis and compare them with those obtained from numerical simulation. The indirect Lyapunov method and Lyapunov exponents are used to examine the dependence on initial conditions. The net direction phase is introduced to measure the symmetry of the system state trajectory. In addition, a real system, which can be modeled by the CML, is presented. In general, this article describes basic elements of environment, which can be used for creating and examining methods of chaos controlling in systems with spatiotemporal dynamics. Keywords: coupled map lattice, spatiotemporal chaos, system stability, Lyapunov exponents, net direction phase. 1. Introduction ticularly important in real, physical systems. For exam- ple, a chaotic behavior in chemical reactors can cause Deterministic chaos is complex behavior which can be ge- unpredictable and negative results. Moreover, many real nerated by a nonlinear dynamic system as a result of sta- systems and their mathematical models represent similar te trajectory evolution when its previous history is known spatiotemporal behaviors, where chaos can be observed (Ott, 2002). The origin of this phenomenon in nonlinear both in time and space. As has been shown, chaos in spa- dynamic systems is connected with their sensitivity to ini- tially distributed dynamical systems can also be control- tial conditions. In fact, chaotic behavior is a common phe- led (Auerbach, 1994; Astakhov et al., 1995; Parmanan- nomenon discovered in physics (liquids (Procaccia and da, 1997; Parekh et al., 1998; Zhu and Chen, 2001; Miha- Meron, 1986; Langenberg et al., 2004), lasers (Yamada liuk et al., 2002; Beck et al., 2002; Boukabou and Manso- and Graham, 1980; Used and Martin, 2010), plasma (Held uri, 2005). This makes this issue even more essential and et al., 1984; Banerjee et al., 2010)), mechanics (Chui and it seems that it could have numerous practical applications Ma, 1982; Ott, 2002), electronics (Gunaratne et al., 1989; (i.e., chemical reactors, lasers, plasma). Yim et al., 2004), chemistry (Argoul et al., 1987; Córdo- The main purpose of the paper is to provide results ba et al., 2006), biology (action of human heart (Govindan of both analytical and numerical research regarding CML et al., 1998) and brain (Andrzejak et al., 2001; Gautama system stability. In addition, we found it very important to et al., 2003)), and in many other branches of science and chose and present proper tools which can be useful during industry. the examination of the system sensitivity to initial con- Chaotic systems, due to their sensitivity to initial ditions. Based on measurements of this property and the conditions, are unpredictable but, as has been proved, level of state trajectory symmetry, some conclusions about they can be effectively controlled (Ott et al., 1990; Sin- the existence of chaos in the system can be made. ger et al., 1991; Dressler and Nitsche, 1992; Chen and The presented model and tools can be helpful while Dong, 1993; Alsing et al., 1994; Pyragas, 2001; Andrie- creating and checking new algorithms of controlling cha- vskii and Fradkov, 2003). The control of chaos in such os for spatiotemporal systems. It is even more essential as a type of system is considered a process which changes its potential, physical application is presented. In general, an irregular and unpredictable behavior to a well-ordered this article describes basic elements of environment which and periodical one. Sensitivity to initial conditions is par- can be used for creating and examining methods of con- 150 Ł. Korus Table 1. Classification of systems with spatiotemporal dynamics. Model Space Time Local structure Partial differential equations (PDEs) C C C Iterated functional equations (IFEs) C D C Oscillator chains (OCs) D C C Lattice dynamical systems (LDSs) D D C Cellular automata (CA) D D D trolling chaos in systems with spatiotemporal dynamics. small lattice sizes, this model is frequently used for mo- deling chaos. In addition, it is worth stating that both CA and CMLs are easy to implement as computer algorithms, 2. Models of systems with spatiotemporal mainly due to their simple structure. In practice, it is much dynamics easier to implement CA or CMLs than to use PDEs on 2.1. Classification. The classification of mathematical computers. methods used for modeling systems with spatiotempo- To sum up, it has to be highlighted that CMLs can ral dynamics was presented by Crutchfield and Kaneko be used as simple dynamic systems where complex col- (1987). This classification is based on choices of the di- lective behaviors, including spatiotemporal chaos, may be scretization of space, time and local state variables (see observed. Table 1). The letter ‘C’ in particular columns of Table 1 stands 3. CML: Model definition for continuous values while the letter ‘D’ for discrete va- A single-dimensional coupled map lattice can be consi- lues. The table contains five most frequently used models dered as a set of simple components numbered as n = for systems with spatiotemporal dynamics. For instance, 1,...,N. The state of each component given by xt(n) partial differential equations (PDEs) require a continu- changes in discrete time steps: t =1,...,T. It is calcu- ous function of time and space to specify a state. Latti- lated based on state values from the previous time step in ce dynamic systems (LDSs) and cellular automata (CA) surrounding components according to the evolution func- (Weimar, 1997; Chopard et al., 2002; Jacewicz, 2002) tion (Crutchfield and Kaneko, 1987; Kaneko, 1990): are constructed using discrete lattices (N-dimensional in general) and their states change according to a discrete xt+1(n)=f(xt(n)) + εLg(xt(n − 1)) (1) clock. The difference between LDSs and CA results from + ε0g(xt(n)) + εRg(xt(n +1)), different types of values in each node of lattice. In LDSs, the set of states is infinite and contains real values, while where the vector in CA only integers are allowed. The LDS class includes ε =(εL,ε0,εR) (2) also the coupled map lattice, i.e., the model which is pre- sented in this article. is the coupling kernel. The function f defines local dyna- When the distribution of nodes in the lattice is consi- mics and it is the most often taken as a logistic map dered, CA and the CML can be put into a single common class known as spatially distributed dynamic systems with f(x)=px(1 − x), (3) a regular structure. This class is special, because there is a symmetry of local connections for all lattice nodes. This or a circular map symmetry remains unchanged for the whole system. f(x)=ω + x + k sin(2πx). (4) The function g defines coupling dynamics and may 2.2. Brief characteristic of classes. Real-valued state be selected as the linear coupling variables in the CML make this model more efficient than CA when it comes to the modeling of real systems. Even g(x)=x, (5) for a small lattice size, behaviors similar to real ones can be observed (Kaneko, 1990). In CA, each cell is treated as or the same as a local function a macro-particle, therefore a bigger lattice size is necessa- ( )= ( ) ry to perform a real simulation (Weimar, 1997; Chopard g x f x . (6) et al., 2002; Jacewicz, 2002; Korus, 2007). For both types The system states in a particular time step can be collected of systems, i.e., CA and the CML, principles of dynamic using the vector systems theory and information theory can be applied. Be- cause of complex behaviors obtained in CMLs even for xt =(xt(1),xt(2),...,xt(N)). (7) Simple environment for developing methods of controlling chaos in spatially distributed systems 151 The paper presents examination results for the CML with diffusive coupling. The kernel for this system is given by the vector =( )= ε 1 − ε ε εL,ε0,εR 2, ε, 2 . (8) Replacing the coupling kernel from the general evolution function (1) with (8) and removing the part representing local dynamics, the evolution function of the system can be written as xt+1(n)=(1− ε)g(xt(n)) (9) + ε [ ( ( − 1)) + ( ( + 1))] 2 g xt n g xt n , where the coupling function is given by the logistic equ- ation g(x)=px(1 − x). (10) Fig. 1. Structure of the coupled map lattice under examination. A detailed analysis of transition to chaos in the logistic equation can be found in the work of Ott (2002). At the beginning of simulation, i.e., for the time step 4.2. Net direction phase. The net direction phase in- t =1, states values for all components are generated from troduced by Wei et al. (2000) is an indicator similar to a uniform distribution. The boundary conditions depend magnetization, which can be defined as on the modeled problem and can be chosen as fixed, pe- 1 T riodic, free, etc. In this paper, a CML model with fixed M = S(t), (13) boundary conditions T t=1 x(N +1)=0, where S(t)=1 for xt+1 − xt ≥ 1 and S(t)=−1 for (11) x(0) = 0 xt+1 − xt < 1,andT is the iteration number.
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