ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 2, Issue 5, November 2012 Measuring Chaos in Some Discrete Nonlinear Systems L.M. Saha, Niteesh Sahni, Til Prasad Sarma The objective of the present work is to observe regular and Abstract— Lyapunov exponents and correlation dimensions, chaotic motions in certain discrete dynamical systems, which as measuring tools for chaos in a dynamical system, are explained have large applications in different areas. Bifurcation in detail. Some discrete systems exhibiting chaotic motion have diagrams are obtained for each system and then, proceeded been discussed for the application of these tools. Bifurcation diagrams are drawn for such systems which provide clear idea of further to calculate numerically LCEs and correlation regular and chaos at various set of parameter values. Also, chaotic dimension of certain chaotic orbit. Our investigation would attractors for these systems have been obtained for certain orbits confined to the following discrete systems: one dimensional at certain set of parameter values. Then, numerical calculations epidemic model, one dimensional exponential (Salmon) map, have been carried out for each model and plot of Lyapunov the gross national product (GNP) model, prey-predator exponents and plot of correlation integral curve have been obtained. Least square linear fit method has been applied to find population model, neural networks model and Ushiki map. correlation dimension to the obtained data for correlation Appearance of chaos is shown through bifurcation diagrams integrals. Some interesting graphics obtained through numerical which indicate the qualitative change in behavior of the simulation indicate very interesting results. system due to certain variations of system parameters. Index Terms— Lyapunov Exponents, Correlation II. SOME DEFINITION Dimensions, Bouncing Ball. A. Bifurcation Phenomena In ordinary sense, bifurcation means splitting into two. I. INTRODUCTION Bifurcation in a dynamical system occurs when a small All evolutionary systems come under the domain of smooth change made to the parameter values of the system dynamical system. A detailed discussion on the subject can be causing a sudden 'qualitative' or topological change in its obtained from pioneer articles on the subject by Devaney, behavior. It is the sudden change in behavior due to sudden (1989), Hao Bai-Lin (1984), Smale (1967),Moon (1987), change of set of parameter values of the system. The point, Stewart (1989), Gleick (1987) Sarkovskii (1964), May (1974, where qualitative change in behavior occur, is known as the 1976), Mandelbrot (1983), Hénon(1976), Chirikov (1979) bifurcation point. The name "bifurcation" was first introduced etc. Chaos is exhibited in nonlinear systems and can be by Henri Poincaré in 1885. Bifurcations occur in both viewed by observing bifurcations by varying a parameter of continuous systems (described by ODEs, DDEs or PDEs), the system. As natural systems are mostly nonlinear, existence and discrete systems (described by maps). of chaos in nature is also quite natural. We say a system B. Lyapunov Exponents evolve chaotically if it shows divergence in behavior of two Relative stability of typical orbits of a system is measured trajectories initiated at slightly different initial conditions, by numbers called Lyapunov exponents, named after the Such sensitivity to initial condition was first noticed by Russian Engineer Alexander M. Lyapunov. A system can Poincaré, Poincaré (1913), and later termed as chaos, Lorenz have as many Lyapunov exponents as there are dimensions in (1963). Lyapunov characteristic exponents, (LCE), are its phase space. Lyapunov exponents are less than zero, considered as very effective tools to distinguish regular and signifies nearby initial conditions all converge on one chaotic motions and provide a clear measure of chaos. If the another, and initial small errors decrease with time. However, divergence is exponential in time, with the constant factor, say if any of the Lyapunov exponents is positive, then , in the exponent, then is a LCE of the system and if > 0, infinitesimally nearby initial conditions diverge from one then the system becomes chaotic. The system is regular as another exponentially fast; means the errors in initial long as ≤ 0, (Ref. Grassberger and Procaccia (1983), Sandri conditions will grow with time. This condition, known as (1996), Martelli (1999), Nagashima and Baba (2005), Saha et sensitive dependence on initial conditions, is one of the few al (2006), Litak et al (2009)). A chaotic system display a set universally agreed-upon conditions defining chaos. called strange attractor which is composed of a complex Since the eigenvalues of a limit cycle characterize the rate pattern. It is a dense set within which all periodic motions are at which nearby trajectories converge or diverge from the unstable and has fractal structure i.e. having self-similar cycle, Lyapunov exponents can be considered generalizations property. Dimension of such a set is positive fractional, (not of the eigenvalues of steady-state and limit-cycle solutions to an integer), and provides a measure of chaos of the system. differential equations. Calculation of Lyapunov exponents Chaos measurement in a system can be given by the measure involves (for nonlinear systems) numerical integration of the of positivity of the LCEs for every orbit and also, by the underlying differential/difference equations of motion, measurement of correlation dimension. together with their associated equations of variation. The 194 ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 2, Issue 5, November 2012 equations of variation govern how the tangent bundle attached 1 n to a system trajectory evolves with time. The largest C(r ) H r - x x , lim i j eigenvalue of a complex dynamical system is an indicator of n n (n 1) i j chaos, Saha and Budhraja (2007). For, a one dimension map the separation at nth iteration can be given by (2.6) n 1 ' l x – y l ≈ f ( t ) x y . 0, x 0 n n 0 0 Where, H(x ) , t 0 1, x 0 (2.1) n n is the unit-step function, (Heaviside function). The where |x0 – y0| << 1 ,|xn – yn| << 1, and xn = f (x0 ), yn = f (y0 th ) are respectively the n iterations of orbits of x0 and y0 under summation indicates the the number of pairs of vectors closer f. Then, the exponential separation rate to r when 1 ≤ i, j ≤ n and i ≠ j. C(r) measures the density of pair log |f '(x)| of two nearby initial conditions, averaged over the entire trajectory, can be given by of distinct vectors xi and xj that are closer to r. n 1 1 ' λ( x ) = lim log f ( x ) , The correlation dimension D of O(x ) is defined as 0 n t c 1 n t 0 (2.2) log C(r ) D lim (2.7) n 1 c log r ' λ ( x )n r 0 Where f ( x ) e 0 , for n >> 1 . t t 0 To obtain Dc, log C(r) is plotted against log r and then we find a straight line fitted to this curve. The y- intercept of this The defined in (2.2) is the LCE of orbit of x . 0 straight line provides the value of the correlation dimension Quantitatively, two trajectories in phase space with initial Dc. separation δx0 diverge (provided that the divergence (can be treated within the linearized approximation) III. DYNAMICAL SYSTEMS EXPLORED λ t δx(t) e δx(0) (2.5) In this paper we consider the following six dynamical systems: the one dimensional epidemic model, one Where λ > 0 is the Lyapunov exponent. dimensional Salmon map, Gross National Product model, C. Correlation Dimension Predator-Prey population model, a neural network model, and the Ushiki map. The study of the above six models forms the Correlation dimension describes the measure of subject matter of sections A-F that follow. dimensionality of the chaotic attractor. It is a positive fractional (non-integer) number. The correlation dimension A. Discrete One Dimensional Epidemic Model was introduced in a work by Grassberger and Procaccia Biological models are of great importance in real life (1983) and letter followed by various researchers. Actually, science, May (1974). The following equation represents the LCE be a positive measure of sensitivity of the initial. A spread of epidemic in society, statistical measure provides us more authenticity in analysis 2 xn + 1 = k xn - 1 (3.1) of the behavior of the models discussed in the previous This model is used quite frequently to model the spreading of sections, Alseda and Costa (2008). measles. The bifurcation diagram for this model, when 0 ≤ k ≤ Being one of the characteristic invariants of nonlinear system 2.0 is represented by left figure of Fig.1.The figure showing dynamics, the correlation dimension actually gives a measure period doubling phenomena of Feigenbaum (1979) followed of complexity for the underlying attractor of the system. It is a by chaos. Also, a plot of Lyapunov exponents, for same range very practical and efficient method of counting dimension of parameter k, is shown by the right side figure of Fig.1. This then other methods, like box counting etc. The procedure to plot provides the range of values of k for the regular obtain correlation dimension follows the following steps, (periodic) motion as well as the chaotic motion. The Martelli (1999): indication of periodic windows appearing in bifurcation Consider an orbit O(x1) = {x1, x2, x3, x4, . ….}, of a map diagram, are nicely shown in the LCE diagram. Exponents n f: U → U, where U is an open bounded set in . To compute (LCEs) of epidemic model. We have also drawn the correlation dimension of O(x1), for a given positive real correlation integral curve, Fig. 2, by numerically calculating number r, we form the correlation integral, Grassberger and the data of correlation integral.