LYAPUNOV EXPONENT AND DIMENSIONS OF THE ATTRACTOR FOR TWO DIMENSIONAL NEURAL MODEL

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1 Volume 1, No. 7, July 2013 Journal of Global Research in Mathematical Archives RESEARCH PAPER Available online at LYAPUNOV EXPONENT AND DIMENSIONS OF THE ATTRACTOR FOR TWO DIMENSIONAL NEURAL MODEL T.K.Dutta 1, A. Kr. Jain 2, D.Bhattacharjee 3 1 Professor (Dr.) Tarini Kumar Dutta, Department of Mathematics, Gauhati University, Guwahati,Assam, Pin code , India. ( tkdutta2001@yahoo.co.in) 2 Anil Kumar Jain, Assistant Professor, Department of Mathematics, Barama College,Barama,Assam,Pincode ,India ( jainanil965@gmail.com) 3 Debasish Bhattacharjee, Assistant Professor, Department of Mathematics, B.Borooah College, Guwahati, Assam, Pin code , India. ( debabh2@gmail.com) Corresponding author: Anil Kumar Jain, Assistant Professor, Department of Mathematics, Barama College, Barama, Assam, Pin code , India. ( jainanil965@gmail.com) Abstract: In this paper a two dimensional non linear neural network model is considered and it is shown that chaotic attractor exists beyond accumulation point. To confirm the existence of chaotic attractor, Lyapunov exponent method is used. Further various fractal dimensions like Correlation dimension, Box-counting and Information dimension of the chaotic attractor were found to assess the geometry of the fractal set. Keywords: Lyapunov Exponent, Strange attractor, fractal dimension, Correlation dimension, Box-counting and Information dimension. 1. INTRODUCTION: In [3] we have seen that two dimensional discrete neural model follows the period doubling bifurcation route as the control parameter vary and reaches from order to chaos beyond some particular value(accumulation point) of the control parameter. It is also seen that with the help of Feigenbaum universal constant, the accumulation point is calculated and it is given as The Lyapunov exponents play a crucial role in the description of the behaviour of dynamical systems. They measure the average rate of divergence or convergence of orbits starting from nearby initial points. Therefore, they can be used to analyse the stability of limits sets and check sensitive dependence on initial conditions, i.e. the presence of chaotic attractors and we see that chaos begins beyond accumulation point [3,11] While the main purpose of Lyapunov exponents is to characterise the dynamical properties of trajectories on attractors, the fractal dimensions focus on the geometry of the chaotic attractor. Chaotic dynamical systems exhibit trajectories in their phase space that converges to a strange attractor. The fractal dimensions of this strange attractor counts the effective number of degree of freedom in the dynamical system and thus quantifies its complexity. [1, 5, 12]. JGRMA 2012, All Rights Reserved 50

2 Various fractal dimensions focus on the geometric structure of attractors in state space which occurs beyond accumulation point. If the system has atleast one positive Lyapunov exponent, we say that the system s behaviour is chaotic. If the attractor exhibits scaling with (in general) a noninteger dimension (scaling index), then we say the attractor is strange. In this paper we calculate maximum lyapunov exponent(mle) of the model and it is observed that chaos occurs beyond accumulation point where mle is positive.we also study various fractal dimensions like Correlation dimension, Box-counting and Information dimension of the chaotic attractor at the onset of chaos, i.e. at the accumulation point (i.e ) [3] which are found to be a non-integer dimension. 2. LYAPUNOV EXPONENTS [4, 11]: Lyapunov exponents provide a quantitative measure of the divergence or convergence of the nearby trajectories for a dynamical system. If we consider a small hypersphere of initial conditions in the phase space, for sufficiently short time scales, the effect will distort this set into a hyperellipsoid, stretched along some directions and contracted along others. The asymptotic rate of expansion of the largest axis, corresponding to the most unstable direction of the flow, is measured by the maximum Lyapunov exponent. [11] Fig 2.1: Describes the average expansions representing Lyapunov exponent[11]. Now if we take the Jacobian matrix of the two dimensional map and calculate it for the nth iterated point and the eigenvalues are calculated which in turn will give the eigenvectors. An eigen value having magnitude less than 1 indicates that the difference of closer points in the direction of corresponding eigen vector is converging i.e. decreasing and on the other hand an eigen value having magnitude greater than 1 indicate that the difference of closer points in the direction of corresponding eigen vector is diverging or increasing. Hence we can say that the eigen values evinces the nature of the magnetism of the nth iterative point. So the logarithm of the magnitude of the eigen values may be termed as the Lyapunov exponents. Let us consider a two dimensional difference equation, which is depending on a parameter a(say) with initial point (x 0,y 0 ).Let the evolved iterated points be (x 1,y 1 ),(x 2,y 2 ).(x n,y n ) at the parameter a=a 0. Suppose that the trajectory moves to the fixed point (x*,y*).since the fixed point is stable and doesn t move under iteration so after finite iterations the trajectory comes very close to the fixed point. Let the Jacobian matrix J * at the fixed point (x *, y * ) of the two dimensional map is JGRMA 2012, All Rights Reserved 51

3 Let the eigenvalue of the J * at the fixed point (x *,y * ) be.now for the nth iterated point (x n,y n ).The Jacobian matrix i.e. J n =J n-1.j n-2..j 0,as n is sufficiently large the Jacobian matrix will be of the form J=J*.J*.J* J n-1.j n-2..j 0, where J* is the Jacobian matrix at the fixed point (x*,y*). So for large iterative value, J* will govern and we may neglect the finite number of J n. Therefore for the large n, the eigen values of J are and then Same argument is valid for periodic points. Thus so long as fixed point or periodic point exists the eigen value will be less than 1 and hence Lyapunov exponent will be negative. At the bifurcation point one of the eigen values is -1 and hence maximum of the Lyapunov exponent will be 0. mle Fig2.2: Showing maximum Lyapunov exponent at the 1 st bifurcation point in y-axes and control parameter in x-axes. However if one of the eigenvalue decreases from positive value to negative and ultimately becomes -1 for the next bifurcation point and if the value is continuously decreasing then it must achieve 0 then the Lyapunov exponent will show negative infinity. However for the parameter value when the maximum Lyapunov exponent becomes positive chaos has started. 3. CALCULATION OF LYAPUNOV EXPONENT OF THE MAP: Here we take our two dimensional discrete model as F(x, y) = F ( ) where f(x, y) = JGRMA 2012, All Rights Reserved 52

4 and g(x, y) = where is the control parameter. The Lyapunov exponent for our two dimensional map is calculated by the eigen values of the limit of the following expression: Where J K is the jacobian given by Now to calculate the lyapunov exponent we first take an initial point and iterate it up to times so that we reach sufficiently nearer to the fixed points. Then we find (J 0.J 1...J n ), where n=100 say, and calculate the eigenvalues of that resultant matrix. Then is the Lyapunov exponent of our model. Fig3.1: Diagram of maximum Lyapunov exponent. Abscissa represents the control parameter while the ordinate represents maximum Lyapunov exponent. In the figure we plot the graph of the maximum lyapunov exponent against parameter values ( ) varies from 0.65 to 1.6. We have considered the initial point as (0.71, 0.71) for the estimate of lyapunov exponent with 100 iterations. Now what is the importance of the graph of the lyapunov exponent? The main importance of the graph is that one can easily identify chaotic region from the periodic one. We see, there are some points of the graph that hit the horizontal line and go to the negative side. These are perioddoubling bifurcation points where lyapunov exponents are almost zero. The points reside on negative side are periodic and the region which lies on the positive side of the parameter is chaotic. Interestingly, the area dominated by chaos occurs beyond the accumulation point Lyapunov exponent near bifurcation point and accumulation point: The following table shows maximum Lyapunov exponent at some parameter values. JGRMA 2012, All Rights Reserved 53

5 Table 2.2: Control parameter maximum Lyapunov exponent Fig 3.2 : Showing part of attractor of the model when initial values are taken as x=1.21 &y=1.21 JGRMA 2012, All Rights Reserved 54

6 Fig 3.3: Showing part of attractor of the model when initial values are taken as x=1.21 &y= GENERALISED CORRELATION DIMENSION [2, 4, 5, 6, 7, 8, 9, 12]: The term "chaotic" and "strange attractor" both has been used to describe the nonperiodic, random like motions in dynamical system. Whereas "chaotic" is meant to convey a loss of information or loss of predictability, the term "strange" is meant to describe the unfamiliar geometric structure on which the motion moves in phase space [9]. In this paper, we will describe a quantitative measure of the strangeness of the attractor. This measure is called the fractal dimension [9]. Since strange attractors are typically characterized by fractal dimensionality, for this purpose, we study different kind of dimensions for our strange attractor [4] Here, we try to determine the dimension of the attractor at the onset of chaos, i.e. at the accumulation point (i.e ) [3] of our two dimensional neural model and.so to study information and box counting dimension,we take the help of generalized dimension, i.e. Renyi s dimension which is defined as follows D q = N ( R) log Pi lim 1 i 1 R 0 q 1 log R q (4.1) where we divide the phase space to N ( R ) boxes of edge R. The probability to find a point in the ith box is denoted by P i and is defined as P i = N i / N The generalization, q, can be any real number; we usually use just an integer q. When we increase q, we give more weight to more populated boxes, and when we decrease q the less occupied boxes become dominant. If q = 0, it gives the box counting dimension For q = 1, by applying the L Hospital rule, equation (4.1) becomes JGRMA 2012, All Rights Reserved 55

7 dimension is where is referred to also as Shanon entropy of the system. The definition of generalised information Note that q = 2 gives the direct arithmetic average that defines the correlation dimension.for a finite set of points G q (N,R) can be approximate as Where H is the Heaviside function and means If, then and If, then. Also here we use the Euclidean norm i.e. if and then Moreover, and Thus, we can see that if we can find G q (N, R), we may get the value of G q (R) and hence we can obtain D q for a particular value of q.for our calculation purpose we calculate G q (30000,r) for different values of q and consider that as G q (r).however it will be justified due to the convergence nature of G q (N,R) as N becomes large. We consider the range of R as 10-7 to 10-1 and get the value of log G q ( R).Then we plot (log R, log G q ( R)) out of which scaling region is selected. In the scaling region we fit a straight line whose slope will give the dimension. 4.1 Box counting dimension [2, 4]: For q = 0 it gives the box counting dimension. We have done the calculation at the parameter with iterations. Calculated value of (log R, log G q (R)) in the scaling region are as follows: JGRMA 2012, All Rights Reserved 56

8 loggq R logr Fig.4.1a: log R vs. log G q (R) in the scaling region JGRMA 2012, All Rights Reserved 57

9 The slope of the above points when fitted with a straight line by least square method is with a mean deviation of The data is obtained from iterated points at the parameter : Information dimension [2, 4] If we take q very near to one i.e. q tends to 1 then that gives the information dimension. At q= at the parameter with iterations. Calculated value of (log R, log G q (R)) in the scaling region are as follows: JGRMA 2012, All Rights Reserved 58

10 loggq R logr Fig.4.2a: log R vs. log G q (R) in the scaling region The slope of the above points when fitted with a straight line by least square method is with a mean deviation of The data is obtained from iterated points at the parameter : Correlation dimension: [2, 4] G q (R) =G q (30000, R) is calculated. The parts of the plotted points (log R, log G q (R)) which follows equation (4.1) is taken. The slope of the fitted straight line in that scaling region is D q. JGRMA 2012, All Rights Reserved 59

11 Calculated value of (log R, log G q (R)) in the scaling region are as follows: JGRMA 2012, All Rights Reserved 60

12 loggq R T.K. Dutta et.al., Journal of Global Research in Mathematical Archives, July-2013, logr Fig4.3.a: log R vs. log G q (R) in the scaling region The slope of the above points when fitted with a straight line by least square method is with a mean deviation of The data is obtained from iterated points at the parameter Conclusion: In this paper we study the Lyapunov exponent of the map and in the figure 3.1, it is seen that at flip bifurcation points [3] maximum Lyapunov exponent are zero. It is also seen that negative Lyapunov exponent signatures stability whereas positive Lyapunov exponent indicates chaos. We also study box counting dimension ( ), Information dimension ( and Correlation dimension ( ) and we see that Correlation dimension D c = Information dimension D I = Box counting dimension D b = Clearly Moreover dimension of the attractors as shown in the figures 3.2 and 3.2 are interestingly found to be less than one. REFERENCES: [1]Crownover,R.M., Introduction to Fractals and Chaos, Jones and Bartlett,Sudbury,MA,1995 [2 ] Dutta, T.K, Jain.A.K and Bhattacharjee.D., Determination of various fractal dimensions in one dimensional models IJST,Vol-2,Issue-3,2013,ISSN JGRMA 2012, All Rights Reserved 61

13 [3]Dutta, T.K, Jain.A.K and Bhattacharjee.D., Period Doubling Scenario Exhibit on Artificial Neural Network Model JGRMA,Vol-1,Issue-6,2013,ISSN [4] Dutta, T.K, and Bhattacharjee.D, Bifurcation, Lyapunov Exponent and fractal Dimensions in a Non-linear Map [2010 AMS Classification: 37G15, 37G35, 37C45]. [5]Falconer, K., Fractal Geometry : Mathematical Foundations and Applications, Wiley, New York,2003. [6]Grassberger.P., Generalised Dimension of the Strange Attractors, Physics Letters,Vol-97A,No-6,1983. [7] Grassberger, P.,Procaccia,I., Characterization of strange Attractors, Physical Review Letters,Vol-50,No-5,1983. [8] Hilborn, R. C., Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers., Oxford University Press, 1994 [9] Moon.F.C., Chaos and fractal dynamics,1992 [10] Ott.E., Strange attractors and chaotic motions of dynamical systems, Rev.Mod.Phys.,Vol- 53,No.4,Part 1,Oct [11] Sandri M., Numerical Calculation of Lyapunov Exponent The Mathematica Journal,Miller freeman Publications,1996, Vol 6,Issue 3. [12] Theiler.J., Estimating Fractal Dimension, J.Opt.Soc.Am.A pp ,vol-7,no.6, June JGRMA 2012, All Rights Reserved 62