CHAPTER 2 FEIGENBAUM UNIVERSALITY IN 1-DIMENSIONAL NONLINEAR ALGEBRAIC MAPS

CHAPTER 2 FEIGENBAUM UNIVERSALITY IN 1-DIMENSIONAL NONLINEAR ALGEBRAIC MAPS The chief aim of this chapter is to discuss the dynamical behaviour of some 1-dimensional discrete maps. In this chapter, we have taken the help of appropriate numerical methods to find out the fixed and periodic points of different periods 2 0, 2 1, 2 2, 2 3 and to find the bifurcation points by using our knowledge of numerical analysis ( Newton-Raphson method, Bisection method etc.) and using C-programming and computer software Mathematica. This chapter is divided into two sections. In section I, we have assumed a 1-dimensional nonlinear algebraic map. Here we discuss a detailed analysis of period- doubling bifurcations of the model, we study some associated universalities, particularly the route from order to chaos, as developed by Mitchell J. Feigenbaum, an American Physicist. M. Feigenbaum discussed the universal behaviors of 1-dimensional unimodel map of the form = ( ) We consider a 1-dimensional nonlinear algebraic mathematical model ()= ( 1 ) where [0,1] and (1,3] is an adjustable positive parameter. Chaos and order ---------------------------------------------------------------------------------------------------------- This chapter is based on our published papers Period- Doubling Scenario In a One Dimensional Non Linear Map Int. J. Math. Sci. & Engg. Appls. No. VI,Vol 6, pp: 259-270, 2012. And Some Aspects Of Dynamical Behavior In a One Dimensional Nonlinear Chaotic Map Int. J. of Statistica and Mathematica, Issue 3, Vol 4, pp: 61-64, 2013. 21

have long been viewed as antagonistic in the science. One of the great surprises revealed through the studies of the cubic iterator = ( 1 ), =0,1,2,3,4 is the discovery that there is a very well defined period-doubling route to chaos. Also we quantify chaos of this map by analysing Lyapunov exponent. In section II, we have considered another 1-dimensional nonlinear rational model of the form ()= ( ) with µ as control parameter and a,b are taken as constants. May and Oster[103] stated about the model, which was taken from biological literature[98,103,104,141,142]. Stone and Hart(1999)[141] showed effect of immigration by adding a constant c on the model. Here we first provide the Feigenbaum tree of bifurcation points along with one of the periodic points, which leads to chaos. Secondly, we determine the accumulation point and draw the bifurcation graph of the model and verify that chaos occur beyond accumulation point. Thirdly the graphs of the time series analysis are confirmed in order to support our periodic orbits of period 2 0, 2 1,2 2,2 3..and lastly the graph of Lyapunov exponent confirms about the existence of the chaotic region[10,24,25,36,42,71]. 2.00 Introduction: Robert May in 1976, in his famous papers [103, 104] showed how simple mathematical models exhibit very complicated dynamics. He talked about different dynamical features of logistic map, since then it has become a role model for showing of the existence of chaos and related properties in 1-dimensional systems. In 1975 the elementary particle theorist, Mitchell J. Feigenbaum exposed, in 1-dimensional iterations with the logistic map = ( 1 ), as one of the key members of its class which follows a universal route to chaos by means of period doubling bifurcation [52, 53]. Period-doubling bifurcations, as a universal route to chaos, are one of the most exciting discoveries of the last few years in the field of nonlinear dynamical systems. This exciting discovery is that if a family f presents period doubling bifurcations then 22

there is an infinite sequence { } of bifurcation values such that lim "#$ % " "#& % "#$ =', where ' is a universal number which is known as Feigenbaum constant. In order to examine the dynamical behaviour of any system, it is necessary to determine its fixed points or periodic points of every order and to discuss their stability as the values of the controlling parameters are varied. It becomes a very tedious job and sometimes not possible to calculate analytically the fixed or periodic point of higher order due to the presence of some higher degree terms or non-algebraic terms in the equation of the dynamical systems. In such cases, we numerical techniques to determine them. require to take up some SECTION I In this section we consider the 1-dimensional nonlinear algebraic map as ()= ( 1 ) where is a control parameter. 2.01 The Range of the Control Parameter: Let ()= ( 1 ) ( ()= (1 )+ ( 2 ) 3 = 3 (1+ ) ( ()=0 =±, () (( ()= 6 (1+ ) Clearly (( ()<0 at x=+, () and (( ()>0 at x=, () So () is minimum at x=, () and this minimum value is µ, µ µ 23

Since population cannot be negative so we discard x=, () Also () is maximum at x=+, () and this maximum value is µ, µ µ Now if we consider =3 then we see that () is maximum at x=, and this maximum value of () is equal to 1. Hence the range may be taken as [ 0,1 ]. We consider the range of the parameter as (1, 3]. Hence the parameter must lie between 1 and upto 3 to have valid population. 2.02 Stability Analysis and Bifurcation Points: The equilibrium points or fixed points for the map are the solution of the equation ()=. So the fixed points for our concerned map is the solution of Solving this we get ( 1 ) = = 9% 9%%, =0, = 9% 9%%. Only last two points are of our interest. Now to examine the stability of these points, we used the stability criterion for a 1-dimensional map which says that the fixed point of the map () will be stable as long as ( () < <1 and unstable if ( () < >1. Bifurcation occurs when ( () < =1 which means that the qualitative behaviour of the fixed point changes from stable to unstable one. For our concerned map ( ()= 3 (1+ ). Now the derivative at =0 is ( () <= =>1, so =0 is an unstable fixed point. ( () < $?@ =3 2. The absolute value remains less than 1 for 1<<2 so?$?@ = 9% 9%% is a stable point for 1<<2 and when >2 this fixed point becomes unstable. Therefore is =2 is the first bifurcation point of our model. 24

3 2.5 2 f x x f x, 0.0 0.0 0.2 0.4 0.6 0.8 x Fig 2.02a: The function (,) plotted as a function of for various values of the parameter. The diagonal line is a plot of ()=. The following command of Mathematica gives the fixed points at the intersection of ()= and our model. [ % ]=( 1 ) /. 2.0; CDEF [G,[]},G,0,1},HIJKLMID G"x", "f(x)"}] 25

f x x 2 x 1 0.2 0.4 0.6 0.8 x Fig 2.02b: Graphs of()= (1 ) and ()= at the parameter =2. Their intersection points give the fixed points of. Now we see at =2, both the fixed points are unstable. So for further discussion on the long term behaviour of the map, we have to shift our interest to the higher order iterations of the map i.e. corresponding periodic points and their nature of stability. Next we consider the iterated map f 2 (x). The fixed points of f 2 are given by the solution of the equation ()= (1+4+6 +4 + P ) Q (3+9 +9 +3 P ) S +(3 +6 +3 P ) T (+ + + P ) + =0 This is 9th degree equation, so gives nine (9) roots. Out of these roots only five roots are non-negative real roots. From these, three roots(x=0, x=1, U= 9V%W ) are unstable, 9%V%W and other two roots are =, % 9& %P and =, 9& %P.Also we see X Y () X =9 2 Y <Z 9& %P 26

and X Y () X =9 2 Y <Z % 9& %P From the of above results it is clear that the nature of stability of the fixed points =, % 9& %P and =, 9& %P of the second iterator map () which are also the periodic points of period 2 of the map () are same. They are stable within the range 2<<2.230679774998. Now as soon as cross the value 2.230679774998, the above two periodic points become unstable. Thus the second bifurcation point for our map is =2.230679774998. f x 0.2 0.4 0.6 0.8 x.. Fig 2.02c: Graphs of () and ()= at the parameter =2.23608. Their intersection gives five fixed points of. 27

f x x Fig 2.02d: Graphs of () and ()= at the parameter =2.23608. Their intersection gives five fixed points of, where x varies from -1 to 1 Now we try to find out third bifurcation point. For this we solve the equation P ()=, which is an 81 th degree equation in x. To solve this equation analytically is cumbersome one. So we use Newton-Raphson method and bisection method respectively. We build up suitable numerical method and C-programming and obtain following bifurcation points of different period, one of the periodic point and Feigenbaum delta(experimental value). We see that the periodic point is =0.381279 and the corresponding bifurcation point is =2.2880317544828. 28

f x 0.2 0.4 0.6 0.8 x Fig 2.02e: Graphs of intersection gives the fixed points of P P () and ()= at the parameter =2.28803. Their 2.03 Numerical Algorithm to Find Periodic Points, Derivatives of Different Iterates of Our Map and the Bifurcation Points: To find a fixed point or a periodic point of our model, we apply Newton-Recurrence formula: = ^( ") [ `^( ")], where n=1, 2, 3, 4 The Newton formula [118] actually gives the zero (es) of a map, and to apply this numerical tools, one needs a number of recurrence formulae which are given below: Let the initial value of x be x 0. Then ( = )= = (1 = ) = = (Say) ( = )=( )= (1 ) = (Say) Proceeding in this manner, the following recurrence formula can be established: 29

= % (1 % ) %, n=1, 2, 3, 4. (ii)again the derivative of f k can be obtained as follows a Y Y a < b = 3 = (1 )+ Again by chain rule of differentiation we get c de& d c < b =c de d c e( b ) cde d c < b =( 3 (1 )+)( 3 = (1 )+) where x 1 =f(x 0 ). Proceeding in this way we can obtain c def c =( 3 d g% (1 )+).. ( 3 = (1 )+) < b We remind that the value of µ will be the bifurcation value for the map g when its derivative def d at a periodic point is equal to 1. Now a table of the bifurcation points, and one of the fixed points (periodic points) at the corresponding bifurcation point and experimental Feigenbaum delta value is given below[71]. Table: 2.03a (Numerical calculation of bifurcation point) Bifurcation Point One of the periodic points Feigenbaum delta (experimental value) =2.0000000000000 77350 =2.2360679774998 0.434016 ' = =2.2880317544828 0.381279 δ 1 =4.5429679808539 P =2.2992279396502 0.368961 δ 2 =4.6667294749943 T =2.3016289140371 0.366447 δ 3 =4.6631839258626 h =2.3021432715815 0.365937 δ 4 =4.6679102536718 S =2.3022534377421 0.365920 δ 5 =4.6689248409437 30

i =2.3022770322596 0.366556 δ 6 =4.6691423378334 Q =2.3022820854965 0.366359 δ 7 =4.6691888717903 = =2.302283167746 0.366550 δ 8 =4.6691913219671 =2.302283399530 0.366362 δ 9 =4.6691913573533 =2.302283449172 0.365875 δ 10 =4.6691914104992 We can see from Table 2.03a that Feigenbaum delta converges to 4.6692011 The following bifurcation diagram indicates the universal route to chaos for our model. x 1.5 2.0 2.5 3.0 Figure 2.03b: Bifurcation diagram of ()= (1 ) 31

2.04 Schwarzian Derivative: The Schwarzian derivative of a function f(x) which is defined in the interval (a,b) having higher order derivatives is given by jk(())= e () e () () le m e () This derivative was first formulated by H.A. Schwarzian and has been used in the theory of differential equation. It has important application in study of bifurcation of periodic orbits, the Schwarzian derivative is used to study the limiting behavior of dynamical systems [73]. If f has a negative Schwarzian derivative on [0, 1], then it turns out that there must be a number c in (0,1) such that ( (n)=0,that is, f has a critical point. We say that f is S-Unimodal if it s Schwarzian derivative is negative [ 64, 78, 132]. A surprise hidden in the above formula is that the Schwarzian is actually not a function.informally speaking Schwarzian derivative is curvature[118]. A sufficient condition for a function from interval to interval to behave chaotically is that its Schwarzian derivative is negative [65]. Here after calculating (), (), ((( () we see jk(())= 36 ( 1+ ) 6 ( 1+) ( 3 (1+)) <0 Eo LDD 2.05 Accumulation Point [71]: As our model follows a period-doubling bifurcation and so we let {µ n } be the sequence of the bifurcation points. Using Feigenbaum δ, if we know the first(µ 1 )and second (µ 2 )bifurcation points then we can expect third bifurcation point (µ 3 ) as &% $ q + (i) (Of course occurrence of first two period-doubling does not give assurance that a third will occur, but if it does occur, then it can be predicted by above equation) Similarly P r% & q So (i) and (ii) implies +. (ii) 32

P ( )s 1 ' + 1 ' t+ Continuing this procedure to obtain µ 5, µ 6 and so on,we get more terms in the sum involving powers of ( )and clearly this sum is familiar with G.P. series. We can sum the series to obtain the following result [71]. ( "#$% " ) + q%, q However, this expression is exact when the bifurcation ratio ' = "#$% " "#& % "#$ is equal for all value of n. In fact {δ n } converges as, that is consider the sequence x, y,. = "#$% " q% uvw value of bifurcation points. Obviously, =. uvw ' ='. So, we +, where are the experimental Using the experimental bifurcation points the sequence of accumulation pointsx, yis calculated for some values of n, as given below µ, 1 =2.3004056739463884416575570157921 µ, 2 =2.3021939026880344523949219097948 µ, 3 =2.3022793351198911317652724768936 µ, 4 =2.3022832728439160215610487135217 µ, 5 = 2.3022834539969671505349403737847 µ, 6 = 2.3022834623012102678688669335085 µ, 7 = 2.3022834626823495879498016069359 µ, 8 = 2.3022834626998914580913863921077 µ, 9 =2.3022834627150379883879328667859 µ, 10 =2.3022834627001456319457838803299 µ, 11 =2.3022834627013737680817166128263 µ, 12=2.3022834627003605521227333570557 µ,13 =2.3022834627005709695403502825713 The above sequence converges to the value 2.302283462700 which is the required accumulation point. [36, 42] 33

2.06 Calculation of Lyapunov Exponent: A quantitative measure of the sensitive dependence to initial conditions is given by the Lyapunov exponents, which measures the exponential separation of nearby orbits. In simple terms, a positive Lyapunov exponent can be considered to be an indicator of chaos, whereas negative exponents are associated with regular behavior (periodic orbits)[29]. There is some standard procedure for obtaining the Lyapunov exponent and the procedure is as follows.we begin by considering an attractor point x 0 and calculate the Lyapunov exponent, which is the average of the sum of logarithm of the derivative of the function at the iteration points. With the help of a computer program we have followed the procedure to get the Lyapunov exponent. The formula may be summarized as follows[10,36,42]: Lyapunov exponent (µ) =lim z (log f/ (x 0 ) + log f / (x 1 ) + log f / (x 2 ) + log f / (x 3 ) + + log f / (x n ) ). We see several points (the 1 st at =2.0, 2 nd at =2.236067977, 3 rd at =2.2880317... ) where Lyapunov exponent hits the horizontal line and then becomes negative. These are the period doubling bifurcation points. Lyapunov exponent at these points are zero (which is almost clear from figure). Interestingly the first chaotic region appears after the value 2.302283....Moreover, in the chaotic region, we see some portions of the graph below the horizontal line.they signify that within the chaotic region also, at certain values of the parameter, there are regular behaviours and after that again chaotic region starts. These are called the periodic windows in the chaotic region. 34

Lyapunov Exponent 2.0 2.2 2.4 2.6 2.8 3.0 mu 1.5 2.0 Fig 2.06a: Graph of Lyapunov exponent for parameter from 1.75 to 3.0 SECTION II In this section we have discussed a 1- dimensional nonlinear rational map of the form ()= ( ) where a, b are constants and µ is a control parameter. 2.07 Analytical Study to Determine First Two Bifurcation Points: Here the model to be discussed is ()= ( ). Solving ()=E, we get = $ ({%) $. At this point we have ()<0, so maximum value for f(x) is ({%)$? } for µ>0. We may take the range as[0, ({%)$? $ ecological models although the main interest is mathematical. } ][42] so as to keep it meaningful to The solution of ()= gives the fixed points of f(x). A fixed point x is said to be a (i) stable fixed point or attractor if ~ ()~<1 (ii) unstable fixed point or repeller if ~ ()~>1. Solving ()=, we get the fixed points as =0 and 35

$ = (%) Also at x = (%) $. Now at x=0,~ ()~=, so x=0 becomes an unstable point for >1., we get ()= (%{){ Now, ( ()= 1 implies (%{){ = 1 or = { {% and ()= µ & ƒ µ ( ) (( $#(ƒ ) ) ) Solving ()=, we get µ & ƒ µ ( ) (( $#(ƒ ) ) )=1 (*) Also de& d = 1 M (L) { (L) { } } axµ 1 Ms 1+(ax) }t axµ s 1+(ax) }t 1 => = 1 ( ) } (1+(L) { ) axµ ( 1+ s 1+(ax) }t ) Let l µ ( ) m} =p and let (L) { =F ( )=> & (ˆ)( ) =1 => =(1+F)(1+Š) ( )=> & ( {ˆ%ˆ%)( { % %) (ˆ) & ( ) & = 1 =>( MF F 1)( MŠ Š 1)+ =0 =>( MF F 1) M( 1 ) (1+F) (1+F) + =0 =>( MF F 1)(M M MF )+ (1+F)=0 which is quadratic in t, solving for t we have and F= 2 +M(M 2)( 1)+9M P 2(M 2)M +(2+(M 2)M) P 2M(M 1) F= 2 +M(M 2)( 1) 9M P 2(M 2)M +(2+(M 2)M) P 2M(M 1) 36

Thus for the particular value of, where the second bifurcation occurs the periodic points are of the form = 1 L {G 2 +M(M 2)( 1)+9M P 2(M 2)M +(2+(M 2)M) P 2M(M 1) } { Also when M =>F 1 Again from (*) we have µ & ( ) (( ƒ µ $#(ƒ ) ) )=1 => µ& ( Œ )( ( ($# ) ) =1 => µ & µ & ( (@&?$)µ µ & =1 =>1+( 1) %{ =1 => 1 = 1, this is an identity i.e. true for all for higher values of b. Analytical study of the map after this stage is cumbersome. So we consider a particular value of the parameters to study dynamical behaviour of the map. Let us assume a=,b =7, then at x = (%) $, ()= %hs Now for 1< µ <1.4, the absolute value of () remains less than 1 and the point is stable. As soon as µ>1.4 the point becomes unstable. Hence µ=1.4 is the 1 st bifurcation point of this model.. f x 4 3 2 1 1 2 3 4 Fig2.07a: Inersection of the model and f(x)=x. x 37

We next consider the periodic points of period-two and higher.the period-2 points are found by solving the equation ()=,where ()= & x( ) ygs Ž ` @ $#(Ž `) t }. Now to solve this equation analytically is cumbersome one. So we use Newton-Raphson method and bisection method respectively. We build up suitable numerical method and obtain following bifurcation points of different period,one of the periodic point and Feigenbaum delta(experimental value). f x 4 f x 4 3 3 2 2 1 1 x x 1 2 3 4 1 2 3 4 Fig2.07b: Graphs of () and ()= Fig2.07c: Graphs of P () and ()= x 2.2 2.0 1.8 1.6 1.4 1.2 1 st bifurcation point at 1.4 1.1 1.2 1.3 1.4 1.5 Fig 2.07d: The period doubling bifurcation at =1.4 38

For numerical procedure first of all I(a,x i *,n) representing the nth derivative of the function f at each of the periodic points is calculated numerically and v< I(a,x,n) is calculated. Now the interval [a 0,a 1 ] is choosen in such a way that ( v< I(a =,x,n) + v< 1)( I(a,y,n)+1)<0, where x i * are the periodic points at the parameter a 0 and y i * are the periodic points at the parameter a 1 for i=1,2,3,n. Then ( v<i(a,z,n)+ 1) is calculated where z i, i=1,2, n are the periodic points at the parameter a= ( b $ ) and the bisection method process is repeated till the bifurcation point up to certain accuracy is achieved. above process. The following are some of the bifurcation points obtained with the help of the Table2.07e:( Calculation of bifucation point ) Bifurcation Point One of the periodic points =1.400000000000000000 1.754613 =1.575706408457226540 1.445574 Feigenbaum (experimental value) ' = delta =1.630043290746602260 1.299966 δ 1 =3.233649032187392615 P =1.642490397413339580 1.641840 δ 2 =4.365422716833262182 T =1.645204772722345780 1.653288 δ 3 =4.585624782797900526 h =1.645788131573193260 1.655620 δ 4 =4.653011272671814281 S =1.645913168853051720 1.656095 δ 5 =4.665479407076307656 i =1.6459399524105507 1.656115 δ 6 =4.668434322956605111 Q =1.645945688834907110 1.656113 δ 7 =4.669033501514143606 = =1.64594691741068755 1.656118 δ 8 =4.669166076449010548 =1.64594718053443330 1.656120 δ 9 =4.669193869117056542 =1.645947236887495 1.652130 δ 10 =4.66915435884768188 =1.64594724895660249 1.655819 δ 11 = 4.66919885165756945 P =1.64594725154143973 1.652396 δ 12 =4.669194293254611296 39

From the above table we can establish the Feigenbaum ' up to 4.6692011 Now the following bifurcation diagram indicates the universal route to chaos for our model [42]. attractar 4 3 2 1.5 2.0 2.5 3.0 mu Fig2.07f:Bifurcation graph of the model.the abcissa represents the control parameter and ordinate represents the iterated points. 2.08 Accumulation Point: Using the experimental bifurcation points the sequence of accumulation pointsx, y is calculated with the help of the following formula[42].. = ' 1 + Table2.08a: Accumulation points for different values of n µ, 1 =1.6235932315645944017782093800194 µ, 2 =1.6458827180086068675824202456858 µ, 3 = 1.6458827175932474774970325246657 µ, 4 =1.6459445454569357623441342339104 µ, 5 = 1.6459471193308955094055767204584 µ, 6 = 1.6459472463699447144735582495296 µ, 7 = 1.6459472519706112002181189171387 40

µ, 8 = 1.6459472522334241941060516283839 µ, 9 = 1.6459472522453221547696101486903 µ, 10 = 1.6459472522458772854595187683096 µ, 11 = 1.6459472522458940791899784228632 µ, 12 = 1.6459472522459031531614616585986 µ, 13 = 1.6459472522459083077829755483082 The above sequence converges to the value 1.64594725224590 which is the required accumulation point[36]. 2.09 Time Series Analysis[42, 130]: The key theoretical tool used for quantifying chaotic behavior is the notion of a time-series of data for the system. By observing data over a period of time, one can easily understand what changes have taken place in the past. Such an analysis is extremely helpful in predicting the future dynamical behaviour[42,130]. We open our journey with a couple of very simple time series experiments. On the horizontal axis, the number of iterations ( time) is marked, while on the vertical axis the amplitudes (ranging from 1 to 3 ) are given for each iteration. Figure 2.09a shows the computed time series of x- values starting at x = 1.4 with the parameter value at µ = 1.35 (which is slightly smaller than ). The points are connected by line segments. Time series graph is non-sensitive, stable behaviour and leads to the same final state of a single fixed point. 3.0 Value of parameter 1.35, Showing Period 1 behavior 2.5 Iteratedvalues 2.0 1.5 0 10 20 30 40 Number of iterations Fig 2.09a: Time series graph for period 1 41

Now, let us look at the second time series in fig 2.09b, which is based on the same formula and the same initial value of x with the parameter value µ = 1.5(which is slightly greater than ) We notice periodicity and oscillate between two fixed points with the same amplitude, and the cycle repeats. 3.0 Value of parameter 1.5,Showing Period 2 behavior 2.5 Iterated values 2.0 1.5 0 10 20 30 40 Number of iterations Fig 2.09b: Time series graph for the period 2 The third time series in fig 2.09c, which is based on the same formula and the same initial value of x with the parameter value µ = 1.617. We notice periodicity and oscillate between four fixed points with the same amplitude, and the cycle repeats. 3.0 Value of parameter 1.617, Showing Period 4 behavior 2.5 Iterated values 2.0 1.5 0 10 20 30 40 50 Number of iterations Fig 2.09c: Time Series graph for the period 4. 42

The fourth time series in fig 2.09d, which is based on the same formula and the same initial value of x with the parameter value µ = 1.64. We notice periodicity and oscillate between eight fixed points with the same amplitude, and the cycle repeats. 3.0 Value of parameter 1.64, Showing Period 8 behavior 2.5 Iterated values 2.0 1.5 0 10 20 30 40 50 Number of iterations Fig 2.09d: Time series graph for the period 8 behaviour. But, if we start with the same initial value of x and the parameter value µ = 1.67, the picture shows an irregular pattern which is difficult to predict meaning thereby the appearance of the chaotic region, Fig 2.09e. Thus, the time series analysis also helps us for full description of bifurcations and chaos for the concerned model. 2.2 Value of parameter 1.67, Showing Period Chaotic behavior 2.0 Iteratedvalues 1.8 1.6 1.4 1.2 0 10 20 30 40 50 60 Number of iterations Fig 2.09e: Time series graph for the Chaotic behaviour 43

2.10 Lyapunov Exponent: In order to verify how much accurate is the accumulation point, the Lyapunov exponent is calculated. Lyapunov exponent at the parameter greater than the accumulation point is found to be positive whereas Lyapunov exponent less than the accumulation point is negative and at the accumulation point it should be equal to zero. We begin by considering an attractor point x 0 and calculate the Lyapunov exponent, which is the average of the sum of logarithm of the derivative of the function at the iteration points. Now we draw the graph of Lyapunov exponent as follows Lyapunov exponent 1.4 1.5 1.6 1.7 mu 1.5 Fig 2.10a: Lyapunov exponent of the map.negative values indicate periodic.almost zero values indicate bifurcation points and positive values indicate chaos. From graph of Lyapunov exponent,we see that some portion lie in the negative side of the parameter axis indicating regular behavior (periodic orbits) and the portion lie on the positive side of the parameter axis confirm us about the existence of chaos for our model [29]. 44

2.11 Results and discussion: In section I, we have seen that simple iterative unimodel mathematical model exhibit period-doubling route to chaos. Also we have seen that our model successfully converges to Feigenbaum universal constant (delta). From the bifurcation graph we can see that the chaotic region occurs beyond accumulation point 2.302283462700 as desired. We see from the graph how Lyapunov exponent changes its sign from negative to positive as the control parameter vary. This negative value indicates about regular behaviour of the periodic points and positive value gives the signal of chaos. In section II, All the features of section I, are also seen in this section. Moreover periodic orbits of period 2 0, 2 1, 2 2, 2 3.. are confirmed by drawing the graphs of the time series analysis. 45