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Some Aspects of Dynamical Behavior in a One Dimensional Non Linear Chaotic Map

T. K. Dutta¹, A. Kr. Jain², D. Bhattacharjee³

¹Professor, Department of Mathematics, Gauhati University, Guwahati ,Assam, Pin code- 781014, INDIA.

²Assistant Professor, Department of Mathematics, Barama College,Barama, Assam, Pincode-781346,INDIA.

³Assistant Professor, Department of Mathematics, B. Borooah College, Guwahati, Assam, Pin code-781007, INDIA.

Corresponding Addresses:

¹[email protected], ²[email protected], ³[email protected]

Abstract: In this paper we consider  an inverse  non linear algebraic ecological model with μ as control parameter and a,b are taken as constants. Here we create a suitable C-programming and use Mathematica software to study period doubling route to chaos. Also to find out an universal route from order to chaos through period doubling bifurcations we set up   proper numerical methods to get periodic points and bifurcation points of different period 2ⁿ, where n=0,1,2,3,4,5…..and we successively achieve Feigenbaum Universal  Constant(δ)=4.66920161029…  with the help of bifurcation points calculated numerically. It is also seen that chaotic region takes place beyond accumulation point. We also confirm about chaotic region by getting positive Lyapunov Exponent at some parametric values.

Keywords: Period –Doubling Bifurcation, Periodic points, Feigenbaum Universal Constant, Time Series, Lyapunov Exponent.

1.Introduction

First-order difference equations, although simple and deterministic, can possess an extraordinarily rich spectrum of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations[9].May and Oster[10] stated about the model in their paper in table 1,which is taken from biological literature[8,9,10,13,14] .Stone and Hart(1999)[12] showed  effect of immigration by adding a constant “c” on the model.In this paper we pay our attention to study period doubling route to chaos of an inverse non linear algebraic one dimensional model. We consider our model as   where a, b are constants and μ is the control parameter. 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 2⁰ ,2¹,2²,2³…..and lastly the graph of Lyapunov exponent confirms   about the existence of the chaotic region.[ 4,5,6]

2. Our vital study

Here the model to be discussed is  .Solving, we get   .At this point we have  ,so maximum value for f(x) is     for μ>0 .We may take the range as[0,   ][ 6]  so as to keep it meaningful to ecological models although the main interest is mathematical.

Also if we consider a= 0.5 ,b =7, then  at x =  , .

 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.

Fig 2.1 Inersection of the model and f(x)=x.Here the abcissa represents x and ordinate represents f.

We next consider the periodic points of period-two and higher.The period-2 points are found by solving the equation   ,where  .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).

Fig2.2 Graphs of    and

Fig2.3 Graphs of    and

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  is calculated. Now the interval [a₀,a₁] is choosen in such a way that   , where x_i* are the periodic points at the parameter a₀ and y_i* are the periodic points at the parameter a₁ for i=1,2,3…,n. Then   is calculated where are the periodic points at the parameter a= and the bisection method process is repeated till the bifurcation point up to certain accuracy is achieved.

The following are some of the bifurcation points obtained with the above process.

3. Accumulation point

Using the experimental bifurcation points the sequence of accumulation pointsis calculated with the help of the following formula.[6]

Table3.1: 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

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. [6]

4. Time Series Analysis: [4,6,15] 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.[5,15]

       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 4.1 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.

                              No of iterations  

Fig 4.1: Time series graph for period 1

  Now, let us look at the second time series in fig 4.2, 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 .

                            No of iterations  

Fig 4.2: Time series graph for the period 2

The third time series in fig 4.3, which is based on the same formula and the same initial value of x with the parameter value   μ = 1.637. We notice periodicity and oscillate between four fixed points with the same amplitude, and the cycle repeats.

                    No of iterations  

Fig 4.3:  Time Series graph for the period 4.

The fourth time series in fig 4.4, which is based on the same formula and the same initial value of x with the parameter value   μ = 1.637. We notice periodicity and oscillate between four fixed points with the same amplitude, and the cycle repeats.

                    No of iterations  

Fig 4.4:  Time series graph for the period 8 behaviour

But, if we start with the same initial value of x and the parameter value μ = 1.644, the picture shows an irregular pattern which is difficult to predict meaning thereby the appearance of the chaotic region, Fig 4.5. Thus, the time series analysis also helps us for full description of bifurcations and chaos for the concerned model.

                    No of iterations  

Fig 4.5:  Time series graph for the Chaotic behaviour

5 .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 x₀  and calculate the Lyapunov exponent, which is the average of the sum of logarithm of the derivative of the function at the iteration points.

The formula may be summarized as follows:[2,7]

Lyapunov exponent (μ) = (log|f^/(x₀)|+ log|f^/(x₁)|+ log|f^/(x₂)|+ log|f^/(x₃)|+……+ log|f^/(x_n)|)

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 .[2]

Fig:5.1  Lyapunov exponent of the map.Negative values indicate periodic.Almost zero values indicate bifurcation points and positive values indicate chaos.

References:

Corresponding author:

Anil Kumar Jain, Assistant Professor, Department of Mathematics, Barama College, Barama, Assam, Pin code-781346, INDIA.

Email: [email protected]