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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]

Research Article


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.



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.

              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 (ii) unstable fixed point or repeller if1.Solving we get the fixed points as.Now at x=0 ,, so x=0 becomes an unstable point for   . Also at x =  , we get 1

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 1st 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,xi*,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 xi* are the periodic points at the parameter a0 and yi* 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.



Table 2.4:  Calculation  of  bifucation point.

Bifurcation Point

One of the periodic points

Feigenbaum delta(experimental value)


























δ7= 4.6690335015141436067738502435894



δ8 = 4.6691660764490105476662516374799



δ9 =4.6691938691170565445429865360879





δ11= 4.6691988516575694472925677338974





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 .[6]

Fig2.4 (a)Bifurcation graph of the model.The abcissa represents the control parameter and ordinate represents the iterated points.


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/(xn)|)

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.



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Corresponding author:

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

Email: [email protected]



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