Chaotic Dynamical System Report

LaGuardiaCommunity College

Roberto Lam

SCP232.1804

Instructor: Dr. Frank Wang

Introduction

The following experiment is about the behavior of some orbits used in a logistic and quadratic function where the use of graphs for each function shows us their behavior. This experiment consists in three parts. The first part is to perform a graphical analysis on each of the graph for each of the seed chosen. The second part is to determine the path of the orbit for each seed; that is, whether the orbit is fixed, periodic, eventually periodic, or has no visible pattern. And the final part is to observed the dynamical behavior of a quadratic function Q(x)=x2+c for a large number of c-values in the interval -2 c 2.5.

Part 1[1]

We introduce a geometric procedure that enables us to use graph of a function to determine the behavior of orbits.

Suppose we have the graph of a function F and wish to display the orbit of a given point X0. We begin by superimposing the diagonal line y=x on the graph of F. To find the orbit of X0, we first draw a vertical line x=X0 to the graph of F. When this vertical line meets the graph, we have reached the point (X0, F(X0)). When then draw a horizontal line from this point to the diagonal. We reach the diagonal at the point whose coordinate y-coordinate is F(X0), and the x-coordinate is also F(X0), which is the next point on the orbit of X0.

Now we continue this procedure. Draw a vertical line from (F(X0), F(X0)) on the diagonal to the graph, which yields the point (F(X0),F2(X0)). Then a horizontal line to the diagonal , from the diagonal it reaches the diagonal at (F2(X0), F2(X0)), directly above the next point in the orbit.

In brief, this iterative procedure is referred to as going “vertically to the curve and horizontally to the line” on the graphs for the following functionsF(x)=x(1-x) and Q(x)=x2+c :

Logistic function with Xo Є (0,1):

a)F(x)=0.8x(1-x)

b)F(x)=2.5x(1-x)

c)F(x)=3.1x(1-x)

d)F(x)=3.8x(1-x)

Quadratic function with Xo= 0 and c Є (-2,0.25):

a)Q(x)=x2-0.4

b)Q(x)=x2-1.3

c)Q(x)= x2-1.38

d)Q(x)= x2-1.44

Observations

Using a pencil and ruler and following the “vertically to the curve and horizontally to the line” instruction on the graphs we get the following results below which can also be seen using Maple:

Logistic function with Xo Є (0,1):

-For the function F(x)=0.8x(1-x), that every seed for lambda= 0.8 the orbit converges to the point (0,0).

-For the function F(x)=2.5x(1-x), every seed for lambda= 2.5 the orbit converges to the point (0.6,0.6).

-For the function F(x)=3.1x(1-x),every seed for lambda= 3.1 the orbit does not converges to certain point but has the following behavior:

-For the function F(x)=3.8x(1-x), every seed for lambda= 3.8 the orbit also does not converges to certain point but has the following behavior( lets say a chaotic behavior):

Quadratic function with Xo= 0 and c Є (-2,0.25):

-For the function Q(x)=x2-0.4 for c= -0.4 the orbit converges approximately to the point (-0.3,-0.3).

-For the function Q(x)=x2-1.3for c= -1.3 the orbit converges does not converges to a point but have a periodic pattern as follow:

-For the function Q(x)= x2-1.38for c= -1.38 the orbit converges does not converges to a point but have also a periodic pattern as follow:

-For the function Q(x)= x2-1.44for c= -1.44 the orbit converges does not converges to a point but have a periodic pattern as follow:

Part 2

In the preceding part we saw the behavior of the orbits for different lambdas () and different c-values for different seeds(x0) using graphical analysis. In this part we are going to determine whether the orbit is fixed, periodic, eventually periodic, or has no visible pattern.

In addition, by using some programs as MS Excel and Maple to calculate and study the behavior for many lambdas and c-values for functions F(x)=x(1-x) and Q(x)=x2 +C we have the following:

Observations

2.1) F(x)=x(1-x) for [1,4] on the interval (0,1)

-For 1<<~2.4 the orbit for any seed Xo Є (0,1) the orbit converges to a fixed point; for instance:

For =2.3 and X0=0.9 we have the following:

-For 2.5<<~3.0 the orbit for any seed Xo Є (0,1) the orbit converges to a fixed point after a periodic pattern; for instance:

For =2.9 and X0=0.9 we have the following:

-For 3.1<~3.4 the orbit for any seed Xo Є (0,1) the orbit is eventually periodic pattern with period 2 ; for instance:

For =3.3 and X0=0.9 we have the following:

-For =3.5 the orbit for any seed Xo Є (0,1) the orbit is eventually periodic pattern with period 4 ; for instance:

For =3.5 and X0=0.9 we have the following:

-For >3.5 the orbit for any seed Xo Є (0,1) the behavior of the orbit eventually becomes sort of chaotic for instance:

For =3.8 and X0=0.9 we have the following:

2.2) Function Q(x)=x2 +C with C values in the interval [-2,0.25]

-For C=-2, the behavior of the orbit eventually converges to a fixed point:

For C=-2, we have the following:

-For -2 < C <- 1.5 , the behavior of the orbit is sort of chaotic, for instance:

For C=-1.5 , we have the following:

-For -1.5 < C <- 1.0 , the behavior of the orbit is eventually periodic from period-16, period-8 and so on through period-2 orbit, for instance:

For C=-1.4 , we have the following:

For C=-1.0 , we have the following:

-For -1.0 < C < 0, the behavior of the orbit is eventually periodic and later converges to a fixed point, for instance:

For C=-0.7 , we have the following:

-For 0 < C < 0.25, the orbit converges to a fixed point:

For C=0.15 , we have the following:

Part 3

In this part we are going to observe the dynamical behavior of Q(x)=x2 +C for a large number of C-values in the interval -2 C 0.25 from the previous part, but this time using Maple Commands given by the professor to generate the following graph which unifies all the ideas from the previous part 2.

The diagram reveals an infinite number of period doubling bifurcations. As C decrease a period-2 orbit becomes a period-4 orbit, and this in turn becomes a period-8, and so on. This sequence of period doubling bifurcation is known as “period doubling ad infinitum”[2]

Conclusions

For the logistic function F(x)=x(1-x),regardless of the value of x0  (0,1) we conclude that:

-For 1<<~2.4the orbit converges to a fixed point

-For 2.5<<~3.0 the orbit converges to a fixed point after a periodic pattern.

-For 3.1<<~3.4 the orbit is eventually periodic pattern with period 2

-For =3.5 the orbit is eventually periodic pattern with period

-For >3.5 the behavior of the orbit eventually becomes sort of chaotic for instance:

For the quadratic function Q(x)=x2 +C with C values in the interval [-2,0.25] we conclude that:

-For C=-2, the behavior of the orbit eventually converges to a fixed point:

-For -2 < C <- 1.5 , the behavior of the orbit is sort of chaotic

-For -1.5 < C <- 1.0 , the behavior of the orbit is eventually periodic from period-16, period-8 and so on through period-2 orbit.

-For -1.0 < C < 0, the behavior of the orbit is eventually periodic and later converges to a fixed point.

-For 0 < C < 0.25, the orbit converges to a fixed point.

In addition, the graph generated in part 3 can give us a summary for the behavior of the orbits for the function Q(x)=x2 +C with C values in the interval [-2,0.25] where “as C decrease a period-2 orbit becomes a period-4 orbit, and this in turn becomes a period-8, and so on”[3]. Besides, this last graph, the previous ones from part 2 can give us an idea of the behavior of the orbits as well.

Work Cited

-Dr.Wang, Frank, “Chaos Experiments Using Maple” Department of Mathematics, LaGuardiaCommunity College, The CityUniversity of New York

[1] Data from “Chaos Experiments Using Maple” by Dr. Frank Wang Department of Mathematics, LaGuardia Community College, The City University of New York

[2]Chapter 3 “Chaos Experiments Using Maple” by Dr. Frank Wang Department of Mathematics, LaGuardia Community College, The City University of New York

[3]Chapter 3 “Chaos Experiments Using Maple” by Dr. Frank Wang Department of Mathematics, LaGuardia Community College, The City University of New York