LAB 1: DISCRETE MAPS AND NUMERICAL CHAOS

** **

__Mathematics: __

The
discrete logistic equation takes the form: ** x_{k+1} = r x_{k}
(1 – x_{k}), **where

Fixed
points ** x_{*} **are defined by the limit:

For
** 0 < r < 1, **the only fixed point is

For
** 1 < r < 4, **there are two fixed points:

* *

**Numerical
chaos** in
mathematical sense was introduced to describe the behavior of deterministic
systems (such as discrete mappings) that generate a random sequence of numbers.
For a chaotic sequence _{}*x _{k }*

__Objectives:__

·
visualize
various sequences of numbers _{}*x _{k }*

·
construct
a bifurcation diagram displaying various regimes of the discrete map for
different values of *r*

·
understand
the difference between fixed points, periodic sequences, and chaotic sequences

·
understand
the sensitivity of the chaotic sequences to the starting value *x _{1}*

* *

__ __

A
typical cobwebbing pattern is shown here:

- Initialize a value of
parameter
*r* - Plot the line:
*x*_{k+1}= x_{k} - Plot the curve:
at the same graph*x*_{k+1 }= r*x_{k}*(1-x_{k}) - Initialize a value for
*x*_{1} - Initialize a number of
elements in a sequence
_{}*x*_{k }_{} - Compute numbers
in a sequence*x*_{k }_{}*x*_{k }_{}and save them in a vector. - Plot horizontal and
vertical segments of the cowebbing pattern at the same graph.

*Exploiting
the MATLAB script:*

- Run the script with
and*r = 0.5; 1.5; 2.5; 3.25; 3.5; 3.75**x*_{1}= 0.25; 0.5; 0.75 - Identify fixed-point,
periodic, and chaotic regimes in (1) as the large-
behaviour of the sequence.*k* - Check that the
fixed-point and periodic regimes do not depend on the starting value
*x*_{1} - Run the script with
and*r = 3.83*. How many periods does the sequence have?*x*_{1}=0.5

__ __

The
discrete logistic equation exhibits the bifurcation diagram called **the
period-doubling route to chaos**. The fixed points become unstable and are
replaced by the 2-period, 4-period and other multi-period sequences, with
larger values of the bifurcation parameter ** r.** For yet larger
values of

- Initialize the minimum
and maximum values for an interval of the parameter
.*r* - Initialize the number
of points in the interval of
*r* - Loop through the values
of
*r* - Initialize the starting
value
and the number of values*x*_{1}in a sequence*x*_{k} - Compute the values
in a sequence to settle at a stationary regime*x*_{k} - Continue the sequence
in a stationary regime, save the values*x*_{k}in a vector*x*_{k} - Plot the values of
versus the same value of*x*_{k}on a graph.*r*

*Exploiting
the MATLAB script:*

- Run the script for
.*0 < r < 4* - Zoom the details of the
chaotic regime by running the script for:
and for even narrow intervals.*3 < r < 4; 3.5 < r < 4;* - Find narrow intervals
of
where there exist 8-periodic and16-period sequences.*r,* - Find stability islands
in the sea of numerical chaos. Check that the sequence may have 3-periods,
5-periods in the stability islands.

__ __

Suppose
two sequences _{}*x _{k }*

- Initialize a value of
the bifurcation parameter r
- Initialize and compute
two sequences
_{}*x*_{k }_{}and_{}*y*_{k }_{} - Compute the error
between the two sequences*e*_{k} - Plot the error
as function of its index*e*_{k}*k*

*Exploiting
the MATLAB script:*

Identify whether the stationary regime is
deterministic or chaotic for ** r = 3.4; 3.6; 3.81; 3.83; 3.85**.