For different value of the paramenter a, we plot a set
of converged values of x. that means, we plot the function x=f(a) with
b=0.3 and x(0)=y(0)=0.
This map receive
a real number between 0 and 1.4, then returns a real number in [-1.5, 1.5]
again. The various sequencies are yielded
depending on the parameter a and the initial values x(0),
y(0). We can see that if the parameter a is taken between 0 and about 0.32,
the sequence x(n) converge to a fixed point xe independent on the initial
value x(0) and y(0). But what happens to the sequence x(n) when parameter
exceeds 0.32? As you
see with the help of the previously graph , the sequence
converge to a periodique orbit of period 2. Such situation happens when
the parameter a
is taken between about 0.32 and 0.9 . If you make
the parameter a larger, the period of the periodic orbit will be doubled,
that is 4,8,16 ...
This is called period
doubling cascade, and beyond this cascade, the stable periodic orbit dissapears
and we will see Chaos if parameter a is bigger than 1.42720. As you see
above, the transition of the orbit structure accordant with the change
of parameter is called bifurcation phenomena. At least, the following
graph shows a zoom on the first lower branch of the bifurcation diagram
:
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