# The Period Doubling Route to Chaos

This is an applet of the period doubling route to chaos,
demonstrating the ideas of iterating one dimensional maps, the
universal scaling of this route to chaos, and the renormalization
group theory.

and while the applet is loading..........
## One Dimensional Maps

Perhaps the simplest "dynamical system" is the iteration of a "one
dimensional map": you give me a value *x* between 0 and 1, and I
give you back *y=f(x)*, also between 0 and 1, where *f* is
a simple function e.g. a quadratic function. This value of *y*
becomes the next input value for the iteration. Symbolically

*x*_{n+1} = f(x_{n})

We can think of *n* as a discrete "time" variable. Even for
simple functions *f* the sequence of numbers generated has a
rich behavior, for example periodic (repeating) with an arbitrary
repeat period that may be chosen by varying a parameter in *f*,
or apparently random, which (loosely) we call chaos.

The iteration of the map is conveniently shown graphically from
the plot of *y=f(x)*. For the input value *x=x*_{n}
the output is the value *y* i.e moving vertically to the graph of
the function to *f(x)*. To get the next input value
*x*_{n+1} we need the value *x=y* given by moving
horizontally to the diagonal. The next iteration
*x*_{n+2} is given by again moving vertically to
*f(x)*, etc.

This process is demonstrated in the accompanying applet. To start,
take the default values (hit the **Reset** button), and iterate
with the **Step** button.

Instructions.

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Last modified 18 August, 2009

Michael Cross