Manual

Buttons

• Reset: Sets the values of the parameters to those shown in the boxes and reinitializes the plot (e.g. resets the initial value to x0).
• Clear: Clears the plot, but does not reset the parameter values and maintains the current value of x .
• Step: Performs and plots one step in the iteration.
• Start: Starts repeated iteration and plotting with a time between plotting set by the delay parameter (in ms). Clear can be used while the iterations are proceeding if the display becomes scrambled.
• Stop: Stops the iteration and allows the parameters to be updated.

Text Boxes

Note that the parameter values are not updated until either Enter is hit on the keyboard or the Reset button is pushed.

• nf: The function f ^{(2 nf}) is iterated and plotted.
• nsc: Rescaling parameter. The central portion of x and f (i.e. near x=0.5 are expanded by the factor (-2.502907875)^nsc. The actual range of x and f plotted are indicated as the "full scale" on the plot.
• x0: Initial value of x expressed as a fraction of the range of the plot (i.e. involves the rescaling set by nsc and scale).
• scale: Further modifies the range of x and f plotted by multiplying by the factor scale - again indicated in the "full scale" shown on the plot. Useful for "fine tuning" the plot.
• trans: Sets the number of iterations done before values are plotted. Useful to eliminate transient behavior if desired.
• a: Sets the value of a in the maps. Value can be typed in or set by the drop down box values below.

Drop-down boxes

It turns out to be useful to look at a sequence of values of a, a₀ , a₁ .... given by

a_m = a_c - C ^-m

where a_c depends on the map function , but = 4.6692016091... for all maps. The value of m can be used to step through values of a yielding 2^m orbits (the symbol I in the list box for m corresponds to infinity i.e. a=a_c). The value of C can be chosen to give particular "types" of orbit.

The list boxes respectively:

• set the map to be iterated and a_c
• set whether a > a_c (+) or a < a_c (-)
• set the value of C giving values of a corresponding to
  • Superstable: one point on orbit at maximum of f.
  • Instability: cycle just unstable.
  • Bandmerging: chaotic bands merge in chaotic region for a > a_c.
• set the value of m

Values can also be typed directly into the text boxes (remember to Enter or Reset).

Scroll bar

• Speed: sets speed of iteration and animation, roughly in iterations per second.

Mouse Click

• Clicking the mouse within the plot area starts a new iteration with an initial value given by the horizontal position. (The vertical position has no effect.)