WaveWarp 2.0 Component
      
Signal Generators:
Chaotic Control Generator 2 (A)
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Functional Description
Generates two
chaotic output sequences (one per control signal ouput)
derived from the state trajectories
of a "strange attractor".
The outputs are nominally scaled between 0 and 1 (for use as "Amplitude" control signals).
NOTE: the sample rate at which this component is executed can be arbitrarily set via the
"Sample-rate setting for Signal Generators and
Input ASCII files" button
(or the "Sample-Rate Setting" command under the "Edit" menu) on the toolbar, which is
activated whenever the component is selected on the DrawingBoard.
Equivalently, it can be set via the dialog box which (i) appears
when the component is initially
dragged on to the DrawingBoard or (ii) is activated using the right-mouse-button when
the component is selected on the DrawingBoard.
This procedure allows the component to either (i) enforce
a user-determined sample rate on the downstream component(s), or (ii) to inherit the sample rate from
the downstream component(s). Different signal generators can run at different
sample rates on a DrawingBoard, as long as the rules of connectivity
for multiple sample rates are adhered to (see the
WaveWarp Users' Guide
for more information.)
Algorithm
Verbatim implementation of
the following system of difference equations (due to Henon, based on the
pioneering work of Lorenz,
see
[St] p. 504-505 ):
| x1n+1 = x2n -(7/5)(x1n)2 +1 |
| x2n+1 = (3/10)x1n |
where x1n+1 and x2n+1
are the next values in the sequences of the two ouput states,
given that
x1n and x2n
are the respective current values.
The phase-space trajectory (i.e. the path traced out by plotting
x1 versus x2) jumps around in a bounded region
(the "strange attractor"), yet never returns to the point where it started.
Furthermore, though bounded by "thick" curves, the detailed trajectory
is extremely sensitive to the initial values of the states.
For tiny differences in the initial conditions, an extremely intricate network of similar
paths are generated, yet no two are the same.
These properties are the hallmark of chaos, and only occur in non-linear systems
(in this case, quadratic).
The initial conditions for the states, plus other parameters of the signal generator,
are
adjustable via the Parameter Window, as summarised in the following table.
| Parameter | Purpose |
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| "Initial state 1"
and "Initial state 2"
sliders |
Sets the initial values for the respective states in the
Henon strange attractor equations.
If the "Random" checkbox is enabled, the sliders are automatically disabled,
and the initial values are computed from a pseudo-random number generator (using the system
clock as a "seed", thereby yielding different
values every time the DrawingBoard is re-started).
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| "Time step" slider |
Sets the time step between successive updates of the Henon equations.
The time step is given by the slider value multiplied by the "Max update increment" selection,
and is displayed in the "Total time increment" window. When the DrawingBoard is playing,
the Henon equations will be updated (and the individual states sent to the individual control signal outputs)
at intervals equal to the "Total time increment".
Between these intervals, the control signal outputs will retain their previous values.
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| "Amplitude" slider |
Adjusts the amplitude of the output signals (after the computation of
the Henon equations).
Depending on the selected "Range", each of the control signal outputs will be scaled either between
-1 and 1, or between 0 and 1.
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| "Modulation" checkbox |
If enabled, the "Range" selection is automatically disabled, and the "Amplitude" slider becomes
a "Depth" slider. The control signal ouputs will each be automatically scaled over the range from 0 to 1.
The "Depth" determines the fraction of this range over which the ouputs of the Henon equations
will be
mapped. Maximum "Depth" (a value of 1) implies that the ouputs of the Henon equations will be
mapped on to the entire range from 0 to 1, thereby maximising the signal variations.
Minimum "Depth" (a value of 0) implies no mapping, thereby minimising the signal variations (i.e. constant
outputs). An intermediate value, say 0.2, implies that the ouputs of the Henon equations will be
mapped on to the upper 20% of the output ranges, and so forth.
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| Plot windows |
The plots display portions of the respective state time-histories
of the Henon equations, corresponding to the chosen
initial conditions.
The data plotted are the raw ouputs of the Henon equations, and do not reflect the
scaling due to the "Amplitude", "Range",
or "Depth" settings (though the time-scale does correspond to the settings of the "Time step").
Note: the Audio Phase-Space Scope
can be used to plot the states versus one another
(after converting them from control to audio using
Control To Audio converters),
thereby clearly revealing the
characteristic pattern of the "strange attractor".
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For additional introductory information on strange attractors,
see
[Gl] p. 149-151.
For an introductory discussion on the use of chaos in computer music, see
[Roa] p. 888-889.
Signal Implementations
| Audio signals | Control signals | Description |
|---|
| n/a | double output | Generates at each ouput a control signal with an amplitude nominally between 0 and 1 |
Related components:
Example DrawingBoards illustrating usage:
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