The trace on the left is generated by a set of three chaotic differential equations that describe the x, y and z coordinates. The three-dimensional attractor produced can be viewed in the x-y, x-z, and z-y planes.
The equations are solved by the Fourth-Order Runge-Kutta method.
A single parameter of each system is bound to the chaos slider, which changes the attractor's behaviour from stable to chaotic.
Two oscillators, one for the left audio channel and one for the right, have their frequencies controlled by one of the three variables of the attractor. The variables are normalized and multiplied by a base frequency set by the L freq and R freq sliders.
The frequencies can also be modulated by a second pair of oscillators attached to another pair of variables. The base frequenices of these oscillators are set by the L mod and R mod sliders. Disable modulation for a channel by setting its variable to 0
The balance between the left and right channels can also be controlled by a variable. Select 0 to equalize the gain of both channels. Alternatively, the left and right channels can be selected on their own.
$$\frac{dx}{dt} = \sigma(y-x), \frac{dy}{dt} = x(\rho-z)-y, \frac{dz}{dt} = xy - \beta z$$ Edward Lorentz described his simplified convection model in "Deterministic Nonperiodic Flow" in the Journal of the Atmospheric Sciences in 1963. The chaos slider varies the value of \(\rho\) between 13 and 28. \( \sigma = 10, \beta = \frac{8}{3}.\)
Chua
$$\frac{dx}{dt} = \alpha\left[y - x - f(x)\right], \frac{dy}{dt} = -x + y +z, \frac{dz}{dt} = -\beta y$$
$$f(x) = m_{1}x + \frac{1}{2}(m_{0}-m_{1})(\left|x+1\right|-\left|x-1\right|)$$
Leon Chua devised his chaotic circuit in 1983, the function \(f(x)\) describes a special nonlinear resistor. The chaos slider varies the value of \(\beta\) between 25 and 50.
\( \alpha =15.6, m_{0}=-\frac{8}{7}, m_{1}=-\frac{5}{7}.\)
Rossler
$$\frac{dx}{dt} = -y -z, \frac{dy}{dt} = x + ay, \frac{dz}{dt} = b + z(x-c)$$
Otto Rössler devised a set of simple differential equations to enable the study of chaotic dynamics
in "An Equation For Continuous Chaos" in Physics Letters in 1976. The chaos slider varies the value of \(c\) between 5.7 and 13. \( a = b = 0.1.\)