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Signal Generators: Models and Emulations

lorenz

  ax, ay,    lorenz      ks, kr, kb, kh, ix, iy, iz, iskip
  az

Description

Implements the Lorenz system of equations. The Lorenz system is a chaotic-dynamic system which was originally used to simulate the motion of a particle in convection currents and simplified weather systems. Small differences in initial conditions rapidly lead to diverging values. This is sometimes expressed as the butterfly effect. If a butterfly flaps its wings in Australia, it will have an effect on the weather in Alaska. This system is one of the milestones in the development of chaos theory. It is useful as a chaotic audio source or as a low frequency modulation source.

Initialization

ix, iy, iz – the initial coordinates of the particle

iskip – used to skip generated values. If iskip is set to 5, only every fifth value generated is output. This is useful in generating higher pitched tones.

Performance

ksv – the Prandtl number or sigma

krv – the Rayleigh number

kbv – the ratio of the length and width of the box in which the convection currents are generated

kh – the step size used in approximating the differential equation. This can be used to control the pitch of the systems. Values of .1-.001 are typical.

The equations are approximated as follows:

  x = x + h*(s*(y - x))
  y = y + h*(-x*z + r*x - y)
  z = z + h*(x*y - b*z)

The historical values of these parameters are:

  ks = 10
  kr = 28
  kb = 8/3

Example

instr 20
  ksv = p4
  krv = p5
  kbv = p6

  ax, ay, az   lorenz   ksv, krv, kbv, .01, .6, .6, .6, 1
endin

;score
  ;   start dur S   R   V 
  i20 5     1   10  28  2.667
e

Author

Hans Mikelson
February 1999 (New in Csound version 3.53)

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Signal Generators: Models and Emulations