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Many dynamical systems in physics, astronomy, chemistry, physiology, meteorology, economics, population
dynamics can be described by
Ordinary Differential Equations.
In the second half of the 20th century much attention has been focussed on the
i.e. unpredictable behaviour of (non-linear) ODE's. A well-known example is the
Lorenz atrractor, illustrating the "Butterfly effect": small causes can have large effects.
Mathgrapher uses an accurate Adams-Bashforth variable order, variable step predictor-corrector
algorithm to integrate systems of up to 20 coupled differential equations.
which we can write as a system of coupled ODEs using
For small d simple limit cycle behaviour is observed. As d increases the cycles bifurcate into
double period cycles and the motion becomes increasingly chaotic. The 2D (F1-F2) graph below
is the result of an integration from T=0 to 50 using a=0.25, d=200, g=1.5. Initial values are
F1=0.8, F2=0,5 and F3=0.25. Output is generated at 3000 intermediate time steps.
Below the power spectrum of F2. The same parameters were used, but in this case T runs from
0 to 250 and output is generated at 256 equal steps.