Differential equation on a doughnut

Here’s a differential equation from [1] that’s interesting to play with.

$\begin{align*} x^\prime &= -my + nxz \\ y^\prime &= \phantom{-} mx + nyz \\ z^\prime &= (n/2)(1 + z^2 - x^2 - y^2) \\ \end{align*}$

Even though it’s a nonlinear system, it has a closed-form solution, namely

$\begin{align*} x(t) &= \frac{2a\cos(mt) - 2b \sin(mt)}{\Delta - 2c \sin(n t) + (2 - \Delta) \cos(n t)} \\ y(t) &= \frac{2a \sin(mt) + 2b \cos(mt)} {\Delta - 2 c \sin(n t) + (2 - \Delta) \cos(n t)} \\ z(t) &= \frac{2c \cos(nt) + (2 - \Delta) \sin(nt)}{\Delta - 2c \sin(nt) + (2 - \Delta) \cos(n t)} \\ \end{align*}$

where (a, b, c) is the solution at t = 0 and Δ = 1 + a² + b² + c².

The solutions lie on the torus (doughnut). If m and n are coprime integers then the solutions form a closed loop. If the ratio m/n is not rational then the solutions are dense on the torus.

Here’s an example with parameters a = 1, b = 1, c = 3, m = 3, and m = 5.

And now with parameters a = 1, b = 1, c = 0.3, m = 4, and m = 5.

And finally with parameters a = 1, b = 1, c = 0.3, m = π, and m = 5.

[1] Richard Parris. A Three-Dimensional System with Knotted Trajectories. The American Mathematical Monthly, Vol. 84, No. 6 (Jun. – Jul., 1977), pp. 468–469

Differential equation on a doughnut

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