What is a minimal, explicit dynamical system (as in, a series of coupled ordinary differential equations) that exhibits quasiperiodic oscillations for some region of parameter space? Two coupled Van der Pol oscillators have this property, for example; however I am hoping to find a dynamical system with a simpler form.

I am imagining something like a system of three variables, in which trajectories trace a spiral on a torus in three dimensional space.

  • 1
    $\begingroup$ Must it be a continuous system or a mapping (discrete time) is also fine? $\endgroup$
    – stafusa
    Aug 7, 2019 at 12:28
  • $\begingroup$ I was hoping for ODEs, since the standard map is a pretty simple discrete time map with quasiperiodicity. Thank you. $\endgroup$
    – wil3
    Aug 7, 2019 at 17:54
  • 1
    $\begingroup$ Yeah, that's where I was getting to. $\endgroup$
    – stafusa
    Aug 7, 2019 at 17:55

1 Answer 1


A minimum, if boring example is that of two uncoupled 1D linear ODE (with incommensurate constants) on a torus, such as can be found in Hasselblatt & Katok:

$$\left\{ \begin{aligned} \dot x &= \omega_1\\ \dot y &= \omega_2, \end{aligned} \right. \tag{4.2.3} $$

whose trivial solution is

$$\left\{ \begin{aligned} x &= x_0 + \omega_1t\\ y &= y_0 + \omega_2t. \end{aligned} \right. \tag{4.2.4} $$

The simplest nontrivial example I could find is the forced van der Pol oscillator (still simpler than the example of two of them coupled, mentioned in the OP), found, e.g., in Guckenheimer's 1980 paper (e-print):

$$\left\{ \begin{aligned} \dot x &= y - \epsilon(x^3/3 -x)\\ \dot y &= -x+b\cos(\omega t), \end{aligned} \right. $$

which can be written as an autonomous system in 3D:

$$\left\{ \begin{aligned} \dot x &= y - \epsilon(x^3/3 -x)\\ \dot y &= -x + b\cos z\\ \dot z &= \omega. \end{aligned} \right. $$

A numerically integrated quasiperiodic trajectory of this system, in normalized variables, can be found in this paper (e-print):

Forced Van der Pol, time integration (ODE suite from Matlab).


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