I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a little about it. There's a weird pattern were some are simple spriograph-like graphs and the others are completely chaotic.

In case anyone enjoys seeing these as much as I do, I'll add an album on imgur for your viewing pleasure. The titles are of the form "$x_0$_$y_0$". These are all rendered with VODE from Scipy with $t_0=0$ and $t_1=1000$.

$$ \begin{eqnarray*} x' &=& \cos(y)+\sin(t) \\ y' &=& \sin(x)+\cos(t) \end{eqnarray*} $$

Here's the album: http://imgur.com/a/lbhrX

If anyone's interested, I uploaded a video of (1,1) plotted from t=0 to 250 with the vector field: https://vimeo.com/88323596.

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    $\begingroup$ Have you investigated the shape of the 'spirograph' region for $x_0$,$y_0$? $\endgroup$ – Neil W Mar 3 '14 at 3:02
  • $\begingroup$ No, I haven't. That's a good idea. I had noticed how they appeared in groups of close initial values. Sounds interesting though, I'll try it. $\endgroup$ – Tyler Mar 3 '14 at 3:05
  • $\begingroup$ I mean the shape of the set $\{(x_0,y_0) \in \mathbb{R}^2 : $ spirographic behaviour results $\}$ $\endgroup$ – Neil W Mar 3 '14 at 3:10
  • $\begingroup$ Building on your suggestion a little, I think I will also search the boundaries of a spirograph regions, to see if/how a continuous transformation from regular to chaotic occurs. Yeah, I figured it out after your edit, but I had already posted before I saw your edit. :) $\endgroup$ – Tyler Mar 3 '14 at 3:10
  • $\begingroup$ This system is very similar to the Peter de Jong map plotted by Fyre. $\endgroup$ – Tyler Jun 23 '14 at 7:02

These are great pictures, but are you sure you aren't seeing artifacts? I tried one of these (1_1) in Mathematica, and when I varied the PrecisionGoal and AccuracyGoal options without changing the equations or initial conditions, I see very different results (though all chaotic).

$\text{ParametricPlot}[\text{Evaluate}[\{x(t),y(t)\}\text{/.}\, s],\{t,0,500\}] \left(s=\text{NDSolve}\left[\left\{x'(t)=\cos (y(t))+\sin (t),y'(t)=\sin (x(t))+\cos (t),x(0)=y(0)=1\right\},\{x,y\},\{t,500\},\text{AccuracyGoal}\to \text{Automatic}\right];\right)$

enter image description here

$\text{ParametricPlot}[\text{Evaluate}[\{x(t),y(t)\}\text{/.}\, s],\{t,0,500\}] \left(s=\text{NDSolve}\left[\left\{x'(t)=\cos (y(t))+\sin (t),y'(t)=\sin (x(t))+\cos (t),x(0)=y(0)=1\right\},\{x,y\},\{t,500\},\text{AccuracyGoal}\to \infty \right];\right)$

enter image description here

$\text{ParametricPlot}[\text{Evaluate}[\{x(t),y(t)\}\text{/.}\, s],\{t,0,500\}] \left(s=\text{NDSolve}\left[\left\{x'(t)=\cos (y(t))+\sin (t),y'(t)=\sin (x(t))+\cos (t),x(0)=y(0)=1\right\},\{x,y\},\{t,500\},\text{PrecisionGoal}\to \infty \right];\right)$

enter image description here

I don't know enough about how Mathematica implements NDSolve to guess what's up, and this still leaves open the question as to whether something chaotic is going on for some initial values, even if it's none of the exact pictures you or I are getting.

  • $\begingroup$ I was just about to make the same comment after varying WorkingPrecision a bit - as well as the initial point. On the the other hand, some of them (the pretty ones) seem to have a persistent structure. $\endgroup$ – Mark McClure Mar 3 '14 at 3:21
  • $\begingroup$ I would guess that Mathematica probably has a better numerical integrator than Scipy, so they probably are artifacts. (But really weird ones.) The one I'm using has automatic time stepping, so that's not it. I'll try with Runge-Kutta and see if I get something different. $\endgroup$ – Tyler Mar 3 '14 at 3:26
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    $\begingroup$ The chaotic behavior seems to happen when the solution strays (whether numerically or actually) from within one $2pi\times2\pi$ region to another. If we call it "spirographic" when it stays within one such region, I can cause a switch from spirographic to semi-chaotic (imgur.com/ZqmMelm) for a similar system by changing just the accuracy goal for some parameter values. I can also get from more to less "all over the place" (i.e., covering more $2pi\times2\pi$ regions) the same way for similar systems. $\endgroup$ – Steve Kass Mar 3 '14 at 3:44
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    $\begingroup$ Yeah, it seems it is likely numerical error. I've tried with any change in absolute or relative error or BDF vs Adam's or VODE vs ZVODE causes changes in the graph. (See here for some idea what those last two pairs mean: docs.scipy.org/doc/scipy/reference/generated/…) $\endgroup$ – Tyler Mar 4 '14 at 2:55
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    $\begingroup$ I'm not really sure that this will help at first, but this system is really a system on torus ($2\pi$ periodic in all spatial dimensions). It's very interesting to see how it acts on $\lbrack 0; 2\pi \rbrack \times \lbrack 0; 2\pi\rbrack$ domain. $\endgroup$ – Evgeny Mar 5 '14 at 6:02

This dynamical system has a time-dependent Hamiltonian $$ H = \cos x + \sin y + \varepsilon \left(y\sin t - x \cos t\right) \, . $$ Indeed, if $x$ is the "generalized position" and $y$ the "generalized momentum", then the corresponding equations are $$ \left\lbrace \begin{aligned} & \dot{x} = \phantom{+}\frac{\partial H}{\partial y} = \cos y + \varepsilon\sin t \, ,\\ & \dot{y} = -\frac{\partial H}{\partial x} = \sin x + \varepsilon\cos t \, . \end{aligned}\right. $$

  • If $\varepsilon = 0$, the system is autonomous and the Hamiltonian is conserved. The surface plot below represents the time-independent Hamiltonian in the phase space $\mathbb{R}^2$. Trajectories are attracted by equilibria (valleys) or by periodic orbits. Note that $H$ is periodic with respect to $x$ and $y$, allowing to reduce the phase space to $\mathbb{T}^2$. The Lagrangian $L = \dot{x}y - H$ deduced from $\dot{x} = \cos y$ reads $$ L\big|_{\varepsilon=0} = \dot{x} \arccos\dot{x} - \sin \arccos\dot{x} - \cos x \, . $$ The corresponding Euler-Lagrange equation is $$ \ddot{x} - \sqrt{1 - \dot{x}^2} \sin x = 0 \, , $$ which can be approximated by $\ddot{x} + x - \pi \simeq 0$ as $(x,\dot{x})$ goes to $(\pi,0)$. This approximation reveals similarities with the dynamics of a $1$-periodic pendulum.

Surface plot of the time-independent Hamiltonian $\varepsilon = 0$

  • If $\varepsilon \neq 0$, the Hamiltonian is not conserved anymore. The time-dependence of the Hamiltonian may be viewed as a vibration of the previous surface plot. Thus, one "particle" can move from one attraction basin to one of its neighbors without being attracted by an equilibrium or by a periodic orbit. This is what happens in the present case $\varepsilon=1$. Note that $H$ is periodic with respect to $t$, allowing to reduce the phase space to $\mathbb{R}^2\times \mathbb{T}$.

Although this system is not among the classical chaotic systems, the analysis above shows similarities with the periodically forced simple pendulum, which is well-known to exhibit chaotic behavior (see e.g. [1]).


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