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It's often said that all continuous chaotic systems must have at least three dimensions of phase space. The Lorenz system has three explicitly, the double pendulum has four (two angles and two angular velocities), and many others have more.

The Duffing oscillator is a second-order forced nonlinear system that obeys, $$ {\ddot x} + A \dot x + B x + Cx^3 = D\cos(\omega t) $$ where $A,B,C,D,$ and $\omega$ are constant parameters of the system. As far as I can tell, this system has a two-dimensional phase space, ($\dot x$ and $x$), and yet it exhibits chaotic behavior[0]. The instantaneous acceleration can be calculated from $\dot x$ and $x$, which means it's not really a dimension of the state. Where have I gotten confused?

[0]https://en.wikipedia.org/wiki/Duffing_equation

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  • $\begingroup$ I guess OP is referring to the Poincaré-Bendixon theorem. en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem But I do not know if it applies to systems which are explicitly time dependent. Maybe this could be a starting point. $\endgroup$ – AlmostClueless Jun 1 at 19:51
  • $\begingroup$ @AlmostClueless Yes, and the important quote is, "In particular, chaotic behaviour can only arise in continuous dynamical systems whose phase space has three or more dimensions. However the theorem does not apply to discrete dynamical systems, [...]". The Duffing oscillator is neither discrete, nor three-dimensional. $\endgroup$ – Retracted Jun 1 at 20:06
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The "dimension of phase space" is not quite the right way to characterize systems that can be chaotic. The proper criterion involves writing the equations in first-order autonomous form: for variables $\xi_1, \dots, \xi_N$, the equations of motion are of the form $$ \dot{\xi}_i = f_i(\xi_1, \dots, \xi_N) $$ where the functions $f_i$ do not depend explicitly on $t$. "First-order" means that the left-hand sides of the $N$ equations are the first derivatives of the configuration; "autonomous" means that the right-hand sides don't depend explicitly on $t$. $N$ is (I think) what you're thinking of as "the dimension of phase space."

Any system of ODEs can be put into this first-order autonomous form. For the system to exhibit chaotic behavior, it is necessary that when the equations are written in this form, the functions $f_i$ are non-linear, and that $N \geq 3$. As you note, the Lorenz equations have $N = 3$, but they are also autonomous; their equations of motion do not depend explicitly on $t$. For the double pendulum, we have $N = 4$, though the fact that we also have a conserved quantity means that the accessible phase space is effectively 3-dimensional instead.

So what about the Duffing oscillator? We can try putting it in first-order form as follows: \begin{align*} \dot{v} &= -Av - Bx - C x^3 - D \cos (\omega t) \\ \dot{x} &= v \end{align*} This is a first-order non-linear set of equations. But it is not autonomous, since $t$ appears on the right-hand side of the first equation. Not to fret, though: we can define an equivalent autonomous system with three variables as follows: \begin{align*} \dot{v} &= -Av - Bx - C x^3 - D \cos (\omega z) \\ \dot{x} &= v \\ \dot{z} &= 1 \end{align*} This system is autonomous, and since it has 3 degrees of freedom it can (and does) exhibit chaotic behavior. The variable $z$ is, of course, secretly equal to $t$.

This trick is quite general, by the way: any non-autonomous first-order system of $N$ equations can be recast as an autonomous first-order system of $N+1$ equations, simply by treating $t$ as one of the variables of the system.

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    $\begingroup$ I'm afraid I don't have a great recollection of where I learned this; my only citation for this is a footnote in Chapter 12 of Taylor's Classical Mechanics. If anyone has better references for the OP to read more on this, I would be grateful. $\endgroup$ – Michael Seifert Jun 1 at 20:28
  • $\begingroup$ "An Introduction to Dynamical Systems" by D.K. Arrowsmith, would be a good starting point. The Trick you mentioned to get to an autonomous system is mentioned in chapter 1.8 Periodic non-autonomous systems. $\endgroup$ – AlmostClueless Jun 1 at 20:52
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    $\begingroup$ Also as a quick comment. The reason why the non-autonomous system is not a good starting point to characterize a system is caused by the fact that, when we talk about chaos we talk about the behavior of flows on manifolds. These flows on a manifold $M$ correspond to an autonomous differential equation. This is also covered by the above book. $\endgroup$ – AlmostClueless Jun 1 at 20:54

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