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Note: in this question when I talk about "phase space," I will be refering to velocity vs. position space, which can also be correctly referred to as "state space." Many sources (including John R. Taylor's Classical Mechanics) allow for both terminologies. This discussion may also apply when the velocity is replaced by the generalized momentum... but I'm specifically interested in velocity vs. position.

In phase space, you will commonly hear the refrain that trajectories cannot cross. This is, of course, true with a simple harmonic oscillator (ellipses forever), a damped harmonic oscillator (an inward elliptical spiral), and more. However, when dealing with some nonlinear oscillators (and certainly when dealing with chaos) the trajectories DO cross regularly. For example, a driven damped pendulum (DDP) clearly has crossing trajectories.

Chaos, however, is not a sufficient criterion for crossing trajectories. A period-2 DDP (i.e. one that flips back and forth between two orbits in a periodic fashion) crosses its own trajectory twice per cycle. A period-4 DDP does it 4 times!

What is the criterion (or criteria) for when a phase space trajectory may cross?

related, but not particularly helpful to me: When can phase trajectories cross?

Qmechanic
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Bunji
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3 Answers3

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The general idea is that trajectories that cross on phase space will have identical initial conditions from that point and will therefore satisfy identical initial-value problems, and from that you can conclude that they will identically track each other.

The example you give is of a driven system, which is subject to an external driving force that depends on time and is therefore not time-translation invariant. For such a system, trajectories can cross at different times in the driving cycle, but you cannot conclude that they are solutions of identical IVPs, because the initial conditions are the same but the driving is not.

If you restrict your attention to systems that are invariant under time translations, this feature will go away.

Emilio Pisanty
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    Should conditions such as an external driving force be considered as extra dimensions of the phase space? It would then include all necessary information, and prevent trajectories crossing. – gidds Aug 22 '22 at 15:29
  • @gidds It generally seems unnatural to me to consider the variation of a driving force as movement in some additional dimension to the phase space ($(x,v) \to (x,v,F)$), but it could be done in principle. It still would not ensure trajectories can't cross, however: it is trivial to contrive a situation with times $t_0$ and $t_1$ satisfying $F(t_0) = F(t_1)$ while $F'(t_0) \neq F'(t_1)$. Two trajectories initiated at any $(x_i,v_i)$ at $t_0$ and $t_1$ will then cross transversally in $(x,v,F)$ space. The best this does is ensure that the $(x,v)$ trajectories will be tangential at such points. – jawheele Aug 22 '22 at 17:34
  • @jawheele Shouldn't ′ then be a dimension, too? From an oversimplified lay perspective, if phase space encodes all the relevant state, then shouldn't that be enough to reconstruct the subsequent evolution of the state? And then trajectories can't cross. Conversely, if trajectories can cross, then the phase space doesn't encode all the relevant state. (And so, presumably, it could be made to do so by adding in further dimensions as necessary. Or is that not always possible?) – gidds Aug 22 '22 at 18:35
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    @gidds: Here's a way to do what you're driving at, I think: A time-dependent system of ODEs in $t$ can always be replaced by a time-independent system by introducing an additional variable $u$, substituting $t \to u$ in the original ODEs, and then adding the ODE $\dot{u} = 1$ to the system. If you do this, the resulting trajectories in the "extended" state space (the original variables plus $u$) don't intersect. But those trajectories are in a higher-dimensional space, and when they're projected back down to the original state space, their projections can intersect. – Michael Seifert Aug 22 '22 at 19:32
  • @gidds That doesn't quite do it. One can pass the buck to the next derivative ad infinitum, and even passing to infinite dimensions to specify all derivatives is not enough in general. One could tack on the relevant infinite dimensional function space directly, but you've encumbered yourself for little gain. The point of phase space is to entirely encode the state of the dynamical objects of interest at each time-- the driving force function is fixed metadata that doesn't need tracking. As Michael suggests, the only missing data between a full trajectory and a phase space point is a time . – jawheele Aug 22 '22 at 20:07
  • @jawheele The formalism where you encode a driven system (without time translation symmetry) as (x, v, t) is used in the book "Nonlinear dynamics and chaos" by Strogatz (and, I beleive, quite widely elsewhere). That formalism does have the feature that trajectories never cross. The formalism has some other advantages. For example in the "include time" formalism one can say "Chaotic attractors require 3 or more dimensions", where as if t is excluded you need to say "They require either 2 dimensions with a time dependent force or else 3 or more dimensions" – Dast Aug 23 '22 at 10:18
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    @Dast Indeed. I agreed with and upvoted Michael's comment-- my comment was responding to gidds' suggestion of adding dimensions to specifically track the the value of the driving force and/or its derivatives. – jawheele Aug 23 '22 at 15:48
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  1. Although Hamiltonian systems and symplectic phase spaces have nicer properties, it is usually not important when we discuss whether trajectories are forbidden to cross. Hence it is fine that OP uses velocities rather than momenta.

  2. It is much more important that the governing ODE (apart from mild regularity requirements) is

    • 1st order

    and

    cf. e.g. my Phys.SE answer here.

  3. In particular, OP's example with a driven pendulum is not autonomous.

Qmechanic
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When you have a phase space where lines can cross, you have forgotten something. Namely some parameter that influences future system behavior, which you have forgotten to include into your phase space.

For the system in the example, that parameter is the phase of the driving force. I.e. the phase space is 3D, not 2D. If you include that third dimension, your crossings will simply disappear.

cmaster - reinstate monica
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