I'm currently 2nd year physics student (undergraduate). I have seminar which theme is double pendulum.
I'm having trouble understanding quasi-periodic motion in general and more importantly in context of double pendulum.
I was hoping you could give me some examples of quasi-periodic motion and how can I identify that type of motion and/or visualize it.
Suppose you have a cart connected to a wall with an ideal spring oscillating frictionless back and towards the wall. Now on that cart, you mount a pendulum that can only oscillate orthogonally to the cart’s direction of motion (with a different frequency). This way, the motion of the cart and pendulum are completely independent of each other. Now, consider the motion of the pendulum’s tip in two dimensions and whether it is periodic. Pick an arbitrary state of the system. Both the cart and the pendulum will return to their respective phases, but never at the same time – unless you tune the frequencies of cart and pendulum just right (more on this later). Thus the motion of the pendulum’s tip is not periodic. However, it is not chaotic either: We can perfectly predict it if we know the initial phases of both oscillations. This superposition of two (or more) oscillations is called quasiperiodic.
Now, you can tune the period lengths $T_\text{c}$ and $T_\text{p}$ of cart and pendulum length such that the motion is periodic, namely if $n T_\text{c} = m T_\text{p}$ with $m,n∈ℕ$. In this case, the period lengths (and frequencies) are rational multiples of each other, which is called commensurate. Of course, in reality, we cannot know whether the ratio of two frequencies of two independent oscillations is a rational or irrational number, but then we also do not want to wait forever for a repetition. Thus, incommensurability is an appropriate default assumption. Mind that in many real applications, the oscillations are not independent, but there is a mechanism that synchronises them – making the ratio of frequencies a rational number (with a small denominator). In fact, in any practical implementation of the above example, the two oscillations will synchronise since you cannot make them perfectly orthogonal to each other.
In the above example, it is obvious how to decompose the dynamics into two oscillations, but now suppose that you can only observe the horizontal position of the pendulum’s tip along a diagonal axis (with respect to the axes of the cart’s and pendulum’s motion). Many quasiperiodic dynamics is like this when you analyse it.
A classical example for quasiperiodic motion is dynamics of the moon. The synodic period (29.53 d), nodal precession (6793 d), and apsidal precession (3233 d) are incommensurate for all practical purposes, originating from processes that are practically uncoupled (thanks to the practical absence of friction in space). As a result, eclipses do not occur regularly, yet we can predict them. By contrast, the synodic period and lunar rotation are not incommensurate (but the same) since tidal locking synchronised them.
In the context of dynamical systems, quasiperiodic motion is characterized by two (or more) incommensurate frequencies (i.e., frequencies which are not rational multiple of each other).
As an example, if you consider a point jumping along a circumference of length $1$ at fixed irrational (say, $1/\sqrt{2}$-long) steps, you also get quasiperiodic motion — which never repeats itself exactly, but where nearby points also don't diverge, as chaotic orbits would. (How this discrete system and a continuous one are connected is probably best seen by means of the Poincaré section technique.)
GOY's classical paper (e-print) on the subject provides a nice introduction to it; some examples and other references can be found in this old answer of mine; and this paper (e-print) is specifically about the double pendulum. Online notes by Yannick Meurice offer a qualitative explanation of the emergence of quasiperiodicity in the double pendulum:
There is always one constant of motion (again the total energy). In the limits where the total energy is very small, the system can be described by two independent oscillators, the energy of each of them being a constant of motion. In the limit where the energy is very large, the effects of gravity are becoming negligible and a new constant of motion appears (the angular momentum). In both limits, the additional constant of motion forces the phases curves to move on a torus (doughnut) and after a suitable change of coordinates, we obtain a dynamical system similar to the example of two independent circular motions discussed in the first section. In general, the ratios of the periods will be irrational and the motion quasiperiodic.
Mathematically (as on the linked wiki), a quasiperiodic function is one which satisfies $f(x + \omega) = g(x, f(x))$, where $\omega$ is some constant, the quasiperiod. Quasiperiodic motion is motion which is describable by a quasiperiodic function.
In common language, which is almost entirely unrelated to the mathematical definition: a system is quasi-periodic if it does more or less the same thing, in more or less the same order, at more or less the same interval, most of the time. A metropolitan bus system, a heartbeat, breathing, a reproductive cycle, a climate cycle, an economic boom-bust cycle, a seasonal migration, etc, are quasi-periodic in this sense. For low amplitude oscillations, a double-pendulum is a quasi-periodic system in this sense.
Ask your instructors to clarify, but I'm pretty sure they mean the common-language definition.
Confusingly, most systems describable by quasiperiodic functions are not, in common language, quasi-periodic systems, and vice-versa. I'm not certain but I do not think that a double pendulum is describable by a quasiperiodic function.
In this context quasi-periodic should be what in mathematics is called almost periodicity. Essentially, a function $f(t)$ is quasi-periodic if there is a quasi-period $T$ such that \begin{equation} |f(t+T)-f(t)|<\varepsilon, \forall x \in \mathbb{R} \end{equation} for some small $\varepsilon>0$. Notice that if $f$ was periodic, above we should have $=0$ instead of $<\varepsilon$.
In other words, this means that the function (or in your case the motion) is not periodic, but it almost is, as there is some scale that defines a certain "repetition".
