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The double pendulum is a famous example of a chaotic system. It consists of one pendulum hanging from the end of another pendulum, which in turn hangs from a fixed point. In the traditional version, both arms are perfectly rigid.

Here's an easy way to generalize the model so that the arms are not perfectly rigid. Let $\mathbf{x}$ and $\mathbf{y}$ be the variable locations of the ends of the two arms, where masses $m_x$ and $m_y$ are located. (The arms themselves are massless.) Take the system's equations of motion to be \begin{align} \newcommand{\bfx}{\mathbf{x}} \newcommand{\bfy}{\mathbf{y}} \newcommand{\bfg}{\mathbf{g}} m_x\ddot\bfx + \nabla_\bfx V(\bfx,\bfy) &= 0 \\ m_y\ddot\bfy + \nabla_\bfy V(\bfx,\bfy) &= 0 \end{align} with potential energy \begin{align} V(\bfx,\bfy) = &- m_x\bfg\cdot\bfx + f\big(k_x,L_x,|\bfx|\big) \\ &- m_y\bfg\cdot\bfy + f\big(k_y,L_y,|\bfy-\bfx|\big) \end{align} where $\bfg$ is the acceleration of gravity (a downward-pointing vector) and the function $f$ is defined by $$ f(k.L,x) = k(L^2-x^2)^2. $$ The $L$s are the nominal arm-lengths, and the $k$s are the degree of rigidity of the arms. For finite $k$s, the potential energy $V$ is a smooth function of $\bfx$ and $\bfy$. The perfectly rigid version corresponds to $k_x,k_y\to\infty$: in that limit, any deviation from the nominal lengths $L_x,L_y$ costs infinite energy.

How chaotic is the system with finite-but-large values of $k_x$ and $k_y$, if chaos is quantified in the standard way(s)? Intuitively, the closer $k_x,k_y$ are to zero, the less chaotic the system should be, because taking the limit $k_x,k_y\to 0$ gives a pair of freely-falling masses that don't interact with each other at all. But can we determine how the degree of chaos (quantified in a standard way) scales with $k_x,k_y$, at least roughly?

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  • $\begingroup$ Would a better way to model the flexibility of the arms be by treating them as springs? Or do you have this particular description in mind for a reason? $\endgroup$
    – fewfew4
    Aug 12, 2021 at 6:36
  • $\begingroup$ Well, we can simulate your equations, compute some measure of chaos (e.g., the Lyapunov exponent) for these simulations, average over an appropriate ensemble of initial conditions and plot the dependency on your parameters. If you know the right tools, it’s mostly legwork. But something tells me that this doesn’t satisfy you. What exactly do you want to do with this information? $\endgroup$
    – Wrzlprmft
    Aug 12, 2021 at 10:38
  • $\begingroup$ @fewfew4 My intent was to model them as something like springs, but with a potential-energy function that is a polynomial function of the coordinates. If using some other continuous function like $f(k,L,x)=k(L-|\mathbf{x}|)^2$ makes the math easier, despite not being differentiable everywhere, then that would be fine. $\endgroup$ Aug 12, 2021 at 13:06
  • $\begingroup$ @Wrzlprmft I'm mostly interested in building intuition. Knowing the results from computer simulations can contribute to intuition, but I was hoping for some kind of analytic/intuitive argument. The reasoning (if sound) is more important to me than the results. $\endgroup$ Aug 12, 2021 at 13:07
  • $\begingroup$ @ChiralAnomaly Standard double pendulum model not depends on $k_x, k_y$, but it depends on $L_x, L_y$. Can you formulate your model so that it has standard model as a limit at large $k_x, k_y$? $\endgroup$ Aug 17, 2021 at 17:29

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It's true that $k_{x,y}\to 0$ will be non chaotic, but, before this limit, the extra degrees of freedom of flexible arms should allow for more complex motion. At any rate I don't expect the degree of chaos to change too smoothly or monotonically with $k_{x,y}$, but rather that, e.g., periodic windows from resonances between spring and pendulum movements show up at intermediate values (resonances in the simple spring-pendulum have been studied for some time, see, e.g., here and here).

As Wrzlprmft comments, some simulations should help with building an intuition — it sounds like a nice undergrad project — if you do perform some, please share here your results. Edit: There are already a number of numerical implementations and simulation results available, for instance: in java (Open Source Physics), Matlab (1), Matlab (2), Mathematica (1), Mathematica (2), Maple, etc.

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  • $\begingroup$ Thank you for the reply! Can you explain the reason for your expectation that "periodic windows from resonances between spring and pendulum movements show up at intermediate values"? Is it based on experience with another model, or on a property of the usual rigid double-pendulum? $\endgroup$ Aug 17, 2021 at 13:08
  • $\begingroup$ (I could do some simulations, but my time is focused on other subjects, so I was hoping to learn from research that others have already done on this particular topic. Not enough time to specialize in everything.) $\endgroup$ Aug 17, 2021 at 13:08
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    $\begingroup$ @ChiralAnomaly I expect resonances mostly because they are generically present in such systems, but I added a reference (and other bits) to the answer. It might also be worthy it to remember that, while there is often chaos associated with these resonances, it might be restricted to a very small region of the phase space, see, e.g., webspace.science.uu.nl/~verhu101/SAMOSpap.pdf . $\endgroup$
    – stafusa
    Aug 17, 2021 at 14:50

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