A lot of papers define a 'diabolical point' as a "double semi-simple eigenvalue." I know a semi-simple eigenvalue is one which has algebraic multiplicity and geometric multiplicity to be equal. However, I could not find any definition of a double semi-simple eigenvalue.

# [Physics] a diabolical point

chaos theorycomplex systemsnon-linear-systemsterminology

#### Related Solutions

It is true that the double pendulum exhibits integrable behavior, when the initial angles are very small, however, in general, it is very difficult to characterize the chaotic behavior of the double pendulum in terms of the initial angles. There are other representations which provide a clearer picture of its chaotic behavior.

The introductory section of the following articleby O. A. Richter, and reference therein describe the main characteristics of the double pendulum integrability (see here for the official journal version). I'll summarize here the main facts:

(Remarks: The numerical values correspond to a standard double pendulum of unit masses and unit rod lengths and unit acceleration due to gravity)

The total energy of the double pendulum is a constant of motion. The double pendulum possesses 4 equilibrium points corresponding to total energies $E= 0, 2, 4, 6$. The total energy determines the topology of the energy hypersurfaces, for $E<2$, the energy hypersurfaces are three spheres, while for $E>6$ , the energy hypersurfaces are three tori.

To further understand this point, in the case of very low energies, the system can be approximated (linearized) to an isotropic harmonic oscillator. The energy hypersurfaces are then of the form $x_1^2+x_2^2+p_1^2+p_2^2=E = const.$, while for the case of very large energies, the kinetic energy dominates and we can neglect gravity. In this case, there are two types of solutions of the equation of motion, one in which the outer rod rotates and the inner rod oscillates, and the second in which both rods rotate. The transition between the two types of solutions is determined by the value of the total angular $L$ momentum which becomes a constant of motion (due to the lack of gravity). For the standard double pendulum the transition occurs at $ L^2 = 2E$.

Both limits (small and large energies) correspond to integrable systems. This is well known, but here is a short explanation. To see that the isotropic harmonic oscillator is integrable, one needs to solve the equations of motion in polar coordinates. The polar angle just rotates with a constant angular velocity, and the radial coordinates oscillates in such a way that the trajectory has the shape of a two trip course through the donut hole drawn on the surface of a two-torus. This is the Liouville-Arnold torus (whose existence indicates the system's integrability) with respect to which the three sphere energy hypersurface is foliated.

In the high energy limit, a similar Liouville-Arnold torus exists when the inner rod oscillates and a torus generated by the two polar angles when the two angles rotate. (Here the exact solution is more difficult, see for example, the following article by Enolskii, Pronine, and Richter.

Now, since both limits of vanishing and very high energies correspond to integrable systems, the total energy also controls the system characteristics, but the dependence here is much more complicated. The transition from integrability to chaos and back as the total energy decreases from infinity to zero is described in figure 2 of Richter's article. There are a lot of details, but here are the main features: The figure correspond to the projection of the trajectories onto the plane spanned by the outer rod angle and the total angular momenta. For very large angular momentum, the projections of the trajectories are horizontal lines with constant angular momenta (which is a constant of motion). As the energy is reduced, two disjoint chaotic regions are formed, the integrable trajectories correspond to rational and irrational tori, together with stable resonances. At about E = 10.352 which corresponds to the golden winding ratio, all irrational tori vanish and a transition to global chaos occurs. The stable resonances also vanish eventually, at the low energies the resonances corresponding to the second integrable region start to appear.

However, what if the time series is multi-dimensional, indeed of the same dimension as the phase space, to begin with?

Well, how would you know that your time series is of the same dimension as the phase space? Usually, because you already know the dynamical equations for your system (as for your pendulum). If you observe a real-life complex system, however, you might be able to obtain a multivariate time series, but there is no way to say whether its dimension corrensponds to the actual dimension of the phase space, since you cannot know the latter. Therefore I am addressing two cases separately:

- You know the dynamical equations for your system. Be very careful to assume this unless your system is simulated.
- You have obtained a multivariate time series from an unknown system.

# 1. You can simulate the system

Roughly speaking, you determine the largest Lyapunov exponent (and also the others) by looking at how quick two trajectories diverge after passing through two points that are close in phase space. If you only have a reconstructed phase space of your system from a time series, the only way to obtain two such nearby trajectories is to look for two points that are close to each other in your reconstructed phase space. However, if you can simulate your system, you can generate such points for yourself simply by applying a slight perturbation to the state of your simulated system. Apart from this, the method is basically the same (and is described in section 3 of the paper by Wolf et al. for example).

Also, there are some cases where you can determine the Lyapunov exponents analytically.

# 2. You have a multivariate time series

Estimating the Lyapunov exponents from a time series happens roughly in two steps:

- Reconstructing the phase space from the time series.
- Estimating the Lyapunov exponent from this reconstructed phase space.

Step 2 does not care about how you reconstructed the phase space – given that you do it properly and that the attractor is maximally unfolded. And in step 1, having more than one observable from your system is usually a huge benefit. A simple approach would be to start with your multivariate time series and add delayed embeddings (as described for example in your quote from Packard et al.) of your component time series until you are confident that you have unfolded the attractor. Keep in mind however, that some of your observables might not be independent or at least strongly correlated. Little surprisingly, there are more sophisticated methods for this (as a start, a quick search yielded this paper).

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## Best Answer

"Double" simply means a degenerate eigenvalue (repeated root of the characteristic equation), thus a "double semi-simple eigenvalue" is a once repeated eigenvalue (i.e., with algebraic multiplicity 2) that spans a 2D vector space (i.e., its geometric multiplicity is also 2).

You can check, e.g., the second section of this paper (e-print), or 4.1 of this (e-print), or section 9.2.4 of this book, or, apparently, chapter 5 of this book.

These points are relevant mostly because they are associated to systems at bifurcations, i.e., structurally unstable systems whose behavior can change qualitatively under small perturbations. Such sensitivity has been used in the construction of very sensitive sensors, and even more sensitive than the diabolical points are the even more degenerate "exceptional points" (where "not only do resonant frequencies coincide but their resonant modes do too") . Both situations are schematically illustrated for light propagation modes (see this article) in the following figure, which shows the modes split with growing perturbation intensity $\epsilon$: