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I’m dealing with a system of ODEs where I think I might have found chaotic solutions. I’ve calculated the largest Lyapunov exponent, and found it to be approximately 0.02. It’s positive, which indicates chaos, however, it’s quite small, compared to say 0.905 for the known Lorenz system.

How reliable is the Lyapunov exponent? At what values does one discard it?

Also I haven’t been able to detect neither a saddle–focus bifurcation (chaos by Shilnikov bifurcation) or a period-doubling cascade around the parameter values where I observe chaos.

Should I simply discard my findings?

Wrzlprmft
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1233023
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  • Actually it's quite interesting that you don't see homoclinic bifurcations or period-doubling cascades. What is system's dimension? If it is four-dimensional in principle it is possible to have a three-dimensional Poincaré section and chaos can rise due to breakdown of invariant torus. – Evgeny Sep 20 '18 at 18:25
  • Hey Evgeny, after more carefull inspection of the phase space, I have been able to detect period-doubling cascades, and a Bogdanov-Takens point also implies that a Homoclinic bifurcation takes place, although I have not yet been able to continue it. – 1233023 Sep 21 '18 at 15:42
  • Okay then! By the way, while Bogdanov-Takens implies that there is a homoclinic bifurcation nearby, it doesn't provide you non-trivial dynamics, only a limit cycle due to separatrix splitting. Is it a part of your research problem? – Evgeny Sep 21 '18 at 16:37
  • Right, but homoclinic bifurcation of a saddle-focus equilibrium with proper eigenvalues implies shilnikov bifurcation, and I think my prof. would be very happy indeed, if I was to discover such a bifurcation. – 1233023 Sep 22 '18 at 10:14
  • Yeah, indeed. Note that while homoclinic to saddle-focus is a sure way to organize complex behaviour, it is not the only way. If your system possesses additional structure (like symmetries), heteroclinic structures might also play role in organizing complex behaviour. – Evgeny Sep 22 '18 at 13:38
  • Mm Okay. I don't know enough about equivariant dynamical systems, but I trust my advisor would have guided me in this direction, if he thought it relevant. But thank you anyway for the interest! Wish I could be a bit more detailed with regards to the equations, but c'est la vie – 1233023 Sep 23 '18 at 13:26
  • Yeah, I agree, usually you know that the system is equivariant from the beginning and use this fact quite a lot :) – Evgeny Sep 28 '18 at 18:26
  • Hey Evgeny. You have been an aid for me a couple of times, and now I'd like to ask you again. Poincaré sections. I'm doing simulations of 3D system in XPP, however the Poincaré sectioning in this software is relatively limited. How do people do e.g. feigenbaum diagrams? As far as I understand coding a Poincaré script is not straightforward. Is there some industry software or similar? Regards – 1233023 Oct 22 '18 at 16:40
  • PyDSTool has such capability (computing intersections of trajectory with Poincaré section), but it fails to compute long trajectories: if you use C or Fortran integrators, it crashes with segfault. Unless you have a way to restart the run, it is useless. I've heard that there is a new ND&DS package in Julia. Here are few examples of how they do it. I've used a lot it myself yet, but in CoCalc this example runs quite smoothly. – Evgeny Oct 22 '18 at 17:14
  • Correction to previous comment: I've only ran examples from their website, I haven't used this package a lot myself yet. But it looks very promising. – Evgeny Oct 22 '18 at 17:53
  • Thank you Evgeny, Julia looks promising, I'll have a go at it. – 1233023 Oct 22 '18 at 18:28
  • Hey Evgeny, thank you for your help so far, I have a final softquestion: How rare is it to find a system which exhibits chaos through a Shilnikov bifurcation? I'm not quite sure if this is the usual road to observing chaos, or whether it is a more rare find outside of the classic systems, like Lorenz, Chua etc. – 1233023 Oct 25 '18 at 15:56
  • It's a tricky question for me. To some extent it is "a chicken and egg problem". If you just calculate attractors, you will quite often see a period doubling of limit cycles leadind to positive LLE. If you have a parameter plane, chaotic domains can be often organized by homoclinic or heteroclinic bifurcations. But Shilnikov bifurcation alone is not a guarantee for a chaotic attractor: Shilnikov bifurcation says that you have a complex hyperbolic set, similar to Smale horseshoe, but it can be non-attracting. That's the caveat. There must be something else to verify that it is attracting. – Evgeny Oct 25 '18 at 19:05
  • Right, but even if it's not an attracting chaotic regime, it stills exists. The reason that I'm interested in Shilnikov bifurcation, is that it of course theoretically explains why the Chaotic regions appear, which is a lot more interesting than just establishing that chaos exist in the considered system. – 1233023 Oct 25 '18 at 19:10
  • If you are okay with non-attractiveness, then it's good :) Many people are concerned with understanding what exactly are these attractors with strange looking shape (and whether they are really attractors or not). – Evgeny Oct 26 '18 at 08:13
  • My theme is a lot more applied, so I'm not really interested in describing the attractor or finding descriptors like i.e. fractal dimension. The reason I'm asking, is that I see a lot of papers on Chaos, of course, however fewer where they actually establish that this is linked to a Shilnikov bifurcation. – 1233023 Oct 26 '18 at 12:44
  • It would be interesting to hear why chaos matters in the problem that you are studying. I probably understand some aspects of chaos (more mathematically than practically), but still learning why it can be important in applications (besides the "hey, guys, not everything is stationary or periodic in our models, some strange non-stationary non-periodic behaviour is also there"). – Evgeny Oct 29 '18 at 18:42
  • Hey Evgeny, I'm in the subfield of synchronization and so called chimera states in groups of oscillators. Chaos in phase space is a so called turbulent chimera, where the synchronicity degree oscillates chaotically. – 1233023 Oct 29 '18 at 22:46
  • That’s funny: I wanted to mention chimeras as an example of something that is a bit unclear what it is in phase space from dynamical point of view. As far as I’ve read, it is still unclear: are some of the observed chimeras just (sometimes chaotic) transients or attractors (maybe in some appropriate sense). – Evgeny Oct 30 '18 at 04:04
  • Even the definition of a chimera is somewhat debated I guess, but in this particular case you can view it from a dynamical system standpoint and apply bifurcation theory to describe the phenomenons. – 1233023 Oct 30 '18 at 10:16
  • Is it a definition by Ashwin and Burylko? Weak chimera? – Evgeny Oct 30 '18 at 10:26
  • Yes right, the one by Ashwin and Burylko – 1233023 Oct 30 '18 at 10:50

1 Answers1

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Let $λ_1$ denote the largest Lyapunov exponent of the system, $λ_2$ the second-largest, and so on.

The absolute value of $λ_1$ says little about its reliability, but mostly depends on how time and the states of your system scale. If $\dot{x} = f(x)$ is the differential equation of the Lorenz system you are considering (with $x∈ℝ^3$), then $\dot{x} = \tfrac{0.02}{0.905} f(x)$ is a dynamical system with $λ_1=0.02$ and the same (chaotic) attractor that just moves along its trajectories more slowly. In fact, many classical chaotic systems have such small Lyapunov exponents in their most popular form.

So, what you can do to ensure that your Lyapunov exponent is really positive is this:

  • Each bounded, non-fixed-point, continuous-time system has at least one zero Lyapunov exponent. Therefore, you can perform conclusions based on differences in the magnitude of $λ_1$ and $λ_2$, more specifically:

    • If $λ_2 ≪ -|λ_1| ≈ 0$, your largest exponent is zero.

    • If $λ_1 ≫ |λ_2| ≈ 0$, your largest exponent is positive.

    • If $|λ_1| ≈ |λ_2| ≈ 0$, you should increase the time used for determining Lyapunov exponents. You may also have a quasi-periodic dynamics.

    • If $|λ_2| ≫ |λ_3| ≈ 0$ or similar, you have multiple positive Lyapunov exponents, i.e., hyperchaos.

  • You usually obtain $λ_1$ by averaging over instantaneous Lyapunov exponents, i.e.,

    $$λ_1 = \frac{1}{n}\sum_{i=1}^n λ_1(t_i).$$

    You can apply a statistical test, to see whether the distribution of the $λ_1(t_i)$ actually has a mean that is significantly different from zero, e.g., Student’s one-sample $t$-test. Note that these tests require that the samples, i.e., the different $λ_1(t_i)$, are independent. Therefore you need to ensure that $t_{i+1}-t_i$ is sufficiently large, e.g., one oscillation of the system.

Wrzlprmft
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  • Tangent question: I understand that a positive Lyapunov exponent is necessary for chaos, but it seems to me that it is necessary but not sufficient. If you take the Lyapunov exponent of an unstable linear system, it will still be positive even though it is not chaotic, no? – Steve Heim Sep 08 '18 at 13:32
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    @SteveHeim: Yes, $\dot{x} = x$ has a positive Lyapunov exponent but is not chaotic. However, such unbounded dynamics is usually easy to distinguish from a chaotic one. – Wrzlprmft Sep 08 '18 at 15:54
  • Thank you very much, for such a well rounded answer – 1233023 Sep 09 '18 at 09:00
  • Wrzlprmft, can you recommend a code for calculating the Lyapunov spectrum? Calculating the largest is relatively straightforward, but I can't seem to find good examples on how to find the whole spectrum, except for datasets, which isn't really relevant in my case. – 1233023 Sep 10 '18 at 09:07
  • @1233023: My own software can calculate more than multiple Lyapunov exponents almost automatically. Note that you often do not need the whole spectrum, but only the two largest ones. Also note that not requiring the second Lyapunov exponent is one of the crucial advantages of the second approach I offered. – Wrzlprmft Sep 10 '18 at 09:45
  • Thanks! Yes, I guess it might be easier just to apply a random number generator element to the calculation, and then do the t-test. – 1233023 Sep 10 '18 at 09:49
  • @1233023: I am not exactly sure what you will use random numbers for. You can use them for the initial condition, but they are not really crucial to the process. – Wrzlprmft Sep 10 '18 at 09:51
  • I thought the independence of the calculated exponents relied on the independence of the initial conditions – 1233023 Sep 10 '18 at 09:54
  • @1233023: That’s one way to do it, but computationally expensive. It suffices to keep enough temporal distance between the instantaneous Lyapunov exponents. – Wrzlprmft Sep 10 '18 at 10:06