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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?

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  • $\begingroup$ 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. $\endgroup$
    – Evgeny
    Sep 20, 2018 at 18:25
  • $\begingroup$ 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. $\endgroup$
    – 1233023
    Sep 21, 2018 at 15:42
  • $\begingroup$ 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? $\endgroup$
    – Evgeny
    Sep 21, 2018 at 16:37
  • $\begingroup$ 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. $\endgroup$
    – 1233023
    Sep 22, 2018 at 10:14
  • $\begingroup$ 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. $\endgroup$
    – Evgeny
    Sep 22, 2018 at 13:38

1 Answer 1

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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.

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  • $\begingroup$ 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? $\endgroup$
    – Steve Heim
    Sep 8, 2018 at 13:32
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    $\begingroup$ @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. $\endgroup$
    – Wrzlprmft
    Sep 8, 2018 at 15:54
  • $\begingroup$ Thank you very much, for such a well rounded answer $\endgroup$
    – 1233023
    Sep 9, 2018 at 9:00
  • $\begingroup$ 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. $\endgroup$
    – 1233023
    Sep 10, 2018 at 9:07
  • $\begingroup$ @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. $\endgroup$
    – Wrzlprmft
    Sep 10, 2018 at 9:45

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