How reliable a measure of chaos is the largest Lyapunov exponent?

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?

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.