(Why) Is there only one Lyapunov exponent?

Question

Lyapunov exponents describe how two (infinitesimally) close initial conditions behave (exponentially) in the long run. If a system is chaotic, the largest Lyapunov exponent is positive. However, as far as I understand it, if we have say, three ODEs depending on time, each state variable has one Lyapunov exponent. But how?

If we consider $$ \lambda = \lim_{t \to \infty} \lim_{\delta \mathbf{Z}_0 \to 0} \frac{1}{t} \ln\frac{| \delta\mathbf{Z}(t)|}{|\delta \mathbf{Z}_0|}, $$ in my understanding, this would always tend to zero as $t\to\infty$ if we consider a chaotic attractor?!

What am I getting wrong?

Thank you!

Answer 1

each state variable has one Lyapunov exponent. But how?

The equation you give is for the maximal Lyapunov exponent, while what you're referring to here is Lyapunov spectrum.

You can find how to calculate it in almost any text on chaos theory - Wikipedia as usual could be a good start for getting the overall picture. This previous answer gives some pointers and references for the calculation, and this answer tries to say something about their meaning.

this would always tend to zero as t→∞ if we consider a chaotic attractor?!

Nope. The factor $1/t$ goes linearly to zero, yes, but $\delta\mathbf{Z}(t)$ diverges exponentially for chaotic orbits, meaning that the factor $\ln | \delta\mathbf{Z}(t)|$ should diverge linearly and the whole thing converges, as also explained in this answer for iterated systems.

Also notice the limit $\delta\mathbf{Z}(0) \to 0$, which means that the values that the equations return are local quantities for a specific orbit - so the divergence is local, that's how you can have finite exponents also for spatially confined orbits.