# (Why) Is there only one Lyapunov exponent?

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!

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.