Largest Lyapunov Exponent

Question

Lyapunov Exponents $\lambda$ are found from $|\Delta\mathbb{X}(x_0,t)| \approx e^{\lambda t}|\Delta\mathbb{X}_0|$ where the initial separation of two trajectories $\mathbb{X}(t)$ and $\mathbb{X}_0(t)$ in phase space is $\Delta\mathbb{X}_0$.

Now there may exist $\geq 1$ possible values for $\lambda$ as the initial separation vector can have different orientations.

What is the relevance of the Largest Lyapunov Exponent in determining chaos of a dynamical system compared to the set of all other possible exponents? Does it follow from the relevance of the largest Eigenvalue in stability analysis?

Thank you!

Answer 1

It follows from the fact that chaos arises from high divergence of points which were close initially. only highest exponent is relevant as it will determine maximal divergence (by monotonicity of $e^{\lambda t}$ in $\lambda$)