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!
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 $\lambda$)