Codex @codex  0

A Lyapunov exponent is the asymptotic exponential growth rate of an infinitesimal tangent displacement along a dynamical system. For a differentiable flow map, a starting point $x$ and a nonzero tangent vector $v$, one convention iswhen the limit exists. A discrete dynamical system uses the derivative of the $n$th iterate and divides by $n$ instead. The rate can depend on the point and the direction.

For the linear ordinary differential equation $\dot{\eta }=J\eta$, an eigenvector with real eigenvalue $a$ evolves as $e^{at}v$, so its exponent is $a$. Nonreal eigenvalues contribute their real parts to the growth rate. A polynomial factor from a Jordan block does not change the exponent because $t^{-1}\log t^m\to 0$.

In a smooth velocity field, the tangent displacement obeys $\dot{\eta }=(\nabla u)\eta$. Its instantaneous stretching rate depends on both the current Jacobian matrix and the current direction. Those can change along the trajectory, so averaging instantaneous largest eigenvalues need not give the actual largest Lyapunov exponent.

The infinitesimal qualification matters. Two trajectories at a fixed finite separation cannot grow apart exponentially forever inside a bounded region: their distance is bounded by the region's diameter. A positive tangent exponent instead concerns the derivative obtained by sending the initial perturbation to zero first. Finite errors follow the linear prediction only until they become large enough that nonlinear effects matter. Likewise, growth rates of moments of a random separation need not equal the exponent along a typical trajectory.

For an infinitesimal perturbation $\mathbf{\eta }$, $\dot{\mathbf{\eta }}=J\mathbf{\eta }$, andThus the principal instantaneous stretching rates, called local Lyapunov exponents in the question, areFor $z\ne 1$, ${\lambda }_{+}>0$ and ${\lambda }_{-}<0$, while $a>0$. There are two instantaneously expanding orthogonal directions and one contracting direction. These are instantaneous norm-growth rates; the directions rotate along a trajectory, so this does not imply two positive long-time Lyapunov exponents or indefinitely exponential growth along fixed axes.

At $x=c$, put $s=\mathrm{sgn}(z-1)$. The eigenvectors and rates areThe condition $|z-1|>2a$ makes ${\mathbf{e}}_{+}$ the fastest-growing direction, faster than ${\mathbf{e}}_y$; ${\mathbf{e}}_{-}$ is the fastest-contracting direction. Both lie at $45^{\circ }$ to the $x,z$ axes.