Consider $V=\mathbb{R}^d\times\mathbb{R}^n$ with coordinate $x^T=[\theta^T,\sigma^T]$. I have an ODE of the form: $\dot{x}=F(x)$, where $F$ is assumed to be sufficiently smooth.

Suppose that there's some $\delta>0$ so that all trajectories of the ODE satisfy the inequality $$ \frac{d}{dt}(x^Tx)=\frac{d}{dt}(\theta^T\theta+\sigma^T\sigma)\le-\delta \theta ^T \theta $$ Barbalat's lemma allows us to conclude that:

- $\theta\to0$ as $t\to\infty$
- There exists some $C>0$ such that $\sigma^T\sigma\to C$

I want to prove that in this case, $\theta \to 0$ exponentially fast, with exponent depending on $\delta$. If $F$ was an affine function, $F(x)=Ax+b$ then one can prove this result by writing $x(0)$ as a linear combination of generalized eigenvectors for $A$. Is there a well-known result for the general, nonlinear case?