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I have this nonlinear differential equation $d\textbf{x}/dt=f(\textbf{x})$, where $\textbf{x}\in \mathbb{R}^n$. There are results which guarantee the convergence of the dynamical system to $\textbf{x}=\textbf{0}$, and the simplest of them deal with linearising the equation about $\textbf{x}=\textbf{0}$.

But I would love to say more about the convergence. For example if $f(\textbf{x})=A\textbf{x}$, then we have results bounding the exponential convergence, with rate of convergence related to the greatest eigenvalue of $A$. Are there linearising or approximating results pertaining to the exponential stability of dynamical systems when $f(\textbf{x})$ is nonlinear? I would love to be directed to textbooks or journal papers.

EDIT: I would like to add one more relevant question. It can be proved that if the linearised system is exponentially stable, then the nonlinear system also is exponentially stable. Does the converse hold? Is there a necessary and sufficient condition using Lyapunov methods?

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  • $\begingroup$ Define $y(t) = e^{at}x(t)$ so that $\dot{y}(t) = e^{at}f(e^{-at}y) + ay = (f'(0) + a)y + O(y^2)$ with $y \equiv 0$ a solution. If all of the eigenvalues of A have real part less than -a, then by the result you quote $y(t) \to 0$ hence $x(t) \to 0$ at the exponential rate $\sim e^{-at}$. The keywords "exponential weight" and "stable manifold" might be useful. $\endgroup$
    – Aaron Hoffman
    Commented Nov 3, 2012 at 14:42

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Here is a statement, due to Lagrange. Take a square system $\dot x=f(x)$ with $f$ of class $C^2$ such that $f(0)=0$ and the spectrum of $df(0)$ is contained in {$z\in \mathbb C , \Re{z}\le -\delta$} for some positive $\delta$. There exists a neighborhood $V$ of 0 and a constant $C$ such that for all $a\in V$, the unique solution of the IVP $\dot x=f(x), x(0)=a$ exists globally in positive times and is such that $$ \Vert x(t)\Vert\le C e^{-\delta t/2}. $$ In other words a strong stability property of the linearized equation pertains to the non-linear system. Of course, one cannot get the same with $\delta=0$: just think about the scalar $\dot x=x^2$ whose all solutions with positive initial value blow-up in a finite time in the future (negative data trigger blow-up in the past). Also a strong unstability property of the linearized equation (existence of an eigenvalue of $df(0)$ with positive real part) triggers unstability for the non-linear equation.

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  • $\begingroup$ Thanks, so is it only a sufficient condition? Does a necessary function involving Lyapunov functions exist? Could you please provide me with a source where I can learn more? $\endgroup$
    – Bravo
    Commented Nov 3, 2012 at 13:42
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I don't understand if you are thinking of the finite dimensional case only; but Chapter 9 in Lunardi's monograph is a rich source of information about exponential convergence of solutions to nonlinear equations.

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