Autonomous convergence theorem

In mathematics, an autonomous convergence theorem is one of a family of related theorems which give conditions for global asymptotic stability of a continuous dynamical system.

History

The Markus-Yamabe conjecture was formulated as an attempt to give conditions for global stability of continuous dynamical systems in two dimensions. However, the Markus-Yamabe conjecture does not hold for dimensions higher than two, a problem which autonomous convergence theorems attempt to address. The first autonomous convergence theorem was constructed by Russell Smith. This theorem was later refined by Michael Li and James Muldowney.

An example autonomous convergence theorem

A recent autonomous convergence theorem is as follows:

Let $x$ be a vector in some space $X\subseteq \mathbb {R} ^{n}$, evolving according to an autonomous differential equation ${\dot {x}}=f(x)$. Suppose that $X$ is convex and forward invariant under $f$, and that there exists a fixed point ${\hat {x}}\in X$ such that $f({\hat {x}})=0$. If there exists a logarithmic norm $\mu$ such that the Jacobian $J(x)=D_{x}f$ satisfies $\mu (J(x))<0$, then ${\hat {x}}$ is the only fixed point, and it is globally asymptotically stable.