On the Jacobian conjecture for global asymptotic stability

Abstract

An old conjecture says that, for the two-dimensional system of ordinary differential equationsx=f(x), wheref: ℝ² → ℝ²,f εC ¹, andf(0)=0 the originx=0 should beglobally asymptotically stable (i.e., a stable equilibrium and all trajectoriesx(t) converge to it ast → +∞) whenever the following conditions on the Jacobian matrixJ(x) off hold: trJ(x) < 0, detJ(x) > 0, ∀x ε ℝ² It is known that if such anf is globallyone-to-one as a mapping of the plane into itself, then the origin is a globally asymptotically stable equilibrium point for the systemx =f(x). In this paper we outline a new strategy to tackle the injectivity off, based on anauxiliary boundary value problem. The strategy is shown to be successful if the norm of the matrixJ(x) ^T J(x)t/det J(x) is bounded or, at least, grows slowly (for instance, linearly) as ¦x¦ → t∞.