ISSN:
1572-9222
Keywords:
Jacobian conditions
;
global injectivity
;
global stability
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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∞.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01048154
