An old conjecture says that, for the two-dimensional system of ordinary differential equations $$\dot x=f(x)\,,\quad \hbox{where $f\colon\R^2\to\R^2$, $f\in C^1$ and $f(0)=0$}\,,$$ the origin $x=0$ should be \it globally asymptotically stable \rm (i.e., a stable equilibrium and all trajectories $x(t)$ converge to it as $t\to+\infty$) whenever the following conditions on the Jacobian matrix $J(x)$ of $f$ hold: $$\tr J(x)<0\,,\quad \det J(x)>0\qquad \forall x\in\R^2\,.$$ It is known that if such an $f$ is globally {\it one-to-one} as a mapping of the plane into itself, then the origin is a globally asymptotically stable equilibrium point for the system $\dot x=f(x)$. In this paper we outline a new strategy to tackle the injectivity of~$f$, based on an {\it auxiliary boundary value problem}. The strategy is shown to be successful if the norm of the matrix $J(x)^TJ(x)/ \det J(x)$ is bounded, or, at least, grows slowly (for instance, linearly) as~$|x|\to+\infty$.
On the Jacobian conjecture for global asymptotic stability
ZAMPIERI, Gaetano;
1992
Abstract
An old conjecture says that, for the two-dimensional system of ordinary differential equations $$\dot x=f(x)\,,\quad \hbox{where $f\colon\R^2\to\R^2$, $f\in C^1$ and $f(0)=0$}\,,$$ the origin $x=0$ should be \it globally asymptotically stable \rm (i.e., a stable equilibrium and all trajectories $x(t)$ converge to it as $t\to+\infty$) whenever the following conditions on the Jacobian matrix $J(x)$ of $f$ hold: $$\tr J(x)<0\,,\quad \det J(x)>0\qquad \forall x\in\R^2\,.$$ It is known that if such an $f$ is globally {\it one-to-one} as a mapping of the plane into itself, then the origin is a globally asymptotically stable equilibrium point for the system $\dot x=f(x)$. In this paper we outline a new strategy to tackle the injectivity of~$f$, based on an {\it auxiliary boundary value problem}. The strategy is shown to be successful if the norm of the matrix $J(x)^TJ(x)/ \det J(x)$ is bounded, or, at least, grows slowly (for instance, linearly) as~$|x|\to+\infty$.
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