Markus-Yamabe conjecture

In mathematics, the Markus-Yamabe conjecture is a conjecture on global asymptotic stability. The conjecture states that if a continuously differentiable map on an $n$-dimensional real vector space has a single fixed point, and its Jacobian matrix is everywhere Hurwitz, then the fixed point is globally stable.

The conjecture is true for the two-dimensional case. However, counterexamples have been constructed in higher dimensions. Hence, in the two-dimensional case "only", it can also be referred to as the Markus-Yamabe theorem.

Related mathematical results concerning global asymptotic stability, which "are" applicable in dimensions higher than two, include various autonomous convergence theorems. A modified version of the Markus-Yamabe conjecture has been proposed, but at present this new conjecture remains unproven. [See, for example, [http://www.math.ualberta.ca/~mli/research/ps_files/stable.pdf] .]

Mathematical statement of conjecture

:Let $f:mathbb\{R\}^n\; ightarrowmathbb\{R\}^n$ be a $C^1$ map with $f(0)\; =\; 0$ and Jacobian $Df(x)$ which is Hurwitz stable for every $x\; in\; mathbb\{R\}^n$.

:Then $0$ is a global attractor of the dynamical system $dot\{x\}=\; f(x)$.

The conjecture is true for $n=2$ and false in general for $n>2$.

Notes

References

* L. Markus and H. Yamabe, "Global Stability Criteria for Differential Systems", "Osaka Math J." 12:305-317 (1960)
* Gary Meisters, " [http://www.math.unl.edu/~gmeisters1/papers/HK1996.pdf A Biography of the Markus-Yamabe Conjecture] " (1996)
* C. Gutierrez, A solution to the bidimensional Global Asymptotic Stability Conjecture, "Ann. Inst. H. Poincaré Anal. Non Linéaire" 12: 627–671 (1995).
* R. Feßler, A proof of the two-dimensional Markus-Yamabe stability conjecture and a generalisation, "Ann. Polon. Math." 62:45-47 (1995)
* A. Cima et al, "A Polynomial Counterexample to the Markus-Yamabe Conjecture", "Advances in Mathematics" 131(2):453-457 (1997)
* Josep Bernat and Jaume Llibre, "Counterexample to Kalman and Markus-Yamabe Conjectures in dimension larger than 3", "Dynam. Contin. Discrete Impuls. Systems" 2(3):337-379, (1996)