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.^[1]

Mathematical statement of conjecture

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ 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

^ See, for example, [1].

References

L. Markus and H. Yamabe, "Global Stability Criteria for Differential Systems", Osaka Math J. 12:305-317 (1960)
Gary Meisters, 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)