Markus-Yamabe conjecture

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In mathematics, the Markus-Yamabe Conjecture is a mathematical conjecture on global asymptotic stability. In words, it says 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 attracting.

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.

Similar tools for proving global asymptotic stability, which are applicable in dimensions higher than two, include variants of an Autonomous Convergence Theorem.^[1] Additionally in this paper, a modified version of the Markus-Yamabe Conjecture is proposed. At present, this new conjecture remains unproven.

[edit] Mathematical statement of conjecture

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

is a $\mathcal{C}^1$ map with

f (0) = 0

and Jacobian

D f (x)

which is Hurwitz $\forall x \in \mathbb{R}^n$ , then $0$ is a global attractor of the dynamical system

$\frac{\mathrm{d}x}{\mathrm{d}t}= f(x)$ .

[edit] Notes

^ See, for example, [1].

[edit] 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)