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
If
is a 
map with
- f(0) = 0
and Jacobian
- Df(x)
which is Hurwitz 
, then 0 is a global attractor of the dynamical system
.
[edit] Notes
[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)
