Markus-Yamabe conjecture

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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]

[edit] Mathematical statement of conjecture

Let f:\mathbb{R}^n\rightarrow\mathbb{R}^n be a C1 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.

[edit] Notes

  1. ^ 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)