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| In [[mathematics]], the '''Markus–Yamabe conjecture''' is a [[conjecture]] on global [[asymptotic stability]]. The conjecture states that if a [[Smooth function|continuously differentiable]] map on an <math>n</math>-dimensional [[Real number|real]] [[vector space]] has a single [[Fixed point (mathematics)|fixed point]], and its [[Jacobian matrix]] is everywhere [[Hurwitz matrix|Hurwitz]], then the fixed point is globally stable.
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| 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'''.
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| Related mathematical results concerning global asymptotic stability, which ''are'' applicable in dimensions higher than two, include various [[autonomous convergence theorem]]s. A modified version of the Markus–Yamabe conjecture has been proposed, but at present this new conjecture remains unproven.<ref>See, for example, [http://www.projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ojm/1200689999].</ref>
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| Analog of the conjecture for nonlinear control system with scalar nonlinearity is known as [[Kalman's conjecture]].
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| == Mathematical statement of conjecture ==
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| :Let <math>f:\mathbb{R}^n\rightarrow\mathbb{R}^n</math> be a <math>C^1</math> map with <math>f(0) = 0</math> and Jacobian <math>Df(x)</math> which is Hurwitz stable for every <math>x \in \mathbb{R}^n</math>.
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| :Then <math>0</math> is a global attractor of the dynamical system <math>\dot{x}= f(x)</math>.
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| The conjecture is true for <math>n=2</math> and false in general for <math>n>2</math>.
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| ==Notes==
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| <references/>
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| == References ==
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| * L. Markus and H. Yamabe, "Global Stability Criteria for Differential Systems", ''Osaka Math J.'' '''12''':305–317 (1960)
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| * Gary Meisters, ''[http://www.math.unl.edu/~gmeisters1/papers/HK1996.pdf A Biography of the Markus–Yamabe Conjecture]'' (1996)
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| * C. Gutierrez, "A solution to the bidimensional Global Asymptotic Stability Conjecture", ''Ann. Inst. H. Poincaré Anal. Non Linéaire'' '''12''': 627–671 (1995).
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| * R. Feßler, "A proof of the two-dimensional Markus–Yamabe stability conjecture and a generalisation", ''Ann. Polon. Math.'' '''62''':45–47 (1995)
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| * A. Cima et al., "A Polynomial Counterexample to the Markus–Yamabe Conjecture", ''Advances in Mathematics'' '''131'''(2):453–457 (1997)
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| * 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)
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| * Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A., [http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalamn-conjectures.pdf "Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits"], ''Journal of Computer and Systems Sciences International'' '''50'''(5):511–543, (2011) ([http://dx.doi.org/10.1134/S106423071104006X doi: 10.1134/S106423071104006X])
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| * Leonov G.A., Kuznetsov N.V., [http://www.worldscientific.com/doi/pdf/10.1142/S0218127413300024 "Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits"], ''International Journal of Bifurcation and Chaos'' '''23'''(1): art. no. 1330002, (2013) ([http://dx.doi.org/10.1142/S0218127413300024 doi: 10.1142/S0218127413300024])
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| {{DEFAULTSORT:Markus-Yamabe conjecture}}
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| [[Category:Conjectures]]
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| [[Category:Stability theory]]
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Theorems in dynamical systems]]
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Nice to meet you, my name is Araceli Oquendo but I don't like when people use my complete title. To play croquet is the hobby I will by no means quit performing. Bookkeeping is what she does. I've always loved living in Idaho.
Feel free to visit my webpage; extended auto warranty