Siegel modular form: Difference between revisions
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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. | |||
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 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> | |||
Analog of the conjecture for nonlinear control system with scalar nonlinearity is known as [[Kalman's conjecture]]. | |||
== Mathematical statement of conjecture == | |||
: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>. | |||
:Then <math>0</math> is a global attractor of the dynamical system <math>\dot{x}= f(x)</math>. | |||
The conjecture is true for <math>n=2</math> and false in general for <math>n>2</math>. | |||
==Notes== | |||
<references/> | |||
== References == | |||
* L. Markus and H. Yamabe, "Global Stability Criteria for Differential Systems", ''Osaka Math J.'' '''12''':305–317 (1960) | |||
* Gary Meisters, ''[http://www.math.unl.edu/~gmeisters1/papers/HK1996.pdf 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) | |||
* 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]) | |||
* 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]) | |||
{{DEFAULTSORT:Markus-Yamabe conjecture}} | |||
[[Category:Conjectures]] | |||
[[Category:Stability theory]] | |||
[[Category:Fixed points (mathematics)]] | |||
[[Category:Theorems in dynamical systems]] | |||
Revision as of 22:47, 5 June 2013
In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. The conjecture states that if a continuously differentiable map on an -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] Analog of the conjecture for nonlinear control system with scalar nonlinearity is known as Kalman's conjecture.
Mathematical statement of conjecture
The conjecture is true for and false in general for .
Notes
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)
- Bragin V.O., Vagaitsev V.I., Kuznetsov N.V., Leonov G.A., "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) (doi: 10.1134/S106423071104006X)
- Leonov G.A., Kuznetsov N.V., "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) (doi: 10.1142/S0218127413300024)