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| In [[mathematics]], in the study of [[dynamical systems]], the '''Hartman–Grobman theorem''' or '''linearization theorem''' is a theorem about the local behaviour of dynamical systems in the [[neighbourhood (mathematics)|neighbourhood]] of a [[hyperbolic equilibrium point]].
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| Basically the theorem states that the behaviour of a dynamical system near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its [[linearization]] near this equilibrium point provided that no eigenvalue of the linearization has its real part equal to 0. Therefore when dealing with such fixed points one can use the simpler linearization of the system to analyze its behaviour. That is, linearisation is 'unreasonably' effective.
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| == Main theorem ==
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| Let <math>f: \mathbb{R}^n \to \mathbb{R}^n</math>
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| be a [[smooth map]] of a dynamical system with differential equation <math>{du}/{dt}=f(u)</math>. Suppose the map has a hyperbolic equilibrium point <math>u_0</math>: that is, <math>f(u_0)=0</math> and the [[Jacobian matrix]] <math>A=[\partial f_i/\partial x_j]</math> of <math>f</math> at point <math>u_0</math> has no [[eigenvalue]] with real part equal to zero. Then there exists a neighborhood <math>N</math> of the equilibrium <math>u_0</math> and a [[homeomorphism]] <math>h : N \to \mathbb{R}^n</math>,
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| such that <math>h(u_0)=0</math>
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| and such that in the neighbourhood <math>N</math> the [[flow (mathematics)|flow]] of <math>du/dt=f(u)</math> is [[topologically conjugate]] by the smooth map <math>U=h(u)</math> to the flow of its linearization <math>dU/dt=AU</math>.<ref>{{cite journal|last = Grobman|first = D. M.|title=О гомеоморфизме систем дифференциальных уравнений|trans_title = Homeomorphisms of systems of differential equations|journal = [[Doklady Akademii Nauk SSSR]]|volume = 128|pages = 880–881|year = 1959}}</ref><ref>{{cite journal|last = Hartman|first = Philip|authorlink=Philip Hartman|title = A lemma in the theory of structural stability of differential equations|journal = Proc. A.M.S.|volume = 11|issue = 4|pages = 610–620|accessdate = 2010-05-28|doi = 10.2307/2034720|date=August 1960|jstor=2034720}}</ref><ref>{{cite journal|last = Hartman|first = Philip|title = On local homeomorphisms of Euclidean spaces|journal = Bol. Soc. Math. Mexicana|volume = 5|pages = 220–241|year = 1960}}</ref>
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| In general, even for infinitely differentiable maps <math>f</math>, the homeomorphism <math>h</math> need not to be smooth, nor even locally Lipschitz. However, it turns out to be [[Hölder continuous]], with an exponent depending on the constant of hyperbolicity of <math>A</math>.
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| ==Example==
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| The algebra necessary for this example is easily carried out by a web service that computes [[Algebraic normal form|normal form]] coordinate transforms of systems of differential equations, autonomous or non-autonomous, deterministic or stochastic [http://www.maths.adelaide.edu.au/anthony.roberts/sdenf.php]. | |
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| Consider the 2D system in variables <math>u=(y,z)</math> evolving according to
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| :<math>dy/dt=-3y+yz\quad\text{and}\quad dz/dt=z+y^2.</math>
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| This system has an equilibrium at the origin, that is <math>u_0=0</math>, among others not analysed here. The coordinate transform, <math>u=h^{-1}(U)</math> where <math>U=(Y,Z)</math>, given by
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| :<math>y\approx Y+YZ+\dfrac1{42}Y^3+\dfrac12YZ^2</math>
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| :<math>z\approx Z-\dfrac17Y^2-\dfrac13Y^2Z</math>
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| is a smooth map between the original <math>u=(y,z)</math> and new <math>U=(Y,Z)</math> coordinates, at least near the equilibrium at the origin. In the new coordinates the dynamical system transforms to its linearisation | |
| :<math>dY/dt=-3Y\quad\text{and}\quad dZ/dt=Z.</math> | |
| That is, a distorted version of the linearization gives the original dynamics in some finite neighbourhood.
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| ==References==
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| <references/>
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| ==External links==
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| *{{cite journal|last = Coayla-Teran|first = E.|coauthors = Mohammed, S. and Ruffino, P.|title = Hartman–Grobman Theorems along Hyperbolic Stationary Trajectories|journal = Discrete and Continuous Dynamical Systems|volume = 17|issue = 2|pages = 281–292|date=February 2007|url = http://sfde.math.siu.edu/Hartmangrobman.pdf|format=PDF|accessdate = 2007-03-09|doi = 10.3934/dcds.2007.17.281}}
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| * {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
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| {{DEFAULTSORT:Hartman-Grobman Theorem}}
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| [[Category:Theorems in analysis]]
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| [[Category:Theorems in dynamical systems]]
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