Hartman-Grobman theorem

The Hartman-Grobman , also known as linearization set , says that the behavior of a dynamic system of in the form of autonomous system of differential equations in the vicinity of a hyperbolic fixed point the behavior of at this point linearized similar system. Hyperbolic fixed point means that none of the eigenvalues ​​of the linearized system has the real part zero.

The phrase is named after the American Philip Hartman and the Russian David Grobman , who published the phrase independently in 1960 and 1959, respectively.

According to the theorem, one can deduce the behavior of a nonlinear system locally from that of the linearized equations in the vicinity of such a fixed point.

sentence

The differential equation system is after development with the Taylor formula around the fixed point, which is, by the figure : ${\displaystyle (x,y)=(0,0)}$

${\displaystyle T:(x,y)\to ({\dot {x}},{\dot {y}})}$

${\displaystyle T:{\dot {x}}=Ax+X(x,y),\,\,\,{\dot {y}}=By+Y(x,y)}$

given with the nonlinear remainder terms ${\displaystyle X,Y\in C^{1}}$

${\displaystyle X,Y=o(\left|x\right|+\left|y\right|)}$for . ${\displaystyle (x,y)\to 0}$

and the constant matrices and . The vector space is already divided so that the eigenvalues with a positive real part of the linearized system are in B, the eigenvalues with a negative real part in A: ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle n}$ ${\displaystyle b_{1},...,b_{\nu }}$ ${\displaystyle m}$ ${\displaystyle a_{1},...,a_{\mu }}$

${\displaystyle Re(a_{m})<0<Re(b_{n})}$for or . ${\displaystyle 1\leqq m\leqq \mu }$ ${\displaystyle 1\leqq n\leqq \nu }$

Then there is a homeomorphism

${\displaystyle R:u=U(x,y),v=V(x,y)}$

between an environment of to an environment of such that ${\displaystyle (x,y)=0}$ ${\displaystyle (u,v)=0}$

${\displaystyle RTR^{-1}=L}$

With

${\displaystyle L:{\dot {u}}=Au,{\dot {v}}=Bv}$.

More generally, a system of the form can with bringing through a linear coordinate transformation always above form if all eigenvalues of non-zero real part have. ${\displaystyle {\dot {x}}=Cx+Z(x)}$ ${\displaystyle Z=o(\left|x\right|)}$ ${\displaystyle C}$

example

Be

${\displaystyle T:{\dot {x}}=-x,{\dot {y}}=y+x^{2}}$.

The only fixed point of the system is . Then ${\displaystyle (0,0)}$

${\displaystyle J_{T}((0,0))={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}}$

the Jacobi matrix at this point, with the linearization of the system accordingly ${\displaystyle r=(u,v)}$

${\displaystyle {\dot {r}}={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}r}$,

so

${\displaystyle {\tilde {T}}:{\dot {u}}=-u,{\dot {v}}=v}$.

The eigenvalues of , ${\displaystyle J_{T}((0,0))}$

${\displaystyle \lambda _{1}=1,\lambda _{2}=-1}$,

have real parts different from zero, thus a hyperbolic fixed point and the requirements of the Hartman-Grobman theorem are fulfilled. Since the eigenvalues ​​have different signs, it is a saddle point and thus an unstable fixed point. According to theorem, this applies not only to the linearized system, but also to the original system. ${\displaystyle (0,0)}$

literature

• DM Grobman: О гомеоморфизме систем дифференциальных уравнений. Docl. Akad. Nauk SSSR 128, 1959, pp. 880-881.
• Philip Hartman: A Lemma in the Theory of Structural Stability of Differential equations . (PDF; 800 kB) In: Proc. Amer. Math. Soc. , 11, 1960, pp. 610-620.
• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.a ).