Hartman–Grobman theorem

Consider a system evolving in time with state ${\displaystyle u(t)\in \mathbb {R} ^{n}}$ that satisfies the differential equation ${\displaystyle du/dt=f(u)}$ for some smooth map ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$. Now suppose the map has a hyperbolic equilibrium state ${\displaystyle u^{*}\in \mathbb {R} ^{n}}$: that is, ${\displaystyle f(u^{*})=0}$ and the Jacobian matrix ${\displaystyle A=[\partial f_{i}/\partial x_{j}]}$ of ${\displaystyle f}$ at state ${\displaystyle u^{*}}$ has no eigenvalue with real part equal to zero. Then there exists a neighbourhood ${\displaystyle N}$ of the equilibrium ${\displaystyle u^{*}}$ and a homeomorphism ${\displaystyle h:N\to \mathbb {R} ^{n}}$, such that ${\displaystyle h(u^{*})=0}$ and such that in the neighbourhood ${\displaystyle N}$ the flow of ${\displaystyle du/dt=f(u)}$ is topologically conjugate by the continuous map ${\displaystyle U=h(u)}$ to the flow of its linearisation ${\displaystyle dU/dt=AU}$.[2][3][4][5]

Even for infinitely differentiable maps ${\displaystyle f}$, the homeomorphism ${\displaystyle h}$ 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 ${\displaystyle A}$.[6]

The Hartman–Grobman theorem has been extended to infinite-dimensional Banach spaces, non-autonomous systems ${\displaystyle du/dt=f(u,t)}$ (potentially stochastic), and to cater for the topological differences that occur when there are eigenvalues with zero or near-zero real-part.[7][8][9][10]

The algebra necessary for this example is easily carried out by a web service that computes normal form coordinate transforms of systems of differential equations, autonomous or non-autonomous, deterministic or stochastic.[11]

Consider the 2D system in variables ${\displaystyle u=(y,z)}$ evolving according to the pair of coupled differential equations

${\displaystyle {\frac {dy}{dt}}=-3y+yz\quad {\text{and}}\quad {\frac {dz}{dt}}=z+y^{2}.}$

By direct computation it can be seen that the only equilibrium of this system lies at the origin, that is ${\displaystyle u^{*}=0}$. The coordinate transform, ${\displaystyle u=h^{-1}(U)}$ where ${\displaystyle U=(Y,Z)}$, given by

${\displaystyle {\begin{aligned}y&\approx Y+YZ+{\dfrac {1}{42}}Y^{3}+{\dfrac {1}{2}}YZ^{2}\\[5pt]z&\approx Z-{\dfrac {1}{7}}Y^{2}-{\dfrac {1}{3}}Y^{2}Z\end{aligned}}}$

is a smooth map between the original ${\displaystyle u=(y,z)}$ and new ${\displaystyle U=(Y,Z)}$ coordinates, at least near the equilibrium at the origin. In the new coordinates the dynamical system transforms to its linearisation

${\displaystyle {\frac {dY}{dt}}=-3Y\quad {\text{and}}\quad {\frac {dZ}{dt}}=Z.}$

That is, a distorted version of the linearisation gives the original dynamics in some finite neighbourhood.

• Linear approximation
• Stable manifold theorem

• Irwin, Michael C. (2001). "Linearization". Smooth Dynamical Systems. World Scientific. pp. 109–142. ISBN 981-02-4599-8.
• Perko, Lawrence (2001). Differential Equations and Dynamical Systems (Third ed.). New York: Springer. pp. 119–127. ISBN 0-387-95116-4.
• Robinson, Clark (1995). Dynamical Systems : Stability, Symbolic Dynamics, and Chaos. Boca Raton: CRC Press. pp. 156–165. ISBN 0-8493-8493-1.

• Coayla-Teran, E.; Mohammed, S.; Ruffino, P. (February 2007). "Hartman–Grobman Theorems along Hyperbolic Stationary Trajectories". Discrete and Continuous Dynamical Systems. 17 (2): 281–292. doi:10.3934/dcds.2007.17.281.
• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
• "The Most Addictive Theorem in Applied Mathematics". Scientific American.