Hartman–Grobman theorem

In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to Philip Hartman and David M. Grobman.

The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearisation near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearisation has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearisation of the system to analyse its behaviour around equilibria.[1]

Main 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 ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$. 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]

Example

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.

See also

• Stable manifold theorem

References

1. Arrowsmith, D. K.; Place, C. M. (1992). "The Linearization Theorem". Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour. London: Chapman & Hall. pp. 77–81. ISBN 978-0-412-39080-7.
2. Grobman, D. M. (1959). "О гомеоморфизме систем дифференциальных уравнений" [Homeomorphisms of systems of differential equations]. Doklady Akademii Nauk SSSR. 128: 880–881.
3. Hartman, Philip (August 1960). "A lemma in the theory of structural stability of differential equations". Proc. A.M.S. 11 (4): 610–620. doi:10.2307/2034720. JSTOR 2034720.
4. Hartman, Philip (1960). "On local homeomorphisms of Euclidean spaces". Bol. Soc. Math. Mexicana. 5: 220–241.
5. Chicone, C. (2006). Ordinary Differential Equations with Applications. Texts in Applied Mathematics. 34 (2nd ed.). Springer. ISBN 978-0-387-30769-5.
6. Belitskii, Genrich; Rayskin, Victoria (2011). "On the Grobman–Hartman theorem in α-Hölder class for Banach spaces" (PDF). Working paper.
7. Aulbach, B.; Wanner, T. (1996). "Integral manifolds for Caratheodory type differential equations in Banach spaces". In Aulbach, B.; Colonius, F. (eds.). Six Lectures on Dynamical Systems. Singapore: World Scientific. pp. 45–119. ISBN 978-981-02-2548-3.
8. Aulbach, B.; Wanner, T. (1999). "Invariant Foliations for Carathéodory Type Differential Equations in Banach Spaces". In Lakshmikantham, V.; Martynyuk, A. A. (eds.). Advances in Stability Theory at the End of the 20th Century. Gordon & Breach. CiteSeerX 10.1.1.45.5229. ISBN 978-0-415-26962-9.
9. Aulbach, B.; Wanner, T. (2000). "The Hartman–Grobman theorem for Caratheodory-type differential equations in Banach spaces". Non-linear Analysis. 40 (1–8): 91–104. doi:10.1016/S0362-546X(00)85006-3.
10. Roberts, A. J. (2008). "Normal form transforms separate slow and fast modes in stochastic dynamical systems". Physica A. 387 (1): 12–38. arXiv:math/0701623. Bibcode:2008PhyA..387...12R. doi:10.1016/j.physa.2007.08.023.
11. Roberts, A. J. (2007). "Normal form of stochastic or deterministic multiscale differential equations". Archived from the original on November 9, 2013.

Further reading

• 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.

External links

• Coayla-Teran, E.; Mohammed, S.; Ruffino, P. (February 2007). "Hartman–Grobman Theorems along Hyperbolic Stationary Trajectories" (PDF). Discrete and Continuous Dynamical Systems. 17 (2): 281–292. doi:10.3934/dcds.2007.17.281. Archived from the original (PDF) on 2007-07-24. Retrieved 2007-03-09.
• 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.