Hartman–Grobman theorem

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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 behavior of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearization—a natural simplification of the system—is effective in predicting qualitative patterns of behavior.

The theorem states that the behavior of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behavior of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyze its behavior 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 neighborhood ${\displaystyle N}$ 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 linearization ${\displaystyle dU/dt=AU}$.[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

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

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

${\displaystyle y\approx Y+YZ+{\dfrac {1}{42}}Y^{3}+{\dfrac {1}{2}}YZ^{2}}$
${\displaystyle z\approx Z-{\dfrac {1}{7}}Y^{2}-{\dfrac {1}{3}}Y^{2}Z}$

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

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

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

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 0-412-39080-9.
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 0-387-30769-9.
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. Six Lectures on Dynamical Systems. Singapore: World Scientific. pp. 45–119. ISBN 981-02-2548-2.
8. ^ Aulbach, B.; Wanner, T. (1999). "Invariant Foliations for Carathéodory Type Differential Equations in Banach Spaces". In Lakshmikantham, V.; Martynyuk, A. A. Advances in Stability Theory at the End of the 20th Century. Gordon & Breach. ISBN 0-415-26962-8.
9. ^ Aulbach, B.; Wanner, T. (2000). "The Hartman–Grobman theorem for Caratheodory-type differential equations in Banach spaces". Non-linear Analysis. 40: 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: 12–38. arXiv:. 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.