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State u(t) == R^t

Main theorem

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Consider a system evolving in time with state that satisfies the differential equation for some smooth map . Suppose the map has a hyperbolic equilibrium state : that is, and the Jacobian matrix of at state has no eigenvalue with real part equal to zero. Then there exists a neighbourhood of the equilibrium and a homeomorphism , such that and such that in the neighbourhood the flow of is topologically conjugate by the continuous map to the flow of its linearisation .[1][2][3][4]

Even for infinitely differentiable maps , the homeomorphism 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 .[5]

The Hartman–Grobman theorem has been extended to infinite-dimensional Banach spaces, non-autonomous systems (potentially stochastic), and to cater for the topological differences that occur when there are eigenvalues with zero or near-zero real-part.[6][7][8][9]

  1. ^ Grobman, D. M. (1959). "О гомеоморфизме систем дифференциальных уравнений" [Homeomorphisms of systems of differential equations]. Doklady Akademii Nauk SSSR. 128: 880–881.
  2. ^ 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.
  3. ^ Hartman, Philip (1960). "On local homeomorphisms of Euclidean spaces". Bol. Soc. Math. Mexicana. 5: 220–241.
  4. ^ Chicone, C. (2006). Ordinary Differential Equations with Applications. Texts in Applied Mathematics. Vol. 34 (2nd ed.). Springer. ISBN 978-0-387-30769-5.
  5. ^ Belitskii, Genrich; Rayskin, Victoria (2011). "On the Grobman–Hartman theorem in α-Hölder class for Banach spaces" (PDF). Working paper.
  6. ^ 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.
  7. ^ 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.
  8. ^ 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.
  9. ^ 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.