Lipschitz conjugacy of linear flows. (English) Zbl 1191.34060

Kawan, C.; Stender, T.

J. Lond. Math. Soc., II. Ser. 80, No. 3, 699-715 (2009).

Let \(\Phi(t,x)=\exp(At)x\) and \(\Psi(t,x)=\exp(Bt)x\) be the linear flows induced by autonomous linear systems \(x'=Ax\) and \(x'=Bx\) in a Euclidean space. It is well known that, in the hyperbolic case, the flows \(\Phi\) and \(\Psi\) are topologically conjugate if and only if the matrices \(A\) and \(B\) have the same number of eigenvalues with negative real parts. At the same time, the flows \(\Phi\) and \(\Psi\) are \(C^1\)-conjugate if and only if they are linearly conjugate.
The authors consider an intermediate case of Lipschitz conjugacy and show that the flows \(\Phi\) and \(\Psi\) are bi-Lipschitz conjugate if and only if the real Jordan forms of the matrices \(A\) and \(B\) coincide with the exception of simple Jordan blocks, where the imaginary parts of the eigenvalues may differ.

Reviewer: Sergei Yu. Pilyugin (St. Petersburg)

MSC:

34C41	Equivalence and asymptotic equivalence of ordinary differential equations
34A30	Linear ordinary differential equations and systems, general