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On finitetime hyperbolicity
A note on almost periodic variational equations
1.  Department of Mathematics, University of Sussex, Brighton, BN1 9RF 
2.  Martin Rasmussen, Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom 
In addition, we also discuss this problem in the discrete case by considering the variational equation $x_{n+1}=D_xF(n,\mu_n)x_n$ of the almost periodic difference equation $x_{n+1}=F(n,x_n)$ along an almost periodic solution $\mu_n$. In particular, we provide an example of a function $F$ which is discrete almost periodic uniformly in $x$ and whose derivative $D_xF$ is not discrete almost periodic.
References:
[1] 
C. Corduneanu, "Almost Periodic Functions,", Interscience Tracts in Pure and Applied Mathematics, (). Google Scholar 
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A. M. Fink, "Almost Periodic Differential Equations,", Springer Lecture Notes in Mathematics, (). Google Scholar 
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P. Giesl and M. Rasmussen, Borg's criterion for almost periodic differential equations,, Nonlinear Analysis. Theory, 69 (2008), 3722. Google Scholar 
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G. R. Sell, Nonautonomous differential equations and dynamical systems  I. The basic theory,, Transactions of the American Mathematical Society, 127 (1967), 241. Google Scholar 
show all references
References:
[1] 
C. Corduneanu, "Almost Periodic Functions,", Interscience Tracts in Pure and Applied Mathematics, (). Google Scholar 
[2] 
A. M. Fink, "Almost Periodic Differential Equations,", Springer Lecture Notes in Mathematics, (). Google Scholar 
[3] 
P. Giesl and M. Rasmussen, Borg's criterion for almost periodic differential equations,, Nonlinear Analysis. Theory, 69 (2008), 3722. Google Scholar 
[4] 
G. R. Sell, Nonautonomous differential equations and dynamical systems  I. The basic theory,, Transactions of the American Mathematical Society, 127 (1967), 241. Google Scholar 
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