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On perturbations of linear systems with quasiperiodic coefficients. (Russian) Zbl 0712.34068

For a linear system \(\dot x=a(t)x+b(t)y,\quad \dot y=c(t)x+d(t)y\) with quasiperiodic coefficients \(a,b,c,d=f_ i(\phi_ 1(t),\phi_ 2(t+\alpha_ 2),...,\phi_ m(t+\alpha_ m),\mu),\) \(i=1,2,3,4\), \(\alpha_ 2,...,\alpha_ m\), \(\mu\) parameters, \(\phi_ j\) periodic with frequency \(\omega_ j\), \(j=1,...,m\), \(k_ 1\omega_ 1+...+k_ m\omega_ m\neq 0\) for each nontrivial set of integers \(k_ j\), the dependence of the characteristic exponents on the parameters \(\alpha\),\(\mu\) and on linear and nonlinear perturbations is investigated. In particular, for the second order equation \[ \ddot x+(\beta \cos t+\gamma \cos \sqrt{2}(t+\alpha))x=0, \] the largest characteristic exponent and the average number of zeros on a segment of length \(2\pi\) are determined in a series of numerical calculations for parameter values \(0\leq \beta \leq 5\), \(0\leq \gamma \leq 8\).
Reviewer: W.Müller

MSC:

34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34A30 Linear ordinary differential equations and systems
34D10 Perturbations of ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
65J99 Numerical analysis in abstract spaces
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