Abstract
We shall be considering a system of equations of the form
, where a \(a,P,f \in {{C}^{r}}\left( {{{\mathcal{T}}_{m}}} \right),\phi = \left( {{{\phi }_{1}}, \ldots ,{{\phi }_{m}}} \right),x = \left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right)\). Underlining its linearity with respect to the variable x we shall call it a linear non-homogeneous system of equations defined on the direct product of the m-dimensional torus T m and the Euclidean space En. An invariant manifold of system of equations (1.1) of the form
where u ∈Cs(T m ), will be called an m-dimensional s times continuously differentiable invariant torus of system (1.1). For s = 0 an invariant manifold of system (1.1) of the form (1.2) will be called an invariant torus of this system.
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© 1991 Springer Science+Business Media Dordrecht
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Samoilenko, A.M. (1991). Some problems of the linear theory. In: Elements of the Mathematical Theory of Multi-Frequency Oscillations. Mathematics and Its Applications, vol 71. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3520-7_3
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DOI: https://doi.org/10.1007/978-94-011-3520-7_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5557-4
Online ISBN: 978-94-011-3520-7
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