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### Open Access Journals

DCDS

Yakubovich, Fradkov, Hill and Proskurnikov have used the Yaku-bovich Frequency Theorem to prove that a strictly dissipative linear-quadratic control process with periodic coefficients admits a storage function, and various related results. We extend their analysis to the case when the coefficients are bounded uniformly continuous functions.

DCDS-B

In recent years, the area of nonautonomous dynamical systems has matured into a field with recognizable contours together with well-defined themes and methods. Its development has been strongly stimulated by various problems of applied mathematics, and it has in its turn influenced such areas of applied and pure mathematics as spectral theory, stability theory, bifurcation theory, the theory of bounded/recurrent motions, etc. Much work in this field concerns the asymptotic properties of the solutions of a nonautonomous differential or discrete system. However, that is by no means always the case, and the reader will find papers in this volume which are concerned only at a distance or not at all with asymptotic matters.

There is a close relation between the field of nonautonomous dynamical systems and that of stochastic dynamical systems. They can be distinguished to a certain extent by the observation that a nonautonomous dynamical system often arises from the study of a differential or discrete system whose coefficients depend on time, but in a non-stochastic way. The limiting case is that of periodic coefficients, but one is also interested in equations whose coefficients exhibit weaker recurrence properties; for example almost periodicity, Birkhoff recurrence, Poisson recurrence, etc. A distinction also occurs on the methodological level in that topological methods tend to find more application in the former field as compared to the latter (while analytical and ergodic tools are heavily used in both). In any case, some people use the term “random dynamics” to refer to both fields in a more or less interchangeable way.

For the full preface, please click on the Full Text "PDF" button above.

There is a close relation between the field of nonautonomous dynamical systems and that of stochastic dynamical systems. They can be distinguished to a certain extent by the observation that a nonautonomous dynamical system often arises from the study of a differential or discrete system whose coefficients depend on time, but in a non-stochastic way. The limiting case is that of periodic coefficients, but one is also interested in equations whose coefficients exhibit weaker recurrence properties; for example almost periodicity, Birkhoff recurrence, Poisson recurrence, etc. A distinction also occurs on the methodological level in that topological methods tend to find more application in the former field as compared to the latter (while analytical and ergodic tools are heavily used in both). In any case, some people use the term “random dynamics” to refer to both fields in a more or less interchangeable way.

For the full preface, please click on the Full Text "PDF" button above.

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DCDS-B

The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.

CPAA

This special issue collects eleven papers in the general area of nonautonomous
dynamical systems. They contain a rich selection of new results on pure and applied
aspects of the eld.

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DCDS

This paper concerns the Sacker-Sell spectral decomposition of a
one-parametric perturbation of a non-autonomous linear Hamiltonian
system with bounded solutions. Conditions ensuring the continuous
variation with respect to the parameter of the spectral intervals
and subbundles are established. These conditions depend on the
perturbation direction and are closely related to the topological
structure of the flows induced by the initial system on the real
and complex Lagrange bundles.

DCDS

Using methods of the theory of nonautonomous linear differential
systems, namely exponential dichotomies
and rotation numbers, we generalize some aspects of Yakubovich's
Frequency Theorem from periodic control
systems to systems with bounded uniformly continuous coefficients.

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