## Journals

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### Open Access Journals

DCDS-S

the class of generalized reflectionless Schrödinger operators was introduced by Lundina in 1985. Marchenko worked out a useful parametrization of these potentials, and Kotani showed that each such potential is of Sato-Segal-Wilson type. Nevertheless the dynamics under translation of a generic generalized reflectionless potential is still not well understood. We give examples which show that certain dynamical anomalies can occur.

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

We study the
phenomenon of stability breakdown for non-autonomous differential
equations whose time dependence is determined by a minimal,
strictly ergodic flow. We find that, under appropriate assumptions,
a new attractor may appear. More generally, almost automorphic
minimal sets are found.

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

We study the minimal subsets of the projective flow defined by a two-dimensional linear differential system with almost periodic coefficients. We show that such a minimal set may exhibit Li-Yorke chaos and discuss specific examples in which this phenomenon is present. We then give a classification of these minimal sets, and use it to discuss the bounded mean motion property relative to the projective flow.

DCDS-B

We study the concept of dissipativity in the sense of Willems for
nonautonomous linear-quadratic (LQ) control systems.
A nonautonomous system of Hamiltonian ODEs is associated with such
an LQ system by way of the Pontryagin Maximum Principle.
We relate the concepts of exponential dichotomy and weak disconjugacy
for this Hamiltonian ODE to that of dissipativity for the LQ system.

DCDS-B

The class of generalized reflectionless Schrödinger potentials was introduced by Marchenko-Lundina and was analyzed by Kotani.
We state and prove various results concerning those stationary ergodic processes of Schrödinger potentials which are contained in this class.

DCDS

We pose and solve an inverse problem of an algebro-geometric type
for the classical Sturm-Liouville operator. We use techniques of
nonautonomous dynamical systems together with methods of classical
algebraic geometry.

DCDS

We study the properties of the error covariance matrix and the asymptotic error covariance
matrix of the Kalman-Bucy filter model with time-varying coefficients. We make use of such
techniques of the theory of nonautonomous differential systems as the exponential dichotomy
concept and the rotation number.

DCDS-S

Generally speaking, the term nonautonomous dynamics refers to the systematic use of dynamical tools to study the solutions of differential or difference equations with time-varying coefficients. The nature of the time variance may range from periodicity at one extreme, through Bohr almost periodicity, Birkhoff recurrence, Poisson recurrence etc. to stochasticity at the other extreme. The ``dynamical tools'' include almost everywhere Lyapunov exponents, exponential splittings, rotation numbers, and the theory of cocycles, but are by no means limited to these. Of course in practise one uses whatever ``works'' in the context of a given problem, so one usually finds dynamical methods used in conjunction with those of numerical analysis, spectral theory, the calculus of variations, and many other fields. The reader will find illustrations of this fact in all the papers of the present volume.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

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