## 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

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.

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CPAA

We show that a generic $SL(2,R)$ valued cocycle in the class of
$C^r$, ($0 < r < 1$) cocycles based on a rotation flow on the $d$-torus, is either uniformly hyperbolic or has zero Lyapunov exponents provided that the components of winding vector $\bar \gamma = (\gamma^1,\cdot \cdot \cdot,\gamma^d)$ of the rotation flow are rationally independent and
satisfy the following super Liouvillian condition :

$ |\gamma^i - \frac{p^i_n}{q_n}| \leq Ce^{-q^{1+\delta}_n}, 1\leq i\leq d, n\in N,$

where $C > 0$ and $\delta > 0$ are some constants and $p^i_n, q_n$ are some sequences of integers with $q_n\to \infty$.

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