Some examples of generalized reflectionless Schrödinger potentials
Russell Johnson Luca Zampogni
Discrete & Continuous Dynamical Systems - S 2016, 9(4): 1149-1170 doi: 10.3934/dcdss.2016046
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.
keywords: reflectionless potential in the sense of Craig glueing property divisor map. Generalized reflectionless potential
On linear-quadratic dissipative control processes with time-varying coefficients
Roberta Fabbri Russell Johnson Sylvia Novo Carmen Núñez
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 193-210 doi: 10.3934/dcds.2013.33.193
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.
keywords: dissipativity Linear-quadratic control system supply rate storage function.
A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles
Russell Johnson Francesca Mantellini
Discrete & Continuous Dynamical Systems - A 2003, 9(1): 209-224 doi: 10.3934/dcds.2003.9.209
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.
keywords: parameter intermittence. almost automorphic minimal set Averaging method bifurcation
Minimal subsets of projective flows
Kristian Bjerklöv Russell Johnson
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 493-516 doi: 10.3934/dcdsb.2008.9.493
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.
keywords: almost periodic coefficients. projective flows minimal subsets Li-Yorke chaos
Remarks on linear-quadratic dissipative control systems
Russell Johnson Carmen Núñez
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 889-914 doi: 10.3934/dcdsb.2015.20.889
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.
keywords: Linear-quadratic control systems Hamiltonian systems dissipativity weak disconjugacy.
Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials
Russell Johnson Luca Zampogni
Discrete & Continuous Dynamical Systems - B 2010, 14(2): 559-586 doi: 10.3934/dcdsb.2010.14.559
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.
keywords: stationary ergodic processes. Sato-Segal-Wilson potentials Generalized reflectionless potentials
On the inverse Sturm-Liouville problem
Russell Johnson Luca Zampogni
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 405-428 doi: 10.3934/dcds.2007.18.405
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.
keywords: nonautonomous differential equations. algebraic curves Sturm-Liouville operator
The Kalman-Bucy filter revisited
Russell Johnson Carmen Núñez
Discrete & Continuous Dynamical Systems - A 2014, 34(10): 4139-4153 doi: 10.3934/dcds.2014.34.4139
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.
keywords: rotation number. Nonautonomous control Kalman-Bucy filter exponential dichotomy error covariance matrix
Russell Johnson Roberta Fabbri Sylvia Novo Carmen Núñez Rafael Obaya
Discrete & Continuous Dynamical Systems - S 2016, 9(4): i-iii doi: 10.3934/dcdss.201604i
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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On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows
Russell Johnson Mahesh G. Nerurkar
Communications on Pure & Applied Analysis 2011, 10(3): 873-884 doi: 10.3934/cpaa.2011.10.873
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 $q_n\to \infty$.

keywords: Lyapunov exponents Cocycles irrational rotations Liouville numbers.

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