A non-autonomous bifurcation theory for deterministic scalar differential equations
Carmen Núñez Rafael Obaya
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 701-730 doi: 10.3934/dcdsb.2008.9.701
In the extension of the concepts of saddle-node, transcritical and pitchfork bifurcations to the non-autonomous case, one considers the change in the number and attraction properties of the minimal sets for the skew-product flow determined by the initial one-parametric equation. In this work conditions on the coefficients of the equation ensuring the existence of a global bifurcation phenomenon of each one of the types mentioned are established. Special attention is paid to show the importance of the non-trivial almost automorphic extensions and pinched sets in describing the dynamics at the bifurcation point.
keywords: pitchfork bifurcation. almost periodic dynamics almost automorphic dynamics Non-autonomus dynamical systems saddle-node bifurcation transcritical bifurcation
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.
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.
Null controllable sets and reachable sets for nonautonomous linear control systems
Roberta Fabbri Sylvia Novo Carmen Núñez Rafael Obaya
Discrete & Continuous Dynamical Systems - S 2016, 9(4): 1069-1094 doi: 10.3934/dcdss.2016042
Under the assumption of lack of uniform controllability for a family of time-dependent linear control systems, we study the dimension, topological structure and other dynamical properties of the sets of null controllable points and of the sets of reachable points. In particular, when the space of null controllable vectors has constant dimension for all the systems of the family, we find a closed invariant subbundle where the uniform null controllability holds. Finally, we associate a family of linear Hamiltonian systems to the control family and assume that it has an exponential dichotomy in order to relate the space of null controllable vectors to one of the Lagrange planes of the continuous hyperbolic splitting.
keywords: reachable sets abnormal systems proper focal points. null controllable sets rotation number linear Hamiltonian systems Nonautonomous linear control systems
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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Skew-product semiflows for non-autonomous partial functional differential equations with delay
Sylvia Novo Carmen Núñez Rafael Obaya Ana M. Sanz
Discrete & Continuous Dynamical Systems - A 2014, 34(10): 4291-4321 doi: 10.3934/dcds.2014.34.4291
A detailed dynamical study of the skew-product semiflows induced by families of AFDEs with infinite delay on a Banach space is carried over. Applications are given for families of non-autonomous quasimonotone reaction-diffusion PFDEs with delay in the nonlinear reaction terms, both with finite and infinite delay. In this monotone setting, relations among the classical concepts of sub and super solutions and the dynamical concept of semi-equilibria are established, and some results on the existence of minimal semiflows with a particular dynamical structure are derived.
keywords: infinite delay. skew-product semiflows Topological dynamics finite delay abstract functional differential equations partial functional differential equations
Roberta Fabbri Carmen Núñez
Communications on Pure & Applied Analysis 2011, 10(3): i-iii doi: 10.3934/cpaa.2011.10.3i
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.
keywords: no
Roberta Fabbri Russell Johnson Carmen Núñez Rafael Obaya
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): i-ii doi: 10.3934/dcdsb.2008.9.3i
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.

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A perturbation theorem for linear Hamiltonian systems with bounded orbits
Roberta Fabbri Carmen Núñez Ana M. Sanz
Discrete & Continuous Dynamical Systems - A 2005, 13(3): 623-635 doi: 10.3934/dcds.2005.13.623
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.
keywords: Linear Hamiltonian systems Sacker-Sell spectral decomposition perturbation theorems.

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