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Communications on Pure & Applied Analysis

2011 , Volume 10 , Issue 3

Special Issue on Nonautonomous Dynamical Systems and Applications

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Preface
Roberta Fabbri and  Carmen Núñez
2011, 10(3): i-iii doi: 10.3934/cpaa.2011.10.3i +[Abstract](47) +[PDF](110.5KB)
Abstract:
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.
A remark about Sil'nikov saddle-focus homoclinic orbits
Flaviano Battelli and  Ken Palmer
2011, 10(3): 817-830 doi: 10.3934/cpaa.2011.10.817 +[Abstract](38) +[PDF](334.6KB)
Abstract:
In this note we study Sil'nikov saddle-focus homoclinic orbits paying particular attention to four and higher dimensions where two additional conditions are needed. We give equivalent conditions in terms of subspaces associated with the variational equation along the orbit. Then we review Rodriguez's construction of a three-dimensional system with Sil'nikov saddle-focus homoclinic orbits and finally show how to construct higher-dimensional systems with such orbits.
Inertial manifolds for stochastic pde with dynamical boundary conditions
Peter Brune and  Björn Schmalfuss
2011, 10(3): 831-846 doi: 10.3934/cpaa.2011.10.831 +[Abstract](43) +[PDF](400.6KB)
Abstract:
In this article we investigate the dynamics of stochastic partial differential equations with dynamical boundary conditions. We prove that such a problem with Lipschitz continuous non--linearity generates a random dynamical system. The main result is to show that this random dynamical system has an inertial manifold. Under additional assumptions on the non--linearity this manifold is differentiable.
Topological conjugacy for affine-linear flows and control systems
Fritz Colonius and  Alexandre J. Santana
2011, 10(3): 847-857 doi: 10.3934/cpaa.2011.10.847 +[Abstract](30) +[PDF](325.6KB)
Abstract:
Hyperbolic affine-linear flows on vector bundles possess unique bounded solutions on the real line. Hence they are topologically skew conjugate to their linear parts. This is used to show a classification of inhomogeneous bilinear control systems.
Spectral properties of limit-periodic Schrödinger operators
David Damanik and  Zheng Gan
2011, 10(3): 859-871 doi: 10.3934/cpaa.2011.10.859 +[Abstract](35) +[PDF](374.0KB)
Abstract:
We investigate the spectral properties of Schrödinger operators in $l^2(Z)$ with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show that for any such base dynamics, the spectrum is of positive Lebesgue measure and purely absolutely continuous for a dense set of sampling functions, and it is of zero Lebesgue measure and purely singular continuous for a dense $G_\delta$ set of sampling functions.
On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows
Russell Johnson and  Mahesh G. Nerurkar
2011, 10(3): 873-884 doi: 10.3934/cpaa.2011.10.873 +[Abstract](27) +[PDF](385.3KB)
Abstract:
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 $q_n\to \infty$.

Robust exponential attractors for non-autonomous equations with memory
Peter E. Kloeden , José Real and  Chunyou Sun
2011, 10(3): 885-915 doi: 10.3934/cpaa.2011.10.885 +[Abstract](66) +[PDF](549.9KB)
Abstract:
The aim of this paper is to consider the robustness of exponential attractors for non-autonomous dynamical systems with line memory which is expressed through convolution integrals. Some properties useful for dealing with the memory term for non-autonomous case are presented. Then, we illustrate the abstract results by applying them to the non-autonomous strongly damped wave equations with linear memory and critical nonlinearity.
Feedback control via inertial manifolds for nonautonomous evolution equations
Norbert Koksch and  Stefan Siegmund
2011, 10(3): 917-936 doi: 10.3934/cpaa.2011.10.917 +[Abstract](31) +[PDF](429.7KB)
Abstract:
In this paper we extend a method to control the dynamics of evolution equations by finite dimensional controllers which was suggested by Brunovsky [3] to nonautonomous evolution equations using nonautonomous inertial manifold theory.
Nonautonomous continuation of bounded solutions
Christian Pötzsche
2011, 10(3): 937-961 doi: 10.3934/cpaa.2011.10.937 +[Abstract](33) +[PDF](509.1KB)
Abstract:
We show the persistence of hyperbolic bounded solutions to nonautonomous difference and retarded functional differential equations under parameter perturbation, where hyperbolicity is given in terms of an exponential dichotomy in variation. Our functional-analytical approach is based on a formulation of dynamical systems as operator equations in ambient sequence or function spaces. This yields short proofs, in particular of the stable manifold theorem.
As an ad hoc application, the behavior of hyperbolic solutions and stable manifolds for ODEs under numerical discretization with varying step-sizes is studied.
On finite-time hyperbolicity
Arno Berger
2011, 10(3): 963-981 doi: 10.3934/cpaa.2011.10.963 +[Abstract](41) +[PDF](314.8KB)
Abstract:
A solution of a nonautonomous ordinary differential equation is finite-time hyperbolic, i.e. hyperbolic on a compact interval of time, if the linearisation along that solution exhibits a strong exponential dichotomy. As a finite-time variant and strengthening of classical asymptotic facts, it is shown that finite-time hyperbolicity guarantees the existence of stable and unstable manifolds of the appropriate dimensions. Eigenvalues and -vectors are often unsuitable for detecting hyperbolicity. A (dynamic) partition of the extended phase space is used to circumvent this difficulty. It is proved that any solution staying clear of the elliptic and degenerate parts of the partition is finite-time hyperbolic. This extends and unifies earlier partial results.
A note on almost periodic variational equations
Peter Giesl and  Martin Rasmussen
2011, 10(3): 983-994 doi: 10.3934/cpaa.2011.10.983 +[Abstract](43) +[PDF](565.5KB)
Abstract:
The variational equation of a nonautonomous differential equation $\dot x= F(t,x)$ along a solution $\mu$ is given by $\dot x=D_x F(t,\mu(t))x$. We consider the question whether the variational equation is almost periodic provided that the original equation is almost periodic by a discussion of the following problem: Is the derivative $D_xF$ almost periodic whenever $F$ is almost periodic? We give a negative answer in this paper, and the counterexample relies on an explicit construction of a scalar almost periodic function whose derivative is not almost periodic. Moreover, we provide a necessary and sufficient condition for the derivative $D_xF$ to be almost periodic.
In addition, we also discuss this problem in the discrete case by considering the variational equation $x_{n+1}=D_xF(n,\mu_n)x_n$ of the almost periodic difference equation $x_{n+1}=F(n,x_n)$ along an almost periodic solution $\mu_n$. In particular, we provide an example of a function $F$ which is discrete almost periodic uniformly in $x$ and whose derivative $D_xF$ is not discrete almost periodic.
Neuronal coding of pacemaker neurons -- A random dynamical systems approach
T. Jäger
2011, 10(3): 995-1009 doi: 10.3934/cpaa.2011.10.995 +[Abstract](31) +[PDF](4602.0KB)
Abstract:
The behaviour of neurons under the influence of periodic external input has been modelled very successfully by circle maps. The aim of this note is to extend certain aspects of this analysis to a much more general class of forcing processes. We apply results on the fibred rotation number of randomly forced circle maps to show the uniqueness of the asymptotic firing frequency of ergodically forced pacemaker neurons. In particular, this allows to treat the forced leaky integrate-and-fire model, which serves as a paradigm example.

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