ISSN:

1937-1632

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1937-1179

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## Discrete & Continuous Dynamical Systems - S

August 2016 , Volume 9 , Issue 4

Issue on Nonautonomous Dynamics

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2016, 9(4): i-iii
doi: 10.3934/dcdss.201604i

*+*[Abstract](2163)*+*[PDF](96.6KB)**Abstract:**

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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2016, 9(4): 895-922
doi: 10.3934/dcdss.2016034

*+*[Abstract](2181)*+*[PDF](524.9KB)**Abstract:**

We study for reversible implicit differential equations the bifurcation of bounded solutions connecting singularities in finite time and their approximation by shadowed periodic solutions. Melnikov like condition is derived. Application is given to planar nonlinear RLC system.

2016, 9(4): 923-958
doi: 10.3934/dcdss.2016035

*+*[Abstract](3091)*+*[PDF](4593.8KB)**Abstract:**

The mesochronic velocity is the average of the velocity field along trajectories generated by the same velocity field over a time interval of finite duration. In this paper we classify initial conditions of trajectories evolving in incompressible vector fields according to the character of motion of material around the trajectory. In particular, we provide calculations that can be used to determine the number of expanding directions and the presence of rotation from the characteristic polynomial of the Jacobian matrix of mesochronic velocity. In doing so, we show that (a) the mesochronic velocity can be used to characterize dynamical deformation of three-dimensional volumes, (b) the resulting mesochronic analysis is a finite-time extension of the Okubo--Weiss--Chong analysis of incompressible velocity fields, (c) the two-dimensional mesochronic analysis from Mezić et al. ``A New Mixing Diagnostic and Gulf Oil Spill Movement'', Science

**330**, (2010), 486-–489, extends to three-dimensional state spaces. Theoretical considerations are further supported by numerical computations performed for a dynamical system arising in fluid mechanics, the unsteady Arnold--Beltrami--Childress (ABC) flow.

2016, 9(4): 959-977
doi: 10.3934/dcdss.2016036

*+*[Abstract](2238)*+*[PDF](422.5KB)**Abstract:**

We prove that a Fredholm--type Alternative holds for recurrent equations with sign, extending a previous result by Cieutat and Haraux in [3]. Moreover, we show that this can be seen a particular case of [1] and we provide a solution to an interesting question raised by Hale in [6]. Finally we characterize the existence of exponential dichotomies also in the nonrecurrent case.

2016, 9(4): 979-994
doi: 10.3934/dcdss.2016037

*+*[Abstract](2765)*+*[PDF](370.4KB)**Abstract:**

The structure of attractors for differential equations is one of the main topics in the qualitative theory of dynamical systems. However, the theory is still in its infancy in the case of multivalued dynamical systems. In this paper we study in detail the structure and internal dynamics of a scalar differential equation, both in the autonomous and non-autonomous cases. To this aim, we will also show a general result on the characterization of a pullback attractor for a multivalued process by the union of all the complete bounded trajectories of the system.

2016, 9(4): 995-1007
doi: 10.3934/dcdss.2016038

*+*[Abstract](3349)*+*[PDF](410.0KB)**Abstract:**

Our aim in this paper is to investigate the openness and denseness for the set of integrally separated systems in the space of bounded linear random differential equations equipped with the $L^{\infty}$-metric. We show that in the general case, the set of integrally separated systems is open and dense. An exception is the case when the base space is isomorphic to the ergodic rotation flow of the unit circle, in which the set of integrally separated systems is open but not dense.

2016, 9(4): 1009-1023
doi: 10.3934/dcdss.2016039

*+*[Abstract](2830)*+*[PDF](389.3KB)**Abstract:**

We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of GL$(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the set of points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.

2016, 9(4): 1025-1038
doi: 10.3934/dcdss.2016040

*+*[Abstract](2338)*+*[PDF](428.7KB)**Abstract:**

In this paper we study the problem of the existence of homoclinic solutions to a Schrödinger equation of the form \[ x''-V(t)x+x^3=0, \] where is a stepwise potential. The technique of proof is based on a topological method, relying on the properties of the transformation of continuous planar paths (the S.A.P. method), together with the application of the classical Conley-Ważewski's method.

2016, 9(4): 1039-1068
doi: 10.3934/dcdss.2016041

*+*[Abstract](2397)*+*[PDF](2036.2KB)**Abstract:**

In this work we attempt to understand what behavior one should expect of a solution trajectory near $\Sigma$ when $\Sigma$ is attractive, what to expect when $\Sigma$ ceases to be attractive (at generic exit points), and finally we also contrast and compare the behavior of some regularizations proposed in the literature, whereby the piecewise smooth system is replaced by a smooth differential system.

Through analysis and experiments in $\mathbb{R}^3$ and $\mathbb{R}^4$, we will confirm some known facts and provide some important insight: (i) when $\Sigma$ is attractive, a solution trajectory remains near $\Sigma$, viz. sliding on $\Sigma$ is an appropriate idealization (though one cannot a priori decide which sliding vector field should be selected); (ii) when $\Sigma$ loses attractivity (at first order exit conditions), a typical solution trajectory leaves a neighborhood of $\Sigma$; (iii) there is no obvious way to regularize the system so that the regularized trajectory will remain near $\Sigma$ while $\Sigma$ is attractive, and so that it will be leaving (a neighborhood of) $\Sigma$ when $\Sigma$ looses attractivity.

We reach the above conclusions by considering exclusively the given piecewise smooth system, without superimposing any assumption on what kind of dynamics near $\Sigma$ should have been taking place.

2016, 9(4): 1069-1094
doi: 10.3934/dcdss.2016042

*+*[Abstract](3042)*+*[PDF](535.1KB)**Abstract:**

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.

2016, 9(4): 1095-1107
doi: 10.3934/dcdss.2016043

*+*[Abstract](2458)*+*[PDF](1176.4KB)**Abstract:**

The purpose of this paper is to describe a new mechanism of destruction of saddle invariant curves in quasiperiodically forced systems, in which an invariant curve experiments a process of fractalization, that is, the curve gets increasingly wrinkled until it breaks down. The phenomenon resembles the one described for attracting invariant curves in a number of quasiperiodically forced dissipative systems, and that has received the attention in the literature for its connections with the so-called Strange Non-Chaotic Attractors. We present a general conceptual framework that provides a simple unifying mathematical picture for fractalization routes in dissipative and conservative systems.

2016, 9(4): 1109-1118
doi: 10.3934/dcdss.2016044

*+*[Abstract](2102)*+*[PDF](419.0KB)**Abstract:**

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is obtained in the case in which the perturbing function exhibits a time decay.

2016, 9(4): 1119-1148
doi: 10.3934/dcdss.2016045

*+*[Abstract](3073)*+*[PDF](543.9KB)**Abstract:**

Under well-known conditions, a one-parameter family of two-dimensional, autonomous ordinary differential equations admits a supercritical\break Andronov-Hopf bifurcation. Let such a family be perturbed by a non-autonomous term. We analyze the sense in which and some conditions under which the Andronov-Hopf pattern persists under such a perturbation.

2016, 9(4): 1149-1170
doi: 10.3934/dcdss.2016046

*+*[Abstract](2369)*+*[PDF](469.1KB)**Abstract:**

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.

2016, 9(4): 1171-1188
doi: 10.3934/dcdss.2016047

*+*[Abstract](1947)*+*[PDF](1453.4KB)**Abstract:**

The authors have recently introduced an extension of the classical one dimensional (doubling) renormalization operator to the case where the one dimensional map is forced quasiperiodically. In the classic case the dynamics around the fixed point of the operator is key for understanding the bifurcations of one parameter families of one dimensional unimodal maps. Here we perform a similar study of the (linearised) dynamics around the fixed point for further application to quasiperiodically forced unimodal maps.

2016, 9(4): 1189-1199
doi: 10.3934/dcdss.2016048

*+*[Abstract](2347)*+*[PDF](351.6KB)**Abstract:**

An integral formula is given representing the generalized principal Lyapunov exponent for random linear parabolic PDEs. As an application, an upper estimate of the exponent is obtained.

2016, 9(4): 1201-1234
doi: 10.3934/dcdss.2016049

*+*[Abstract](2397)*+*[PDF](522.9KB)**Abstract:**

We study the generic dynamical behaviour of skew-product extensions generated by cocycles arising from equations of forced linear oscillators of special form. This work extends our earlier work on cocycles into compact Lie groups arising from differential equations of special form, (cf. [21]), to the case of non-compact fiber groups of Euclidean type. The earlier techniques do not work in the non-compact case. In the non-compact case one of the main obstacle is the lack of `recurrence'. Thus, our approach to studying Euclidean group extensions is : (i) first, to use a `twisted version' of the so called `conjugation approximation method' and then (ii) to use `geometric-control theoretic methods' developed in our earlier work (cf. [20] and [21]). Even then, our arguments only work for base flows that admit a global Poincaé section, (e.g. for the irrational rotation flows on tori and for certain nil flows). We apply these results to study generic spectral behaviour of the forced quantum harmonic oscillator with time dependent stationary force restricted to satisfy given constraints.

2016, 9(4): 1235-1268
doi: 10.3934/dcdss.2016050

*+*[Abstract](2231)*+*[PDF](666.0KB)**Abstract:**

Topological linearization results typically require solution flows rather than merely semiflows. An exception occurs when the linearization fulfills spectral assumptions met e.g. for scalar reaction-diffusion equations. We employ tools from the geometric theory of nonautonomous dynamical systems in order to extend earlier work by Lu [12] to time-variant evolution equations under corresponding conditions on the Sacker-Sell spectrum of the linear part. Our abstract results are applied to nonautonomous reaction-diffusion and convection equations.

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