Mini-workshop on Dynamical Systems and Nonautonomous Differential Equations

Flaviano Battelli

Homoclinic orbits, Transversality and Chaos in Singular Systems

Abstract: We consider the problem of existence of homoclinic orbits in singular systems of differential equations. We relate this to some kind of chaotic behaviour of the system (Sil'nikov chaos) collecting some old and new results on this topic.

Irene Benedetti

Delay evolution semilinear differential inclusions with impulses: existence of solutions on compact and noncompact domains

Abstract: This talk deals with an impulsive Cauchy problem with delay governed by the semilinear evolution differential inclusion x'(t) ∈ A(t)x(t) + F(t,x_t), where {A(t)}_t∈[0,b] is a family of linear operators (not necessarily bounded) in a Banach space E generating an evolution operator and F is a Carathéodory type multifunction. At first a theorem on the compactness of the set of all mild solutions for the problem is given.
Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on non compact domains.

Pierluigi Benevieri

Global Branches of Periodic Solutions for Forced Delay Differential Equations on Compact Manifolds

Abstract: We study a nonautonomous delay differential equation of the first order,

x'(t) = a f(t,x(t),x(t-1)),

with constant time lag, given in a multidimensional Euclidean space and depending on the nonnegative real parameter a. We focus our attention on solutions lying entirely on a given compact ANR (absolute neighborhood retract) and we prove a global bifurcation theorem for periodic solutions. More precisely, we assume the ANR has nonzero Euler-Poincaré characteristic, we suppose that f is periodic with respect to the first variable (with period T greater than 1) and satisfies some inward conditions along the boundary of the ANR.
This work is directly motivated by the purpose of extending previous results obtained by M. Furi and M.P. Pera in which global bifurcation is proved for the undelayed equation

x'(t) = a f(t,x(t)).

In their works a topological method based on the fixed point index for maps on ANR's is applied. Our approach here makes use of the same tool.
We point out that the assumption T>1 is crucial for the method used here, based on fixed point theory for locally compact maps. It is possible tackle the case when T<1, by a slight different methond since the Poincaré-type translation operator is not locally compact (actually not locally condensing).

Alessandro Calamai

A degree theory for a class of perturbed Fredholm maps

Abstracts:In a recent paper we gave a notion of degree for a class of perturbations of nonlinear Fredholm maps of index zero between real infinite dimensional Banach spaces, called α-Fredholm maps. These maps are defined in terms of α, i.e. the Kuratowski measure of noncompactness.
Our purpose here is to extend that notion in order to include the degree introduced by Nussbaum for local α-condensing perturbations of the identity, as well as the degree for locally compact perturbations of Fredholm maps of index zero (quasi-Fredholm maps for short) recently defined by Benevieri and Furi.
We present also an application of the degree for α-Fredholm maps. We prove a bifurcation result for a boundary value problem depending on a parameter, establishing conditions for the existence of atypical bifurcation points (in the sense of Prodi and Ambrosetti) for the given problem.
This is a joint work with P. Benevieri and M. Furi

Roberta Fabbri

A perturbation theorem for linear Hamiltonian systems with bounded orbits

Abstract:The Sacker-Sell spectral decomposition of a one-parametric perturbation of a nonautonomous linear Hamiltonian system with bounded solutions is considered: 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 they are related to the the topological structure of the flow induced by the solutions of the system on the real and complex Langrange bundles.
Joint work with Carmen Núñez and Ana Sanz, from Universidad de Valladolid.

Russell Johnson

Integral manifolds for nonautonomous systems with fast oscillations

Abstract: We use methods of ergodic theory and topological dynamics together with results of Hirsh-Pugh-Shub and Irwin-Foster to prove an integral manifold theorem for differential systems of the form

x' = ε f(t, x, ε)

The talk is based on joint work with R. Fabbri and K.Palmer.

Serena Matucci

Travelling wave solutions for reaction diffusion-aggregation equations

Abstract:In this talk, the existence of travelling wave solutions for a class of reaction-diffusion equations is analyzed and necessary and sufficient conditions for existence are given. In the considered equations, both the reaction and the diffusion term are density dependent and may change sign to take into account different biological phenomena such as aggregation of individuals. The problem of existence of wave fronts leads to a BVP on the real line for a second order nonlinear differential equation, which is analyzed by means of phase-plane techniques, comparison theorems and upper and lower solutions method. The degenerate case is also considered and the appearence of different types of sharp profiles is proved.

Carmen Núñez:

Almost Automorphic Minimal Sets and Strange Non-Chaotic Attractors

Abstract: During the last years a lot of attention has been paid to the analysis of the so-called strange non-chaotic attractors (SNA). In this work we show their connection with the well known almost automorphic but not almost periodic minimal sets appearing in the dynamical description of the projective flows induced by two-dimensional linear systems of ODEs, including discrete and continuous one-dimensional Schrödinger equations.
The work has been made in collaboration with Àngel Jorba, Rafael Obaya, and Joan Carles Tatjer.

Rafael Ortega

Stability and index of periodic solutions of a nonlinear telegraph equation

Abstract:Given a non-autonomous evolution equation and a periodic solution, the index is defined in terms of the topological degree. This index is usually employed in proofs of existence and multiplicity but sometimes it is also useful to discuss the stability properties of the periodic solution. This fact will be applied to certain wave equations with dissipation.

Stefan Siegmund

Hyperbolicity and Invariant Manifolds for Planar Nonautonomous Systems on Finite Time Intervals

Abstract: The method of invariant manifolds was originally developed for hyperbolic rest points of autonomous equations. It was then extended from fixed points to arbitrary solutions and from autonomous equations to nonautonomous dynamical systems by either the Lypunov-Perron approach or Hadamard's graph transformation.
We go one step further and study meaningful notions of hyperbolicity and stable and unstable manifolds for equations which are defined or known only for a finite time, together with matching notions of attraction and repulsion. As a consequence, hyperbolicity and invariant manifolds will describe the dynamics on the finite time interval.
We prove an analog of the Theorem of Linearized Asymptotic Stability on finite time intervals, generalize the Okubo-Weiss criterion from fluid dynamics and prove a theorem on the location of periodic orbits. Several examples are treated, including a double gyre flow and symmetric vortex merger.

Marco Spadini

Branches of harmonic solutions to periodically perturbed coupled equations on differentiable manifolds

Abstract: Let M⊂R^k and N⊂R^s be boundaryless smooth manifolds, let f:R×M×N→R^k be tangent to M, and let g:M×N→R^s and h:R×M×N→R^s be tangent to N. Given T>0, we assume also that f and h are T-periodic in the first variable. Consider the following system of differential equations for λ≥ 0:

x'=λ f(t,x,y),

y'=g(x,y)+λ h(t,x,y).

Using fixed point index methods, we study the structure of the set of its T-periodic solutions when λ≥0. This leads to unifying results for several known facts about equations of both the forms

x'=λ f(t,x),

and

y'=g(y)+λ h(t,y)

on differentiable manifolds.
The techniques used are inspired by previous work done on the first equation by M. Furi, M.P. Pera, and on the second one by M. Furi, M.P. Pera and myself.
As a bonus, the results obtained can be used to investigate the set of T-periodic solutions of some second order differential equations.

Valentina Taddei

Periodic and bounded trajectories of multivalued maps in Banach spaces

Abstract:We investigate the solvability of the Floquet boundary value problems for an upper Carathéodory semilinear differential inclusion in a separable and reflexive real Banach space. We employ a continuation principle due to Andres-Gabor-Górniewicz, embedding the equation into a one parameter family of linearized problems. The transversality condition required for all the linearized problems is obtained by means of a C¹ Lyapunov like bounding function. An approximation argument of Scorza-Dragoni type allows us to localize the transversality condition. The existence of an entirely bounded solution is obtained in a sequential way. The results are obtained in collaboration with J. Andres and L. Malaguti.

Massimo Villarini

Normalization of Poincaré singularities via variation of constants

Abstract:We present a geometric proof of the Poincaré-Dulac Theorem for analytic vector fields with singularities of Poincaré type.
Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field.
An analogous approach leading to analogous results is considered in the problem of linearization of maps in a neighborhood of a hyperbolic fixed point.

Luca Zampogni

Inverse problem for the Sturm-Liouville operator

Abstract:We use methods of nonautonomous differential equations together with basic techniques of the classical theory of algebraic curves to solve an inverse problem for (p φ')' + q φ = - λ y φ