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Discrete & Continuous Dynamical Systems - A

2008 , Volume 21 , Issue 4

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Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices
Eric Bedford and  Kyounghee Kim
2008, 21(4): 977-1013 doi: 10.3934/dcds.2008.21.977 +[Abstract](26) +[PDF](795.4KB)
Abstract:
We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set $\mathcal{SC}_q$ of symmetric, cyclic matrices is invariant under $K$. In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$ restricted to $\mathcal{SC}_q$.
Hausdorff dimension of self-affine limit sets with an invariant direction
Krzysztof Barański
2008, 21(4): 1015-1023 doi: 10.3934/dcds.2008.21.1015 +[Abstract](34) +[PDF](229.3KB)
Abstract:
We determine the Hausdorff dimension of self-affine limit sets for some class of iterated function systems in the plane with an invariant direction. In particular, the method applies to some type of generalized non-self-similar Sierpiński triangles. This partially answers a question asked by Falconer and Lammering and extends a result by Lalley and Gatzouras.
Characterization of stable manifolds for nonuniform exponential dichotomies
Luis Barreira and  Claudia Valls
2008, 21(4): 1025-1046 doi: 10.3934/dcds.2008.21.1025 +[Abstract](33) +[PDF](266.9KB)
Abstract:
We establish the existence of smooth stable manifolds for nonautonomous differential equations $v'=A(t)v+f(t,v)$ in a Banach space, obtained from sufficiently small perturbations of a linear equation $v'=A(t)v$ admitting a nonuniform exponential dichotomy. In addition to the exponential decay of the flow on the stable manifold we also obtain the exponential decay of its derivative with respect to the initial condition. Furthermore, we give a characterization of the stable manifold in terms of the exponential growth rate of the solutions.
Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach
Mariane Bourgoing
2008, 21(4): 1047-1069 doi: 10.3934/dcds.2008.21.1047 +[Abstract](46) +[PDF](303.5KB)
Abstract:
In this article, we continue the study of viscosity solutions for second-order fully nonlinear parabolic equations, having a $L^1$ dependence in time, associated with nonlinear Neumann boundary conditions, which started in a previous paper (cf [2]). First, we obtain the existence of continuous viscosity solutions by adapting Perron's method and using the comparison results obtained in [2]. Then, we apply these existence and comparison results to the study of the level-set approach for front propagations problems when the normal velocity has a $L^1$-dependence in time.
Rotation numbers and Lyapunov stability of elliptic periodic solutions
Jifeng Chu and  Meirong Zhang
2008, 21(4): 1071-1094 doi: 10.3934/dcds.2008.21.1071 +[Abstract](35) +[PDF](327.9KB)
Abstract:
Using the relation between the Hill's equations and the Ermakov-Pinney equations established by Zhang [27], we will give some interesting lower bounds of rotation numbers of Hill's equations. Based on the Birkhoff normal forms and the Moser twist theorem, we will prove that two classes of nonlinear, scalar, time-periodic, Newtonian equations will have twist periodic solutions, one class being regular and another class being singular.
Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space
Hongjie Dong and  Dapeng Du
2008, 21(4): 1095-1101 doi: 10.3934/dcds.2008.21.1095 +[Abstract](52) +[PDF](146.9KB)
Abstract:
By adapting a method in [11] with a suitable modification, we show that the critical dissipative quasi-geostrophic equations in $R^2$ has global well-posedness with arbitrary $H^1$ initial data. A decay in time estimate for homogeneous Sobolev norms of solutions is also discussed.
Generic points in systems of specification and Banach valued Birkhoff ergodic average
Aihua Fan , Lingmin Liao and  Jacques Peyrière
2008, 21(4): 1103-1128 doi: 10.3934/dcds.2008.21.1103 +[Abstract](52) +[PDF](297.2KB)
Abstract:
We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure.
Decay and local eventual positivity for biharmonic parabolic equations
Alberto Ferrero , Filippo Gazzola and  Hans-Christoph Grunau
2008, 21(4): 1129-1157 doi: 10.3934/dcds.2008.21.1129 +[Abstract](48) +[PDF](338.0KB)
Abstract:
We study existence and positivity properties for solutions of Cauchy problems for both linear and semilinear parabolic equations with the biharmonic operator as elliptic principal part. The self-similar kernel of the parabolic operator $\partial_t+\Delta^2$ is a sign changing function and the solution of the evolution problem with a positive initial datum may display almost instantaneous change of sign. We determine conditions on the initial datum for which the corresponding solution exhibits some kind of positivity behaviour. We prove eventual local positivity properties both in the linear and semilinear case. At the same time, we show that negativity of the solution may occur also for arbitrarily large given time, provided the initial datum is suitably constructed.
Nonlinear delay equations with nonautonomous past
Genni Fragnelli and  Dimitri Mugnai
2008, 21(4): 1159-1183 doi: 10.3934/dcds.2008.21.1159 +[Abstract](25) +[PDF](314.1KB)
Abstract:
Inspired by a biological model on genetic repression proposed by P. Jacob and J. Monod, we introduce a new class of delay equations with nonautonomous past and nonlinear delay operator. With the aid of some new techniques from functional analysis we prove that these equations, which cover the biological model, are well--posed.
Long-time behaviour of wave equations with nonlinear interior damping
A. Kh. Khanmamedov
2008, 21(4): 1185-1198 doi: 10.3934/dcds.2008.21.1185 +[Abstract](30) +[PDF](174.5KB)
Abstract:
We prove the existence of attractors for higher dimensional wave equations with nonlinear interior damping which grows faster than polynomials at infinity.
Identifying a BV-kernel in a hyperbolic integrodifferential equation
Alfredo Lorenzi and  Eugenio Sinestrari
2008, 21(4): 1199-1219 doi: 10.3934/dcds.2008.21.1199 +[Abstract](40) +[PDF](246.3KB)
Abstract:
This paper is devoted to determining the scalar relaxation kernel $a$ in a second-order (in time) integrodifferential equation related to a Banach space when an additional measurement involving the state function is available. A result concerning global existence and uniqueness is proved.
    The novelty of this paper consists in looking for the kernel $a$ in the Banach space $BV(0,T)$, consisting of functions of bounded variations, instead of the space $W^{1,1}(0,T)$ used up to now to identify $a$.
    An application is given, in the framework of $L^2$-spaces, to the case of hyperbolic second-order integrodifferential equations endowed with initial and Dirichlet boundary conditions.
Unfolding of resonant saddles and the Dulac time
Pavao Mardešić , David Marín and  Jordi Villadelprat
2008, 21(4): 1221-1244 doi: 10.3934/dcds.2008.21.1221 +[Abstract](27) +[PDF](300.9KB)
Abstract:
In this work we study unfoldings of planar vector fields in a neighbourhood of a resonant saddle. We give a $\mathcal C^k$ normal form for the unfolding with respect to the conjugacy relation. Using our normal form we determine an asymptotic development, uniform with respect to the parameters, of the Dulac time of a resonant saddle deformation. Conjugacy relation instead of weaker equivalence relation is necessary when studying the time function. The Dulac time of a resonant saddle can be seen as the basic building block of the total period function of an unfolding of a hyperbolic polycycle.
Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations
Gabriela Planas and  Eduardo Hernández
2008, 21(4): 1245-1258 doi: 10.3934/dcds.2008.21.1245 +[Abstract](27) +[PDF](171.8KB)
Abstract:
We consider the two-dimensional Navier-Stokes equations with a time-delayed convective term and a forcing term which contains some hereditary features. Some results on existence and uniqueness of solutions are established. We discuss the asymptotic behaviour of solutions and we also show the exponential stability of stationary solutions.
Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems
Shengfan Zhou , Caidi Zhao and  Yejuan Wang
2008, 21(4): 1259-1277 doi: 10.3934/dcds.2008.21.1259 +[Abstract](32) +[PDF](267.5KB)
Abstract:
In this paper, we first establish a criteria for finite fractal dimensionality of a family of compact subsets of a Hilbert space, and apply it to obtain an upper bound of fractal dimension of compact kernel sections to first order non-autonomous lattice systems. Then we consider the upper semicontinuity of kernel sections of general first order non-autonomous lattice systems and give an application.

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