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Discrete & Continuous Dynamical Systems - A

2005 , Volume 13 , Issue 1

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On the blow-up for a discrete Boltzmann equation in the plane
Alberto Bressan and  Massimo Fonte
2005, 13(1): 1-12 doi: 10.3934/dcds.2005.13.1 +[Abstract](43) +[PDF](186.7KB)
We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed.
Necessary conditions for the existence of wandering triangles for cubic laminations
Alexander Blokh
2005, 13(1): 13-34 doi: 10.3934/dcds.2005.13.13 +[Abstract](36) +[PDF](278.2KB)
In his 84 preprint W. Thurston proved that quadratic laminations do not admit so-called wandering triangles and asked a deep question concerning their existence for laminations of higher degrees. Recently it has been discovered by L. Oversteegen and the author that some closed laminations of the unit circle invariant under $z\mapsto z^d, d>2$ admit wandering triangles. This makes the problem of describing the criteria for the existence of wandering triangles important because solving this problem would help understand the combinatorial structure of the family of all polynomials of the appropriate degree.
In this paper for a closed lamination on the unit circle invariant under $z\mapsto z^3$ (cubic lamination) we prove that if it has a wandering triangle then there must be two distinct recurrent critical points in the corresponding quotient space ("topological Julia set") $J$ with the same limit set coinciding with the limit set of any wandering vertex (wandering vertices in $J$ correspond to wandering gaps in the lamination).
A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses
Michael L. Frankel and  Victor Roytburd
2005, 13(1): 35-62 doi: 10.3934/dcds.2005.13.35 +[Abstract](26) +[PDF](292.1KB)
We study a two-phase modified Stefan problem modeling solid combustion and nonequilibrium phase transitions. The problem is known to exhibit a variety of non-trivial dynamical scenarios. In the presense of heat losses we develop a priori estimates and establish well-posedness of the problem in weighted spaces of continuous functions. The estimates secure sufficient decay of solutions that allows us to carry out analysis in Hilbert spaces. We demonstrate existence of a compact attractor in the weighted spaces and prove that the attractor consist of sufficiently regular functions. We show that the Hausdorff dimension of the attractor is finite.
Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori
Denis G. Gaidashev
2005, 13(1): 63-102 doi: 10.3934/dcds.2005.13.63 +[Abstract](28) +[PDF](433.7KB)
The paper presents a study of a renormalization group transformation acting on an appropriate space of Hamiltonian functions in two angle and two action variables. In particular, we study the existence of real invariant tori, on which the flow is conjugate to a rotation with a rotation number equal to a quadratic irrational ($\omega$-tori). We demonstrate that the stable manifold of the renormalization operator at the "simple" fixed point contains isoenergetically degenerate Hamiltonians possessing shearless $\omega$-tori. We also show that one-parameter families of Hamiltonians transverse to the stable manifold undergo a bifurcation: for a certain range of the parameter values the members of these families posses two distinct $\omega$-tori, the members of such families lying on the stable manifold posses one shearless $\omega$-torus, while the members corresponding to other parameter values do not posses any.
Viscosity solution methods and the discrete Aubry-Mather problem
Diogo A. Gomes
2005, 13(1): 103-116 doi: 10.3934/dcds.2005.13.103 +[Abstract](34) +[PDF](200.6KB)
In this paper we study a discrete multi-dimensional version of Aubry-Mather theory using mostly tools from the theory of viscosity solutions. We set this problem as an infinite dimensional linear programming problem. The dual problem turns out to be a discrete analog of the Hamilton-Jacobi equations. We present some applications to discretizations of Lagrangian systems.
On the uniqueness of bounded solutions to singular parabolic problems
Maria Assunta Pozio and  Alberto Tesei
2005, 13(1): 117-137 doi: 10.3934/dcds.2005.13.117 +[Abstract](44) +[PDF](256.5KB)
We provide criteria for uniqueness or nonuniqueness of bounded solutions for a wide class of second order parabolic problems with singular coefficients.
Two-parameter families of implicit differential equations
Farid Tari
2005, 13(1): 139-162 doi: 10.3934/dcds.2005.13.139 +[Abstract](30) +[PDF](388.4KB)
We study in this paper local codimension 2 singularities of (first order) implicit differential equations $F(x,y,p)=0$, where $F$ is a germ of a smooth function, $p=\frac{dy}{dx}$, $F_p=0$ and $F_{p p}\ne 0$ at the singular point. We obtain topological models of these singularities and deal with their bifurcations in generic 2-parameter families of equations.
Notions of sublinearity and superlinearity for nonvariational elliptic systems
Boyan Sirakov
2005, 13(1): 163-174 doi: 10.3934/dcds.2005.13.163 +[Abstract](28) +[PDF](210.2KB)
We study existence of solutions of boundary-value problems for elliptic systems of type ($\po$) below. We introduce notions of sublinearity and superlinearity for such systems and show that sublinear systems always have a positive solution, while superlinear systems admit a positive solution provided the set of their positive solutions is bounded in the uniform norm. These facts have long been known for scalar equations.
Critical points for a class of nondifferentiable functions and applications
P. Candito , S. A. Marano and  D. Motreanu
2005, 13(1): 175-194 doi: 10.3934/dcds.2005.13.175 +[Abstract](27) +[PDF](240.9KB)
Some critical point theorems involving functionals that are the sum of a locally Lipschitz continuous term and of a convex, proper, besides lower semicontinuous, function are established. A recent existence result of Adly, Buttazzo, and Théra [1, Theorem 2.3] is improved. Applications to elliptic variational-hemivariational inequalities are then examined.
Homogeneity of surjective cellular automata
T.K. Subrahmonian Moothathu
2005, 13(1): 195-202 doi: 10.3934/dcds.2005.13.195 +[Abstract](34) +[PDF](193.4KB)
We bring out some similarities among one-dimensional surjective cellular automata. Four main results are the following: (i) all periodic points of a cellular automata are shift-periodic if and only if the set of periodic points of any fixed period is finite, (ii) forward recurrent points as well as backward recurrent points are residual for every onto cellular automata, (iii) every onto cellular automata is semi-open, and (iv) all transitive cellular automata are weak mixing and hence maximally sensitive (which improves an existing result).
Nonlinear stability of solitary waves for a 2-d Benney--Luke equation
José R. Quintero
2005, 13(1): 203-218 doi: 10.3934/dcds.2005.13.203 +[Abstract](34) +[PDF](225.8KB)
We study orbital stability of solitary wave of least energy for a nonlinear Benney-Luke equation that models long water waves with small amplitude.
Multi-dimensional dynamical systems and Benford's Law
Arno Berger
2005, 13(1): 219-237 doi: 10.3934/dcds.2005.13.219 +[Abstract](33) +[PDF](303.7KB)
One-dimensional projections of (at least) almost all orbits of manymulti-dimensional dynamical systems are shown to follow Benford's law,i.e. their (base $b$) mantissa distribution is asymptotically logarithmic,typically for all bases $b$. As a generalization and unificationof known results it is proved that under a (generic) non-resonance conditionon $A\in \mathbb C^{d\times d}$, for every $z\in \mathbb C^d$ real and imaginary part of each non-trivialcomponent of $(A^nz)_{n\in N_0}$ and $(e^{At}z)_{t\ge 0}$ follow Benford's law. Also,Benford behavior is found to be ubiquitous for several classes of non-linear maps anddifferential equations. In particular, emergence of the logarithmic mantissadistribution turns out to be generic for complex analytic maps $T$ with $T(0)=0$, $|T'(0)|<1$.The results significantly extend known facts obtained by other, e.g. number-theoretical methods,and also generalize recent findings for one-dimensional systems.

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