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Discrete & Continuous Dynamical Systems - A

2011 , Volume 29 , Issue 1

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Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations
Regina Martínez and  Carles Simó
2011, 29(1): 1-24 doi: 10.3934/dcds.2011.29.1 +[Abstract](38) +[PDF](2247.8KB)
Abstract:
Non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations (VEk), are applied to prove the non-integrability of the Swinging Atwood's Machine for values of the parameter which can not be decided using first order variational equations (VE1).
Existence of solutions for a Boussinesq system on the half line and on a finite interval
Karine Adamy
2011, 29(1): 25-49 doi: 10.3934/dcds.2011.29.25 +[Abstract](34) +[PDF](263.7KB)
Abstract:
The initial boundary-value problem for a Boussinesq system is studied on the half line and on a finite interval. Global existence of weak solutions satisfying the boundary conditions is proven and uniqueness for solutions in a suitable class is studied. A proof of the persistence of finite regularity for solutions in the whole space is also presented.
Multiplicity results for extremal operators through bifurcation
Alejandro Allendes and  Alexander Quaas
2011, 29(1): 51-65 doi: 10.3934/dcds.2011.29.51 +[Abstract](45) +[PDF](222.5KB)
Abstract:
We study non-proper uniformly elliptic fully nonlinear equations involving extremal operators of Pucci type. We prove the existence of all radial spectrum for this type of operators and establish a multiplicity existence results through global bifurcation.
Dynamics of birth-and-death processes with proliferation - stability and chaos
Jacek Banasiak and  Marcin Moszyński
2011, 29(1): 67-79 doi: 10.3934/dcds.2011.29.67 +[Abstract](45) +[PDF](203.3KB)
Abstract:
We provide a detailed description of long time dynamics in $l^1$ of the semigroup associated with constant coefficient infinite birth-and-death systems with proliferation. In particular, we discuss and slightly extend earlier stability results of [8] and also identify a range of parameters for which the semigroup is both stable in the sense of op. cit. and topologically chaotic. Moreover, for a range of parameters, we provide an explicit description of subspaces of $l^1$ which cannot generate chaotic orbits.
Rich quasi-linear system for integrable geodesic flows on 2-torus
Misha Bialy and  Andrey E. Mironov
2011, 29(1): 81-90 doi: 10.3934/dcds.2011.29.81 +[Abstract](38) +[PDF](151.7KB)
Abstract:
Consider a Riemannian metric on two-torus. We prove that the question of existence of polynomial first integrals leads naturally to a remarkable system of quasi-linear equations which turns out to be a Rich system of conservation laws. This reduces the question of integrability to the question of existence of smooth (quasi-) periodic solutions for this Rich quasi-linear system.
Morphisms of discrete dynamical systems
Mădălina Roxana Buneci
2011, 29(1): 91-107 doi: 10.3934/dcds.2011.29.91 +[Abstract](49) +[PDF](213.4KB)
Abstract:
The purpose of this paper is to introduce a category whose objects are discrete dynamical systems $( X,P,H,\theta ) $ in the sense of [6] and whose arrows will be defined starting from the notion of groupoid morphism given in [10]. We shall also construct a contravariant functor $( X,P,H,\theta ) \rightarrow $C* $( X,P,H,\theta ) $ from the subcategory of discrete dynamical systems for which $PP^{-1}$ is amenable to the category of C* -algebras, where C* $( X,P,H,\theta ) $ is the C* -algebra associated to the groupoid $G( X,P,H,\theta)$.
Detectable canard cycles with singular slow dynamics of any order at the turning point
P. De Maesschalck and  Freddy Dumortier
2011, 29(1): 109-140 doi: 10.3934/dcds.2011.29.109 +[Abstract](33) +[PDF](718.2KB)
Abstract:
This paper deals with the study of limit cycles that appear in a class of planar slow-fast systems, near a "canard'' limit periodic set of FSTS-type. Limit periodic sets of FSTS-type are closed orbits, composed of a Fast branch, an attracting Slow branch, a Turning point, and a repelling Slow branch. Techniques to bound the number of limit cycles near a FSTS-l.p.s. are based on the study of the so-called slow divergence integral, calculated along the slow branches. In this paper, we extend the technique to the case where the slow dynamics has singularities of any (finite) order that accumulate to the turning point, and in which case the slow divergence integral becomes unbounded. Bounds on the number of limit cycles near the FSTS-l.p.s. are derived by examining appropriate derivatives of the slow divergence integral.
Time-dependent attractor for the Oscillon equation
Francesco Di Plinio , Gregory S. Duane and  Roger Temam
2011, 29(1): 141-167 doi: 10.3934/dcds.2011.29.141 +[Abstract](48) +[PDF](361.4KB)
Abstract:
We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation

tt $u(x,t) +H $ t$ u(x,t) -\e^{-2Ht}$ xx $ u(x,t) + V'(u(x,t)) =0, \quad (x,t)\in (0,1) \times \R,$

with periodic boundary conditions, where $H>0$ is the Hubble constant and $V$ is a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular global attractor $\A=\A(t)$. The kernel sections $\A(t)$ have finite fractal dimension.

Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force
Alessandro Fonda and  Antonio J. Ureña
2011, 29(1): 169-192 doi: 10.3934/dcds.2011.29.169 +[Abstract](39) +[PDF](375.9KB)
Abstract:
We consider planar systems driven by a central force which depends periodically on time. If the force is sublinear and attractive, then there is a connected set of subharmonic and quasi-periodic solutions rotating around the origin at different speeds; moreover, this connected set stretches from zero to infinity. The result still holds allowing the force to be attractive only in average provided that an uniformity condition is satisfied and there are no periodic oscillations with zero angular momentum. We provide examples showing that these assumptions cannot be skipped.
Dynamics and abstract computability: Computing invariant measures
Stefano Galatolo , Mathieu Hoyrup and  Cristóbal Rojas
2011, 29(1): 193-212 doi: 10.3934/dcds.2011.29.193 +[Abstract](53) +[PDF](275.4KB)
Abstract:
We consider the question of computing invariant measures from an abstract point of view. Here, computing a measure means finding an algorithm which can output descriptions of the measure up to any precision. We work in a general framework (computable metric spaces) where this problem can be posed precisely. We will find invariant measures as fixed points of the transfer operator. In this case, a general result ensures the computability of isolated fixed points of a computable map.
     We give general conditions under which the transfer operator is computable on a suitable set. This implies the computability of many "regular enough" invariant measures and among them many physical measures.
     On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two examples of computable dynamics, one having a physical measure which is not computable and one for which no invariant measure is computable, showing some subtlety in this kind of problems.
Expansive homeomorphisms of the plane
Jorge Groisman
2011, 29(1): 213-239 doi: 10.3934/dcds.2011.29.213 +[Abstract](36) +[PDF](277.4KB)
Abstract:
This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques involve topological and metric aspects of the plane. The use of a Lyapunov metric function which defines the same topology as the one induced by the usual metric but that, in general, is not equivalent to it is an example of such techniques. The discovery of a hypothesis about the behavior of Lyapunov functions at infinity allows us to generalize some results that are valid in the compact context. Additional local properties allow us to obtain another classification theorem.
$C^{\alpha}$-Hölder classical solutions for non-autonomous neutral differential equations
Eduardo Hernández and  Donal O'Regan
2011, 29(1): 241-260 doi: 10.3934/dcds.2011.29.241 +[Abstract](41) +[PDF](257.4KB)
Abstract:
In this paper we discuss the existence of α-Hölder classical solutions for non-autonomous abstract partial neutral functional differential equations. An application is considered.
Reducibility of skew-product systems with multidimensional Brjuno base flows
Saša Kocić
2011, 29(1): 261-283 doi: 10.3934/dcds.2011.29.261 +[Abstract](34) +[PDF](323.3KB)
Abstract:
We develop a renormalization method that applies to the problem of the local reducibility of analytic skew-product flows on Td $\times$ SL(2,R). We apply the method to give a proof of a reducibility theorem for these flows with Brjuno base frequency vectors.
The domain of analyticity of solutions to the three-dimensional Euler equations in a half space
Igor Kukavica and  Vlad C. Vicol
2011, 29(1): 285-303 doi: 10.3934/dcds.2011.29.285 +[Abstract](55) +[PDF](244.3KB)
Abstract:
We address the problem of analyticity up to the boundary of solutions to the Euler equations in the half space. We characterize the rate of decay of the real-analyticity radius of the solution $u(t)$ in terms of exp$\int_{0}^{t} $||$ \nabla u(s) $|| L ds , improving the previously known results. We also prove the persistence of the sub-analytic Gevrey-class regularity for the Euler equations in a half space, and obtain an explicit rate of decay of the radius of Gevrey-class regularity.
Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions
Khalid Latrach and  Hatem Megdiche
2011, 29(1): 305-321 doi: 10.3934/dcds.2011.29.305 +[Abstract](52) +[PDF](222.9KB)
Abstract:
In this paper we discuss the large time behavior of the solution to the Cauchy problem governed by a transport equation with Maxwell boundary conditions arising in growing cell population in $L^1$-spaces. Our result completes previous ones established in [3] in $L^p$-spaces with $1 < p < \infty$.
An approximation theorem for maps between tiling spaces
Betseygail Rand and  Lorenzo Sadun
2011, 29(1): 323-326 doi: 10.3934/dcds.2011.29.323 +[Abstract](33) +[PDF](102.4KB)
Abstract:
We show that every continuous map from one translationally finite tiling space to another can be approximated by a local map. If two local maps are homotopic, then the homotopy can be chosen so that every interpolating map is also local.
Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability
Jesenko Vukadinovic
2011, 29(1): 327-341 doi: 10.3934/dcds.2011.29.327 +[Abstract](31) +[PDF](211.3KB)
Abstract:
The global well-posedness, the existence of globally absorbing sets and the existence of inertial manifolds are investigated in a class of diffusive (viscous) Burgers equations. The class includes diffusive Burgers equation with nontrivial forcing, Burgers-Sivashinsky equation and Quasi-Steady equation of cellular flames. Global dissipativity is proven in two space dimensions for periodic boundary conditions. For the proof of the existence of inertial manifolds, the spectral-gap condition, which Burgers-type equations do not satisfy in their original form, is circumvented by employing the Cole-Hopf transform. The procedure is valid in both one and two space dimensions.
Spatial dynamics of a nonlocal and delayed population model in a periodic habitat
Peixuan Weng and  Xiao-Qiang Zhao
2011, 29(1): 343-366 doi: 10.3934/dcds.2011.29.343 +[Abstract](61) +[PDF](319.8KB)
Abstract:
We derived an age-structured population model with nonlocal effects and time delay in a periodic habitat. The spatial dynamics of the model including the comparison principle, the global attractivity of spatially periodic equilibrium, spreading speeds, and spatially periodic traveling wavefronts is investigated. It turns out that the spreading speed coincides with the minimal wave speed for spatially periodic travel waves.
The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion
Yaping Wu and  Qian Xu
2011, 29(1): 367-385 doi: 10.3934/dcds.2011.29.367 +[Abstract](42) +[PDF](295.4KB)
Abstract:
This paper is concerned with the existence of large positive spiky steady states for S-K-T competition systems with cross-diffusion. Firstly by detailed integral and perturbation estimates, the existence and detailed fast-slow structure of a class of spiky steady states are obtained for the corresponding shadow system, which also verify and extend some existence results on spiky steady states obtained in [10] by different method of proof. Further by applying special perturbation method, we prove the existence of large positive spiky steady states for the original competition systems with large cross-diffusion rate.
Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers
Kazuyuki Yagasaki
2011, 29(1): 387-402 doi: 10.3934/dcds.2011.29.387 +[Abstract](40) +[PDF](238.9KB)
Abstract:
We consider two-degree-of-freedom Hamiltonian systems with saddle-centers, and develop a Melnikov-type technique for detecting creation of transverse homoclinic orbits by higher-order terms. We apply the technique to the generalized Hénon-Heiles system and give a positive answer to a remaining question of whether chaotic dynamics occurs for some parameter values although it is known to be nonintegrable in a complex analytical meaning.

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