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Discrete & Continuous Dynamical Systems - A

2007 , Volume 17 , Issue 1

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On the first positive Neumann eigenvalue
Wei-Ming Ni and Xuefeng Wang
2007, 17(1): 1-19 doi: 10.3934/dcds.2007.17.1 +[Abstract](83) +[PDF](209.9KB)
We study the first positive Neumann eigenvalue $\mu_1$ of the Laplace operator on a planar domain $\Omega$. We are particularly interested in how the size of $\mu_1$ depends on the size and geometry of $\Omega$. A notion of the intrinsic diameter of $\Omega$ is proposed and various examples are provided to illustrate the effect of the intrinsic diameter and its interplay with the geometry of the domain.
Intermittency and Jakobson's theorem near saddle-node bifurcations
Ale Jan Homburg and Todd Young
2007, 17(1): 21-58 doi: 10.3934/dcds.2007.17.21 +[Abstract](75) +[PDF](507.0KB)
We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to an atomic measure supported on the orbit of the saddle-node point. Using these measures we analyze the intermittent time series that result from the destruction of the periodic attractor in the saddle-node bifurcation and prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the periodic attractor.
Characterizing asymptotic stability with Dulac functions
Marc Chamberland, Anna Cima, Armengol Gasull and Francesc Mañosas
2007, 17(1): 59-76 doi: 10.3934/dcds.2007.17.59 +[Abstract](49) +[PDF](237.3KB)
This paper studies questions regarding the local and global asymptotic stability of analytic autonomous ordinary differential equations in $\mathbb{R}^n$. It is well-known that such stability can be characterized in terms of Liapunov functions. The authors prove similar results for the more geometrically motivated Dulac functions. In particular it holds that any analytic autonomous ordinary differential equation having a critical point which is a global attractor admits a Dulac function. These results can be used to give criteria of global attraction in two-dimensional systems.
Topological quasi-group shifts
Marcelo Sobottka
2007, 17(1): 77-93 doi: 10.3934/dcds.2007.17.77 +[Abstract](87) +[PDF](226.9KB)
In this work we characterize those shift spaces which can support a 1-block quasi-group operation and show the analogous of Kitchens result: any such shift is conjugated to a product of a full shift with a finite shift. Moreover, we prove that every expansive automorphism on a compact zero-dimensional quasi-group that verifies the medial property, commutativity and has period 2, is isomorphic to the shift map on a product of a finite quasi-group with a full shift.
Characteristic equation approach to stability measures of linear neutral systems with multiple time delays
D. Q. Cao, Y. R. Yang and Y. M. Ge
2007, 17(1): 95-105 doi: 10.3934/dcds.2007.17.95 +[Abstract](99) +[PDF](161.6KB)
Using the characteristic equation approach, the problem of asymptotic stability of linear neutral systems with multiple time delays is investigated in this paper. New delay-independent stability criteria are derived in terms of the spectral radius of corresponding modulus matrices. The structure information of the system matrices are taken into consideration in the proposed stability criteria, thus the conservatism found in the literature can be significantly reduced. The explicit nature of the construction permits us to directly express the algebraic criteria in terms of the plant parameters, thus checking of stability by our criteria can be carried out rather simply. Numerical examples are given to demonstrate the validity of the new criteria and to compare them with the previous results.
Two-point pre-image entropy
Wen-Chiao Cheng
2007, 17(1): 107-119 doi: 10.3934/dcds.2007.17.107 +[Abstract](63) +[PDF](148.4KB)
The aim of this paper is to define and study a new kind of entropy-like invariants in the case of probability space and compact metric topological group of continuous endomorphisms. These new invariants are only non-zero for non-invertible maps, but many propositions can be described and the analogue of the well-known variational principle can be established.
On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompact global attractor
Rolci Cipolatti and Otared Kavian
2007, 17(1): 121-132 doi: 10.3934/dcds.2007.17.121 +[Abstract](63) +[PDF](185.0KB)
We study a Schrödinger equation with a nonlocal nonlinearity, which has been considered as a model for ultra-short laser pulses. An interesting feature of this equation is that the underlying dynamical system possesses a bounded non compact global attractor, actually a ball in $L^2(R)$. Existence and instability of standing waves are also proved.
Entropy dimensions and a class of constructive examples
Sébastien Ferenczi and Kyewon Koh Park
2007, 17(1): 133-141 doi: 10.3934/dcds.2007.17.133 +[Abstract](59) +[PDF](143.1KB)
Motivated by the study of actions of $\Z^{2}$ and more general groups, and their non-cocompact subgroup actions, we investigate entropy-type invariants for deterministic systems. In particular, we define a new isomorphism invariant, the entropy dimension, and look at its behaviour on examples. We also look at other natural notions suitable for processes.
Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential
Leszek Gasiński
2007, 17(1): 143-158 doi: 10.3934/dcds.2007.17.143 +[Abstract](85) +[PDF](205.7KB)
In this paper we study boundary value problem with one dimensional $p$-Laplacian. Assuming complete resonance at $+\infty$ and partial resonance at $0^+$, an existence of at least one positive solution is proved. By strengthening our assumptions we can guarantee strict positivity of the obtained solution.
The global attractor for the solutions to the 3D viscous primitive equations
Ning Ju
2007, 17(1): 159-179 doi: 10.3934/dcds.2007.17.159 +[Abstract](100) +[PDF](247.9KB)
Existence of the global attractor is proved for the strong solutions to the 3D viscous Primitive Equations (PEs) modeling large scale ocean and atmosphere dynamics. This result is obtained under the natural assumption that the external heat source $Q$ is square integrable. Furthermore, it is shown in [20] that the fractal and Hausdroff dimensions of the global attractor for 3D viscous PEs are both finite.
Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations
Changxing Miao and Bo Zhang
2007, 17(1): 181-200 doi: 10.3934/dcds.2007.17.181 +[Abstract](73) +[PDF](268.7KB)
In this paper the global well-posedness in $L^2$ and $H^m$ of the Cauchy problem is proved for nonlinear Schrödinger-type equations. This we do by establishing regular Strichartz estimates for the corresponding linear equations and some nonlinear a priori estimates in the framework of Besov spaces. We further establish the regularity of the $H^m$-solution to the Cauchy problem.
Dimension and ergodic decompositions for hyperbolic flows
Luis Barreira and Christian Wolf
2007, 17(1): 201-212 doi: 10.3934/dcds.2007.17.201 +[Abstract](60) +[PDF](186.7KB)
For conformal hyperbolic flows, we establish explicit formulas for the Hausdorff dimension and for the pointwise dimension of an arbitrary invariant measure. We emphasize that these measures are not necessarily ergodic. The formula for the pointwise dimension is expressed in terms of the local entropy and of the Lyapunov exponents. We note that this formula was obtained before only in the special case of (ergodic) equilibrium measures, and these always possess a local product structure (which is not the case for arbitrary invariant measures). The formula for the pointwise dimension allows us to show that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimension of the measures in an ergodic decomposition.
Persistence of Bowen-Ruelle-Sinai measures
Maria Pires De Carvalho
2007, 17(1): 213-221 doi: 10.3934/dcds.2007.17.213 +[Abstract](63) +[PDF](144.5KB)
We study the changes on the Bowen-Ruelle-Sinai measures along an arc that starts at an Anosov diffeomorphism on a two-torus and reaches the boundary of its stability component while a flat homoclinic tangency or a first cubic heteroclinic tangency is happening. The outermost diffeomorphisms of such arcs are not hyperbolic but are conjugate to the original Anosov diffeomorphism and share similar ergodic traits. In particular, the torus is a global attractor with a full supported physical measure.

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