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Discrete & Continuous Dynamical Systems - A

2007 , Volume 19 , Issue 4

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Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals
Satoshi Kosugi, Yoshihisa Morita and Shoji Yotsutani
2007, 19(4): 609-629 doi: 10.3934/dcds.2007.19.609 +[Abstract](539) +[PDF](268.8KB)
Abstract:
We investigate stationary solutions of the one-dimensional Cahn-Hilliard equation with the diffusion coefficient and the total mass of the density as two given parameters. We solve the equation completely in the whole parameter space by using the Jacobi elliptic functions and complete elliptic integrals. In addition to counting the stationary solutions, which was studied by Grinfeld and Novick-Cohen, we provide an exact expression of the solutions. We also illustrate global bifurcation diagrams together with the asymptotic behavior of the solutions as the diffusion coefficient vanishes.
Bifurcation of relaxation oscillations in dimension two
Freddy Dumortier and Robert Roussarie
2007, 19(4): 631-674 doi: 10.3934/dcds.2007.19.631 +[Abstract](562) +[PDF](523.7KB)
Abstract:
The paper deals with the bifurcation of relaxation oscillations in two dimensional slow-fast systems. The most generic case is studied by means of geometric singular perturbation theory, using blow up at contact points. It reveals that the bifurcation goes through a continuum of transient canard oscillations, controlled by the slow divergence integral along the critical curve. The theory is applied to polynomial Liénard equations, showing that the cyclicity near a generic coallescence of two relaxation oscillations does not need to be limited to two, but can be arbitrarily high.
Action functionals that attain regular minima in presence of energy gaps
Alessandro Ferriero
2007, 19(4): 675-690 doi: 10.3934/dcds.2007.19.675 +[Abstract](512) +[PDF](192.6KB)
Abstract:
We present three simple regular one-dimensional variational problems that present the Lavrentiev gap phenomenon, i.e.,

inf$\{\int_a^b L(t,x,\dot x): x\in W_0^{1,1}(a,b)\} $< inf$\{\int_a^bL(t,x,\dot x): x\in W_0^{1,\infty}(a,b)\}$

(where $ W_0^{1,p}(a,b)$ denote the usual Sobolev spaces with zero boundary conditions), in which in the first example the two infima are actually minima, in the second example the infimum in $ W_0^{1,\infty}(a,b)$ is attained while the infimum in $ W_0^{1,1}(a,b)$ is not, and in the third example both infimum are not attained. We discuss also how to construct energies with a gap between any space and energies with multi-gaps.

Mild mixing property for special flows under piecewise constant functions
Krzysztof Frączek, M. Lemańczyk and E. Lesigne
2007, 19(4): 691-710 doi: 10.3934/dcds.2007.19.691 +[Abstract](429) +[PDF](371.8KB)
Abstract:
We give a condition on a piecewise constant roof function and an irrational rotation by $\alpha$ on the circle to give rise to a special flow having the mild mixing property. Such flows will also satisfy Ratner's property. As a consequence we obtain a class of mildly mixing singular flows on the two-torus that arise from quasi-periodic Hamiltonians flows by velocity changes.
Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation
N. I. Karachalios, Hector E. Nistazakis and Athanasios N. Yannacopoulos
2007, 19(4): 711-736 doi: 10.3934/dcds.2007.19.711 +[Abstract](564) +[PDF](361.9KB)
Abstract:
We study the asymptotic behavior of complex discrete evolution equations of Ginzburg- Landau type. Depending on the nonlinearity and the data of the problem, we find different dynamical behavior ranging from global existence of solutions and global attractors to blow-up in finite time. We provide estimates for the blow-up time, depending not only on the initial data but also on the size of the lattice. Some of the theoretical results are tested by numerical simulations.
Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients
Nobuyuki Kato and Norio Kikuchi
2007, 19(4): 737-760 doi: 10.3934/dcds.2007.19.737 +[Abstract](539) +[PDF](319.7KB)
Abstract:
The aim of this paper is to formulate Campanato-type boundary estimates for solutions of the Rothe approximate scheme to parabolic partial differential systems with constant coefficients. The core observation is that such estimates hold independently of the approximate systems.
On the intersection of homoclinic classes on singular-hyperbolic sets
S. Bautista, C. Morales and M. J. Pacifico
2007, 19(4): 761-775 doi: 10.3934/dcds.2007.19.761 +[Abstract](451) +[PDF](312.9KB)
Abstract:
We know that two different homoclinic classes contained in the same hyperbolic set are disjoint [12]. Moreover, a connected singular-hyperbolic attracting set with dense periodic orbits and a unique equilibrium is either transitive or the union of two different homoclinic classes [6]. These results motivate the questions of if two different homoclinic classes contained in the same singular-hyperbolic set are disjoint or if the second alternative in [6] cannot occur. Here we give a negative answer for both questions. Indeed we prove that every compact $3$-manifold supports a vector field exhibiting a connected singular-hyperbolic attracting set which has dense periodic orbits, a unique singularity, is the union of two homoclinic classes but is not transitive.
The geometry of mesoscopic phase transition interfaces
Matteo Novaga and Enrico Valdinoci
2007, 19(4): 777-798 doi: 10.3934/dcds.2007.19.777 +[Abstract](573) +[PDF](277.1KB)
Abstract:
We consider a mesoscopic model of phase transitions and investigate the geometric properties of the interfaces of the associated minimal solutions. We provide density estimates for level sets and, in the periodic setting, we construct minimal interfaces at a universal distance from any given hyperplane.
Rates of convergence towards the boundary of a self-similar set
L. Olsen
2007, 19(4): 799-811 doi: 10.3934/dcds.2007.19.799 +[Abstract](596) +[PDF](231.5KB)
Abstract:
The geometry of self-similar sets $K$ has been studied intensively during the past 20 years frequently assuming the so-called Open Set Condition (OSC). The OSC guarantees the existence of an open set $U$ satisfying various natural invariance properties, and is instrumental in the study of self-similar sets for the following reason: a careful analysis of the boundaries of the iterates of $\overline U$ is the key technique for obtaining information about the geometry of $K$. In order to obtain a better understanding of the OSC and because of the geometric significance of the boundaries of the iterates of $\overline U$, it is clearly of interest to provide quantitative estimates for the "number" of points close to the boundaries of the iterates of $\overline U$. This motivates a detailed study of the rate at which the distance between a point in $K$ and the boundaries of the iterates of $\overline U$ converge to $0$. In this paper we show that for each $t\in I$ (where $I$ is a certain interval defined below) there is a significant number of points for which the rate of convergence equals $t$. In fact, for each $t\in I$, we show that the set of points whose rate of convergence equals $t$ has positive Hausdorff dimension, and we obtain a lower bound for this dimension. Examples show that this bound is, in general, the best possible and cannot be improved.
New maximum principles for fully nonlinear ODEs of second order
Wenmin Sun and Jiguang Bao
2007, 19(4): 813-823 doi: 10.3934/dcds.2007.19.813 +[Abstract](432) +[PDF](189.4KB)
Abstract:
In the paper, we give the positive answer of an open problem of Li-Nirenberg under the weaker conditions, and we prove a new variation of the boundary point lemma for second order fully nonlinear ODEs by a new method. A simpler proof of Li-Nirenberg Theorem is also presented.

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