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Discrete & Continuous Dynamical Systems - A

2011 , Volume 31 , Issue 2

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Exponentially small splitting of separatrices in the perturbed McMillan map
Pau Martín, David Sauzin and Tere M. Seara
2011, 31(2): 301-372 doi: 10.3934/dcds.2011.31.301 +[Abstract](223) +[PDF](1074.8KB)
Abstract:
The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coefficients involved in the expansion have a resurgent origin, as shown in [14].
Periodic solutions of resonant systems with rapidly rotating nonlinearities
Pablo Amster and Mónica Clapp
2011, 31(2): 373-383 doi: 10.3934/dcds.2011.31.373 +[Abstract](202) +[PDF](388.5KB)
Abstract:
We obtain existence of $T$-periodic solutions to a second order system of ordinary differential equations of the form \[ u^{\prime\prime}+cu^{\prime}+g(u)=p \] where $c\in\mathbb{R},$ $p\in C(\mathbb{R},\mathbb{R}^{N})$ is $T$-periodic and has mean value zero, and $g\in C(\mathbb{R}^{N},\mathbb{R}^{N})$ is e.g. sublinear. In contrast with a well known result by Nirenberg [6], where it is assumed that the nonlinearity $g$ has non-zero uniform radial limits at infinity, our main result allows rapid rotations in $g$.
On $C^0$-variational solutions for Hamilton-Jacobi equations
Olga Bernardi and Franco Cardin
2011, 31(2): 385-406 doi: 10.3934/dcds.2011.31.385 +[Abstract](253) +[PDF](460.0KB)
Abstract:
For evolutive Hamilton-Jacobi equations, we propose a refined definition of $C^0$-variational solution, adapted to Cauchy problems for continuous initial data. This weaker framework enables us to investigate the semigroup property for these solutions. In the case of $p$-convex Hamiltonians, when variational solutions are known to be identical to viscosity solutions, we verify directly the semigroup property by using minmax techniques. In the non-convex case, we construct a first explicit evolutive example where minmax and viscosity solutions are different. Provided the initial data allow for the separation of variables, we also detect the semigroup property for convex-concave Hamiltonians. In this case, and for general initial data, we finally give new upper and lower Hopf-type estimates for the variational solutions.
Rates of decay for the wave systems with time dependent damping
Moez Daoulatli
2011, 31(2): 407-443 doi: 10.3934/dcds.2011.31.407 +[Abstract](217) +[PDF](539.6KB)
Abstract:
We study the rate of decay of solutions of the wave systems with time dependent nonlinear damping. The damping is modeled by a continuous monotone function without any growth restrictions imposed at the origin and infinity. The decay rate of the energy functional is obtained by solving a nonlinear non-autonomous ODE.
Exponential attractors for lattice dynamical systems in weighted spaces
Xiaoying Han
2011, 31(2): 445-467 doi: 10.3934/dcds.2011.31.445 +[Abstract](238) +[PDF](472.7KB)
Abstract:
We first present some sufficient conditions for the existence of exponential attractors for locally coupled lattice dynamical systems in weighted spaces of infinite sequences. Then we apply this result to discuss the existence of exponential attractors for first order lattice systems, partly dissipative lattice systems, and second order lattice systems in weighted spaces of infinite sequences.
On well-posedness of the Degasperis-Procesi equation
A. Alexandrou Himonas and Curtis Holliman
2011, 31(2): 469-488 doi: 10.3934/dcds.2011.31.469 +[Abstract](349) +[PDF](447.3KB)
Abstract:
It is shown in both the periodic and the non-periodic cases that the data-to-solution map for the Degasperis-Procesi (DP) equation is not a uniformly continuous map on bounded subsets of Sobolev spaces with exponent greater than 3/2. This shows that continuous dependence on initial data of solutions to the DP equation is sharp. The proof is based on well-posedness results and approximate solutions. It also exploits the fact that DP solutions conserve a quantity which is equivalent to the $L^2$ norm. Finally, it provides an outline of the local well-posedness proof including the key estimates for the size of the solution and for the solution's lifespan that are needed in the proof of the main result.
Simple waves and pressure delta waves for a Chaplygin gas in two-dimensions
Geng Lai, Wancheng Sheng and Yuxi Zheng
2011, 31(2): 489-523 doi: 10.3934/dcds.2011.31.489 +[Abstract](239) +[PDF](679.1KB)
Abstract:
We present two new types of self-similar solutions to the Chaplygin gas model in two space dimensions: Simple waves and pressure delta waves, which are absent in one space dimension, but appear in the solutions to the two-dimensional Riemann problems. A simple wave is a flow in a physical region whose image in the state space is a one-dimensional curve. The solutions to the interaction of two rarefaction simple waves are constructed. Comparisons with polytropic gases are made. Pressure delta waves are Dirac type concentration in the pressure variable, or impulses of the pressure on discontinuities. They appear in the study of Riemann problems of four rarefaction shocks. This type of discontinuities and concentrations are different from delta waves for the pressureless gas flow model, for which the delta waves are associated with convection and concentration of mass. By re-interpreting the terms in the Chaplygin gas system into new forms we are able to define distributional solutions that include the pressure delta waves. Generalized Rankine-Hugoniot conditions for pressure delta waves are derived.
Two dimensional invisibility cloaking via transformation optics
Hongyu Liu and Ting Zhou
2011, 31(2): 525-543 doi: 10.3934/dcds.2011.31.525 +[Abstract](186) +[PDF](447.6KB)
Abstract:
We investigate two-dimensional invisibility cloaking via transformation optics approach. The cloaking media possess much more singular parameters than those having been considered for three-dimensional cloaking in literature. Finite energy solutions for these cloaking devices are studied in appropriate weighted Sobolev spaces. We derive some crucial properties of the singularly weighted Sobolev spaces. The invisibility cloaking is then justified by decoupling the underlying singular PDEs into one problem in the cloaked region and the other one in the cloaking layer. We derive some completely novel characterizations of the finite energy solutions corresponding to the singular cloaking problems. Particularly, some `hidden' boundary conditions on the cloaking interface are shown for the first time. We present our study for a very general system of PDEs, where the Helmholtz equation underlying acoustic cloaking is included as a special case.
Topological pressure and topological entropy of a semigroup of maps
Dongkui Ma and Min Wu
2011, 31(2): 545-556 doi: 10.3934/dcds.2011.31.545 +[Abstract](256) +[PDF](185.5KB)
Abstract:
By using the Carathéodory-Pesin structure(C-P structure), with respect to arbitrary subset, the topological pressure and topological entropy, introduced for a single continuous map, is generalized to the cases of semigroup of continuous maps. Several of their basic properties are provided.
On some exotic Schottky groups
Marc Peigné
2011, 31(2): 559-579 doi: 10.3934/dcds.2011.31.559 +[Abstract](201) +[PDF](489.9KB)
Abstract:
We construct a Cartan-Hadamard manifold with pinched negative curvature whose group of isometries possesses divergent discrete free subgroups with parabolic elements that do not satisfy the so-called "parabolic gap condition'' introduced in [4]. This construction relies on the comparaison between the Poincaré series of these free groups and the potential of some transfer operator which appears naturally in this context.
Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data
Joana Terra and Noemi Wolanski
2011, 31(2): 581-605 doi: 10.3934/dcds.2011.31.581 +[Abstract](223) +[PDF](491.2KB)
Abstract:
We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^{\alpha}u_0(x)\rightarrow A>0$ as $|x|\rightarrow\infty$. One of our main goals is the study of the critical case $p=1+2/\alpha$ for $0 < \alpha < N$, left open in previous articles, for which we prove that $t^{\alpha/2}|u(x,t)-U(x,t)|\to 0$ where $U$ is the solution of the heat equation with absorption with initial datum $U(x,0)=C_{A,N}|x|^{-\alpha}$. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data $u_0$ in the supercritical case and also in the critical case ($p=1+2/N$) for bounded and integrable $u_0$.

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