# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

August 2011 , Volume 30 , Issue 3

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2011, 30(3): 573-621 doi: 10.3934/dcds.2011.30.573 +[Abstract](1203) +[PDF](791.0KB)
Abstract:
We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2 \approx 0.866$. This extends previous work of Klainerman-Machedon [24] on finite energy data $s \geq 1$, and Eardley-Moncrief [11] for still smoother data. We use the method of almost conservation laws, sometimes called the "I-method", to construct an almost conserved quantity based on the Hamiltonian, but at the regularity of $H^s_x$ rather than $H^1_x$. One then uses Strichartz, null form, and commutator estimates to control the development of this quantity. The main technical difficulty (compared with other applications of the method of almost conservation laws) is at low frequencies, because of the poor control on the $L^2_x$ norm. In an appendix, we demonstrate the equations' relative lack of smoothing - a property that presents serious difficulties for studying rough solutions using other known methods.
2011, 30(3): 623-639 doi: 10.3934/dcds.2011.30.623 +[Abstract](1034) +[PDF](252.1KB)
Abstract:
The aim of this work is to characterize the discrete gradient vector fields on infinite and locally finite simplicial complexes which are induced by a proper discrete Morse function. This characterization is essentially given by the non-existence of closed trajectories and the absence of a certain kind of incidence between monotonous rays in the given field.
2011, 30(3): 641-670 doi: 10.3934/dcds.2011.30.641 +[Abstract](1287) +[PDF](1447.5KB)
Abstract:
We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. We provide conditions which imply the existence of the manifold within an investigated region of the phase space. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method to obtain an invariant manifold within an explicit range of parameters for the rotating Hénon map.
2011, 30(3): 671-685 doi: 10.3934/dcds.2011.30.671 +[Abstract](1017) +[PDF](482.4KB)
Abstract:
We prove that generic symmetric $C^{r}$-vector field families on $\mathbb{R}^{3}$ unfolding a contracting singular cycle, exhibits singular attractors for a positive lebesgue measure set of parameter values. Essentially the cycle is formed by a real contracting singularity, like those in the geometric contracting Lorenz attractor, whose unstable branches go to periodic orbits in the cycle. We obtain a lower estimate for the density of this set at the first bifurcation value. Furthermore, for parameter values in this set the corresponding vector field admits a unique SRB measure, whose support coincides with the attractor.
2011, 30(3): 687-697 doi: 10.3934/dcds.2011.30.687 +[Abstract](1046) +[PDF](382.5KB)
Abstract:
In this paper we investigate smooth conjugacy of $C^\infty$ expanding maps on certain nilmanifolds. We show that several rigidity results about expanding maps on the circle can not be generalized directly to higher dimensions. For example the following result is obtained: Let $\Gamma_1\backslash N_1$ and $\Gamma_2\backslash N_2$ be two nilmanifolds of homogeneous type. We show that for any positive integer $k$ there exist on the product nilmanifold $\Gamma_1\times \Gamma_2\backslash N_1\times N_2$ a $C^\infty$ expanding map $\varphi$ and an expanding nilendomorphism $\psi$ which are $C^k$ conjugate, but not $C^{k,lip}$ conjugate. While in the case of dimension one, it was shown by M. Shub and D. Sullivan that if two $C^\infty$ expanding maps on $\mathbb S^1$ are absolutely continuously conjugate, then they must be $C^\infty$ conjugate.
2011, 30(3): 699-708 doi: 10.3934/dcds.2011.30.699 +[Abstract](1210) +[PDF](368.5KB)
Abstract:
Let $\{M_i\}_{i=1}^l$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1\cdots i_n\in \{1,\ldots, l\}^n$ such that $M_{i_1}\cdots M_{i_n}\ne$0. Let P : $(0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^l$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.
2011, 30(3): 709-735 doi: 10.3934/dcds.2011.30.709 +[Abstract](1167) +[PDF](342.9KB)
Abstract:
In this paper we present a general framework for applications of the twisted equivariant degree (with one free parameter) to an autonomous $\Gamma$-symmetric system of functional differential equations in order to detect and classify (according to their symmetric properties) its periodic solutions. As an example we establish the existence of multiple non-constant periodic solutions of delay Lotka-Volterra equations with $\Gamma$-symmetries. We also include some computational examples for several finite groups $\Gamma$.
2011, 30(3): 737-765 doi: 10.3934/dcds.2011.30.737 +[Abstract](1036) +[PDF](530.2KB)
Abstract:
We study the well-posedness of initial value problems for scalar functional algebraic and differential functional equations of mixed type. We provide a practical way to determine whether such problems admit unique solutions that grow at a specified rate. In particular, we exploit the fact that the answer to such questions is encoded in an integer n#. We show how this number can be tracked as a problem is transformed to a reference problem for which a Wiener-Hopf splitting can be computed. Once such a splitting is available, results due to Mallet-Paret and Verduyn-Lunel can be used to compute n#. We illustrate our techniques by analytically studying the well-posedness of two macro-economic overlapping generations models for which Wiener-Hopf splittings are not readily available.
2011, 30(3): 767-777 doi: 10.3934/dcds.2011.30.767 +[Abstract](978) +[PDF](353.8KB)
Abstract:
For a continuous selfmap $f$ of a compact metric space $X$ we study the set of its continuous extensions $F$ on the space $X\times I$, where $I$ is a compact interval. In particular, we have solved an open problem (raised in [Ll. Alseda, S. Kolyada, J. Llibre, and L. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999)]) by proving that any continuous transitive map $f$ on $X$ can be extended to a continuous transitive triangular map $F=(f,g_x)$ on $X\times I$ without increasing topological entropy.
2011, 30(3): 779-790 doi: 10.3934/dcds.2011.30.779 +[Abstract](1295) +[PDF](319.3KB)
Abstract:
We study the Hopf bifurcation from the singular point with eigenvalues $a$ε$\ \pm\ bi$ and $c$ε located at the origen of an analytic differential system of the form $\dot x= f( x)$, where $x \in \R^3$. Under convenient assumptions we prove that the Hopf bifurcation can produce $1$, $2$ or $3$ limit cycles. We also characterize the stability of these limit cycles. The main tool for proving these results is the averaging theory of first and second order.
2011, 30(3): 791-806 doi: 10.3934/dcds.2011.30.791 +[Abstract](1231) +[PDF](302.6KB)
Abstract:
Let D:$=\{(x,y);\ x^2+y^2 < l^2\}\subset\R^2$. We study the shape of a local minimizer of the problem

$E(u)=\int_{D}(\frac{|\nabla u|^2}{2}+\lambda W(u)) dx\ \$subject to$\ \ m=\frac{1}{|D|}\int_Dudx$

and study the global structure of critical points. We show that for an arbitrary potential $W\in C^4$ every level set of every nonconstant local minimizer is a $C^1$-curve and it divides $D$ into exactly two simply connected subdomains. Next we consider the case $W(u)=(u^2-1)^2/4$ (Cahn-Hilliard equation). When $\lambda$ varies and $m$ is fixed, we show that this problem has an unbounded continuum of critical points. When $m$ varies and $\lambda$ is fixed, we show that this problem has a bounded continuum meeting at two different points on the trivial branch. Moreover, we show that in each case a bifurcating critical point is stable (a local minimizer) near the bifurcation point in a certain parameter range. The main technique is the nodal curve which relates the shape with the Morse index. We do not use a small parameter or the $\Gamma$-convergence technique.

2011, 30(3): 807-834 doi: 10.3934/dcds.2011.30.807 +[Abstract](1312) +[PDF](493.8KB)
Abstract:
We consider the Cauchy problem for the critical nongauge invariant nonlinear Schrödinger equations

$iu_{t}+\frac{1}{2}$uxx$=i\mu\overline{u}^{\alpha}u^{\beta},\text{ } x\in\mathbf{R},\text{ }t>0,$
$\ \ \ \ \ \ \ \ u(0,x) =u_{0}(x) ,\text{ }x\in\mathbf{R,} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

where $\beta>\alpha\geq0,$ $\alpha+\beta\geq2,$ $\mu=-i^{\frac{\omega}{2} }t^{\frac{\theta}{2}-1},$ $\omega=\beta-\alpha-1,$ $\theta=\alpha+\beta-1.$ We prove that there exists a unique solution $u\in\mathbf{C}( [ 0,\infty) ;\mathbf{H}^{1}\cap\mathbf{H}^{0,1})$ of the Cauchy problem (1). Also we find the large time asymptotics of solutions.

2011, 30(3): 835-849 doi: 10.3934/dcds.2011.30.835 +[Abstract](1334) +[PDF](380.0KB)
Abstract:
We prove the existence of periodic and solitary traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities. The approach is based on variational techniques and concentration compactness.
2011, 30(3): 851-871 doi: 10.3934/dcds.2011.30.851 +[Abstract](1484) +[PDF](507.8KB)
Abstract:
In this work we establish new results about the existence of smooth, explicit families of periodic traveling waves for the modified and fourth-order Benjamin-Bona-Mahony equations. We also prove, under certain conditions, that these families are nonlinearly stable in the energy space. The techniques employed may be of further use in the study of the stability of periodic traveling-wave solutions of other nonlinear evolution equations.
2011, 30(3): 873-890 doi: 10.3934/dcds.2011.30.873 +[Abstract](1302) +[PDF](418.9KB)
Abstract:
The cyclicity of the period annulus of a quadratic reversible and non-Hamiltonian system under quadratic perturbations is studied. The centroid curve method and other mathematical techniques are combined to prove that the related Abelian integral has at most two zeros. This gives a proof of Conjecture 1 in [8] for one case.
2011, 30(3): 891-916 doi: 10.3934/dcds.2011.30.891 +[Abstract](1132) +[PDF](498.5KB)
Abstract:
Uniqueness and nonuniqueness of solutions to the first initial-boundary value problem for degenerate semilinear parabolic equations, with possibly unbounded coefficients, are studied. Sub- and supersolutions of suitable auxiliary problems, such as the first exit time problem, are used to determine on which part of the boundary Dirichlet data must be given. As an application of the general results, we study uniqueness and nonuniqueness of bounded solutions to a semilinear Cauchy problem in the hyperbolic space $\mathbb{H}^n$ using the Poincaré model.
2011, 30(3): 917-944 doi: 10.3934/dcds.2011.30.917 +[Abstract](1203) +[PDF](542.2KB)
Abstract:
We consider piecewise $\C^{1+\alpha}$ uniformly expanding maps on a Riemannian manifold, and study their invariant physical measures. We study the Perron-Frobenius operator on Sobolev spaces and bounded variation spaces, and prove that it is quasicompact if some conditions on the Lyapunov exponent and the combinatorial complexities are satisfied. Then, we get strong results concerning the existence of physical ergodic measures, and the exponential mixing of smooth observables.
2011, 30(3): 945-963 doi: 10.3934/dcds.2011.30.945 +[Abstract](1191) +[PDF](372.8KB)
Abstract:
By using the local active coordinates consisting of tangent vectors of the invariant subspaces, as well as the Silnikov coordinates, the simple normal form is established in the neighborhood of the double homoclinic loops with bellows configuration in a general system, then the dynamics near the homoclinic bellows is investigated, and the existence, uniqueness of the homoclinic orbits and periodic orbits with various patterns bifurcated from the primary orbits are demonstrated, and the corresponding bifurcation curves (or surfaces) and existence regions are located.
2011, 30(3): 965-994 doi: 10.3934/dcds.2011.30.965 +[Abstract](1119) +[PDF](566.3KB)
Abstract:
We consider the following singularly perturbed elliptic problem

ε2Δ ũ + (ũ –a(ŷ))(1- ũ2)=0 in M

&ytilde; where M is a two dimensional smooth compact Riemannian manifold associated with metric ğ, ε is a small parameter. The inhomogeneous term -1 < a(ŷ) < 1 takes maximum value b with 0 < b < 1. Assume that Γ = { ŷ ∈ M : a(ŷ) = 0} is a closed, smooth curve that Γ separate M into two disjoint components M+ and M- and also ∂a/∂v > 0 on Γ, where v is the normal of Γ pointing to the interior of M-. Moreover the maximum value loop Γ = { ŷ ∈ M : a(ŷ) = b} is a closed, smooth geodesic contained in M in such a way and Γ separate M- into two disjoint components. We will show the existence of solution possessing both transition and concentration phenomenon, i.e.

uε → + 1 in M-\ Γδ, uε → -1 in M+, uε → 1 – C along Γ as ε → 0,

where Γδ is a small neighborhood of Γ and C is a fixed positive constant.

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