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Discrete & Continuous Dynamical Systems - A

2006 , Volume 15 , Issue 3

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Spectral theory and nonlinear partial differential equations: A survey
Wilhelm Schlag
2006, 15(3): 703-723 doi: 10.3934/dcds.2006.15.703 +[Abstract](452) +[PDF](313.2KB)
Abstract:
This paper reviews some of the recent work of the author on stable manifolds for unstable evolution equations. In particular, we discuss such a result, jointly with Joachim Krieger, for the critical focusing nonlinear wave equation in three dimensions.
Periodic solutions for a 3x 3 competitive system with cross-diffusion
Salomé Martínez and Wei-Ming Ni
2006, 15(3): 725-746 doi: 10.3934/dcds.2006.15.725 +[Abstract](326) +[PDF](312.9KB)
Abstract:
In this paper we study the role of cross-diffusion in the existence of spatially non-constant periodic solutions for the Lotka-Volterra competition system for three species. By properly choosing cross-diffusion coefficients, we show that Hopf bifurcation occurs at a constant steady state. Furthermore, these spatially nonhomogeneous periodic solutions are stable if diffusion rates are in appropriate ranges.
Collisionless orbits of singular and nonsingular dynamical systems
Daniel Franco and J. R. L. Webb
2006, 15(3): 747-757 doi: 10.3934/dcds.2006.15.747 +[Abstract](363) +[PDF](235.8KB)
Abstract:
In this paper we study the existence of periodic solutions for some nonlinear systems of differential equations. We assume the nonlinearity satisfies suitable properties in one direction and we consider both the singular and the nonsingular case. As an application we present the existence of collisionless orbits for singular Lagrangian systems where the singular potential can have an attractive or repulsive behaviour near the singularity and we do not need to consider so-called strong force conditions. Our method is based on fixed point index theory for completely continuous operators, involving a new type of cone. In contrast with previous work using this type of technique, the nonlinearity neither needs to be positive nor to have a constant sign behaviour. The results improve recent work even in the scalar case.
Non-autonomous boundary value problems on the real line
Barbara Bianconi and Francesca Papalini
2006, 15(3): 759-776 doi: 10.3934/dcds.2006.15.759 +[Abstract](332) +[PDF](275.4KB)
Abstract:
We study the existence of solutions for the problem $(\Phi(x'(t)))'=f(t,x(t),x'(t))$, $x(-\infty)=0, x(+\infty)=1$, where $\Phi$: IR $\to$ IR is a monotone function which generalizes the one-dimensional p-Laplacian operator. When the right-hand side of the equation has the product structure $f(t,x,x')=a(t,x)b(x,x')$, we deduce operative criteria for the existence and non-existence of solutions.
Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models
Igor Chueshov and Irena Lasiecka
2006, 15(3): 777-809 doi: 10.3934/dcds.2006.15.777 +[Abstract](354) +[PDF](454.2KB)
Abstract:
We study dynamics of a class of nonlinear Kirchhoff-Boussinesq plate models. The main results of the paper are: (i) existence and uniqueness of weak (finite energy) solutions, (ii) existence of weakly compact attractors.
Hyperbolic invariant sets with positive measures
Zhihong Xia
2006, 15(3): 811-818 doi: 10.3934/dcds.2006.15.811 +[Abstract](328) +[PDF](190.5KB)
Abstract:
In this short paper we prove some results concerning volume-preserving Anosov diffeomorphisms on compact manifolds. The first theorem is that if a $C^{1 + \alpha}$, $\alpha >0$, volume-preserving diffeomorphism on a compact connected manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. The same result had been obtained by Bochi and Viana [2]. This result is not necessarily true for $C^1$ maps. The proof uses a Pugh-Shub type of dynamically defined measure density points, which are different from the standard Lebesgue density points. We then give a direct proof of the ergodicity of $C^{1+\alpha}$ volume preserving Anosov diffeomorphisms, without using the usual Hopf arguments or the Birkhoff ergodic theorem. The method we introduced also has interesting applications to partially hyperbolic and non-uniformly hyperbolic systems.
Global stability of traveling curved fronts in the Allen-Cahn equations
Hirokazu Ninomiya and Masaharu Taniguchi
2006, 15(3): 819-832 doi: 10.3934/dcds.2006.15.819 +[Abstract](481) +[PDF](274.2KB)
Abstract:
This paper is concerned with the global stability of a traveling curved front in the Allen-Cahn equation. The existence of such a front is recently proved by constructing supersolutions and subsolutions. In this paper, we introduce a method to construct new subsolutions and prove the asymptotic stability of traveling curved fronts globally in space.
On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics
Flavia Antonacci and Marco Degiovanni
2006, 15(3): 833-842 doi: 10.3934/dcds.2006.15.833 +[Abstract](274) +[PDF](203.9KB)
Abstract:
We provide a qualitative description of curves minimizing the energy functional on a Riemannian manifold whose metric is discontinuous along a hypersurface $\Sigma$. Such a study is motivated by the variational description of refraction phenomena.
On stochastic stabilization of difference equations
John A. D. Appleby, Xuerong Mao and Alexandra Rodkina
2006, 15(3): 843-857 doi: 10.3934/dcds.2006.15.843 +[Abstract](340) +[PDF](274.9KB)
Abstract:
We consider unstable scalar deterministic difference equation

$x_{n+1}=x_n(1+a_nf(x_n))$, $n\ge 1$, $x_0=a$.

We show how this equation can be stabilized by adding the random noise term $\sigma_ng(x_n)\xi_{n+1}$ where $\xi_n$ takes the values +1 or -1 each with probability $1/2$. We also prove a theorem on the almost sure asymptotic stability of the solution of a scalar nonlinear stochastic difference equation with bounded coefficients, and show the connection between the noise stabilization of a stochastic differential equation, and a discretization of this equation.

$Z^d$ Toeplitz arrays
María Isabel Cortez
2006, 15(3): 859-881 doi: 10.3934/dcds.2006.15.859 +[Abstract](314) +[PDF](392.4KB)
Abstract:
In this paper we give a definition of Toeplitz sequences and odometers for $\mathbb{Z}^d$ actions for $d\geq 1$ which generalizes that in dimension one. For these new concepts we study properties of the induced Toeplitz dynamical systems and odometers classical for $d=1$. In particular, we characterize the $\mathbb{Z}^d$-regularly recurrent systems as the minimal almost 1-1 extensions of odometers and the $\mathbb{Z}^d$-Toeplitz systems as the family of subshifts which are regularly recurrent.
Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$
Pietro Baldi
2006, 15(3): 883-903 doi: 10.3934/dcds.2006.15.883 +[Abstract](313) +[PDF](317.2KB)
Abstract:
We consider 1D completely resonant nonlinear wave equations of the type $v_{t t}$ - $v_{x x}$$ = -v^3 + \mathcal{O}(v^4)$ with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, depending only on time.
Dynamics of the degree six Landen transformation
Marc Chamberland and Victor H. Moll
2006, 15(3): 905-919 doi: 10.3934/dcds.2006.15.905 +[Abstract](330) +[PDF](265.4KB)
Abstract:
We establish the basin of attraction for the fixed point $(3,3)$ of a dynamical system arising from the evaluation of a definite integral.
Asymptotic selection of viscosity equilibria of semilinear evolution equations by the introduction of a slowly vanishing term
Felipe Alvarez and Alexandre Cabot
2006, 15(3): 921-938 doi: 10.3934/dcds.2006.15.921 +[Abstract](374) +[PDF](290.1KB)
Abstract:
The behavior at infinity is investigated of global solutions to some nonautonomous semilinear evolution equations with conservative and convex nonlinearities. It is proved that the trajectories converge to viscosity stationary solutions as time goes to infinity, that is, they evolve towards stationary solutions that are minimal with respect to a generalized viscosity criterion. Hierarchical viscosity selections and applications to specific nonlinear PDE are given.
Periodic solutions for discrete convex Hamiltonian systems via Clarke duality
Jianshe Yu, Honghua Bin and Zhiming Guo
2006, 15(3): 939-950 doi: 10.3934/dcds.2006.15.939 +[Abstract](364) +[PDF](216.7KB)
Abstract:
Based on the Legendre transform, the dual action functional which corresponds to discrete Hamiltonian systems is given. In this paper, the existence of periodic solution for discrete convex Hamiltonian systems with forcing terms is obtained by using the dual least action principle and the perturbation technique.
Branches of harmonic solutions to periodically perturbed coupled differential equations on manifolds
Marco Spadini
2006, 15(3): 951-964 doi: 10.3934/dcds.2006.15.951 +[Abstract](285) +[PDF](269.5KB)
Abstract:
Fixed point index methods are used to explore the structure of the set of harmonic solutions to periodically perturbed coupled differential equations on differentiable manifolds. The results obtained generalize existing theorems for single differential equation by gathering them in an unique framework.
Ill-Posedness for the Benney system
Adán J. Corcho
2006, 15(3): 965-972 doi: 10.3934/dcds.2006.15.965 +[Abstract](291) +[PDF](138.1KB)
Abstract:
We discuss ill-posedness issues for the initial value problem associated to the Benney system. To prove our results we use the method introduced by Kenig, Ponce and Vega [10] to show ill-posedness for some canonical dispersive equations.
Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity
Hironobu Sasaki
2006, 15(3): 973-981 doi: 10.3934/dcds.2006.15.973 +[Abstract](367) +[PDF](140.7KB)
Abstract:
We consider the scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity. We give some estimates for the nonlinearity, and prove the existence of the scattering operator, which improves the known results in some sense. Our proof is based on the Strichartz estimates for the inhomogeneous Klein-Gordon equation.
Applied equivariant degree, part I: An axiomatic approach to primary degree
Zalman Balanov, Wieslaw Krawcewicz and Haibo Ruan
2006, 15(3): 983-1016 doi: 10.3934/dcds.2006.15.983 +[Abstract](412) +[PDF](361.8KB)
Abstract:
An axiomatic approach to the primary equivariant degree is discussed and a construction of the primary equivariant degree via fundamental domains is presented. For a class of equivariant maps, which naturally appear in one-parameter equivariant Hopf bifurcation, effective computational primary degree formulae are established.

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