ISSN:
1078-0947
eISSN:
1553-5231
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Discrete and Continuous Dynamical Systems
August 2006 , Volume 15 , Issue 3
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2006, 15(3): 703-723
doi: 10.3934/dcds.2006.15.703
+[Abstract](3956)
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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.
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.
2006, 15(3): 725-746
doi: 10.3934/dcds.2006.15.725
+[Abstract](2705)
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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.
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.
2006, 15(3): 747-757
doi: 10.3934/dcds.2006.15.747
+[Abstract](2882)
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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.
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.
2006, 15(3): 759-776
doi: 10.3934/dcds.2006.15.759
+[Abstract](2464)
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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.
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.
2006, 15(3): 777-809
doi: 10.3934/dcds.2006.15.777
+[Abstract](2845)
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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.
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.
2006, 15(3): 811-818
doi: 10.3934/dcds.2006.15.811
+[Abstract](2892)
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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.
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.
2006, 15(3): 819-832
doi: 10.3934/dcds.2006.15.819
+[Abstract](3518)
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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.
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.
2006, 15(3): 833-842
doi: 10.3934/dcds.2006.15.833
+[Abstract](2348)
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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.
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.
2006, 15(3): 843-857
doi: 10.3934/dcds.2006.15.843
+[Abstract](3069)
+[PDF](274.9KB)
Abstract:
We consider unstable scalar deterministic difference equation
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.
2006, 15(3): 859-881
doi: 10.3934/dcds.2006.15.859
+[Abstract](2348)
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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.
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.
2006, 15(3): 883-903
doi: 10.3934/dcds.2006.15.883
+[Abstract](2526)
+[PDF](317.2KB)
Abstract:
We consider 1D completely resonant nonlinear wave equations of the type $v_{t t}$ - $v_{x x}$$ = -v^3 + \mathcal{O}(v^4)$ with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, depending only on time.
We consider 1D completely resonant nonlinear wave equations of the type $v_{t t}$ - $v_{x x}$$ = -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.
2006, 15(3): 905-919
doi: 10.3934/dcds.2006.15.905
+[Abstract](2793)
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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.
We establish the basin of attraction for the fixed point $(3,3)$ of a dynamical system arising from the evaluation of a definite integral.
2006, 15(3): 921-938
doi: 10.3934/dcds.2006.15.921
+[Abstract](2403)
+[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.
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.
2006, 15(3): 939-950
doi: 10.3934/dcds.2006.15.939
+[Abstract](2795)
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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.
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
2006, 15(3): 951-964
doi: 10.3934/dcds.2006.15.951
+[Abstract](2257)
+[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.
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.
2006, 15(3): 965-972
doi: 10.3934/dcds.2006.15.965
+[Abstract](2273)
+[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.
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.
2006, 15(3): 973-981
doi: 10.3934/dcds.2006.15.973
+[Abstract](2879)
+[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.
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.
2006, 15(3): 983-1016
doi: 10.3934/dcds.2006.15.983
+[Abstract](2937)
+[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.
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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