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1078-0947
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Discrete & Continuous Dynamical Systems - A
January 2010 , Volume 26 , Issue 1
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2010, 26(1): 1-42
doi: 10.3934/dcds.2010.26.1
+[Abstract](2367)
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Abstract:
Given a $C^1$-diffeomorphism $f$ of a compact manifold, we show that if the stable/unstable dominated splitting along a saddle is weak enough, then there is a small $C^1$-perturbation that preserves the orbit of the saddle and that generates a homoclinic tangency related to it. Moreover, we show that the perturbation can be performed preserving a homoclinic relation to another saddle. We derive some consequences on homoclinic classes. In particular, if the homoclinic class of a saddle $P$ has no dominated splitting of same index as $P$, then a $C^1$-perturbation generates a homoclinic tangency related to $P$.
Given a $C^1$-diffeomorphism $f$ of a compact manifold, we show that if the stable/unstable dominated splitting along a saddle is weak enough, then there is a small $C^1$-perturbation that preserves the orbit of the saddle and that generates a homoclinic tangency related to it. Moreover, we show that the perturbation can be performed preserving a homoclinic relation to another saddle. We derive some consequences on homoclinic classes. In particular, if the homoclinic class of a saddle $P$ has no dominated splitting of same index as $P$, then a $C^1$-perturbation generates a homoclinic tangency related to $P$.
2010, 26(1): 43-61
doi: 10.3934/dcds.2010.26.43
+[Abstract](2727)
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Abstract:
In this paper we prove a general result concerning continuity of the blow-up time and the blow-up set for an evolution problem under perturbations. This result is based on some convergence of the perturbations for times smaller than the blow-up time of the unperturbed problem together with uniform bounds on the blow-up rates of the perturbed problems.
We also present several examples. Among them we consider changing the spacial domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow-up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set.
In this paper we prove a general result concerning continuity of the blow-up time and the blow-up set for an evolution problem under perturbations. This result is based on some convergence of the perturbations for times smaller than the blow-up time of the unperturbed problem together with uniform bounds on the blow-up rates of the perturbed problems.
We also present several examples. Among them we consider changing the spacial domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow-up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set.
2010, 26(1): 63-74
doi: 10.3934/dcds.2010.26.63
+[Abstract](2077)
+[PDF](170.4KB)
Abstract:
We characterize completely in terms of strict quadratic Lyapunov sequences when a linear nonautonomous dynamics admits a nonuniform exponential dichotomy. We allow asymptotic rates of the form $e^{c\rho(m)}$ determined by an arbitrary increasing sequence $\rho(m)$, which thus may correspond to infinite Lyapunov exponents, and not only the usual exponential behavior with $\rho(m)=m$. In particular, we obtain inverse theorems in this general setting, by constructing explicitly a strict quadratic Lyapunov sequence for each nonuniform exponential dichotomy. Furthermore, for a large class of perturbations and using only quadratic Lyapunov sequences, we show in a simple manner that if the linear dynamics admits a nonuniform exponential dichotomy, then the perturbed dynamics remains unstable.
We characterize completely in terms of strict quadratic Lyapunov sequences when a linear nonautonomous dynamics admits a nonuniform exponential dichotomy. We allow asymptotic rates of the form $e^{c\rho(m)}$ determined by an arbitrary increasing sequence $\rho(m)$, which thus may correspond to infinite Lyapunov exponents, and not only the usual exponential behavior with $\rho(m)=m$. In particular, we obtain inverse theorems in this general setting, by constructing explicitly a strict quadratic Lyapunov sequence for each nonuniform exponential dichotomy. Furthermore, for a large class of perturbations and using only quadratic Lyapunov sequences, we show in a simple manner that if the linear dynamics admits a nonuniform exponential dichotomy, then the perturbed dynamics remains unstable.
2010, 26(1): 75-87
doi: 10.3934/dcds.2010.26.75
+[Abstract](1908)
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Abstract:
Let $F:\R^2\to \R^2$, $F=(p,q)$, be a polynomial mapping such that $\det DF$ never vanishes. In this paper it is shown that if either $p$ or $q$ has degree less or equal 3, then $F$ is injective. The technique relates solvability of appropriate vector fields with injectivity of the mapping.
Let $F:\R^2\to \R^2$, $F=(p,q)$, be a polynomial mapping such that $\det DF$ never vanishes. In this paper it is shown that if either $p$ or $q$ has degree less or equal 3, then $F$ is injective. The technique relates solvability of appropriate vector fields with injectivity of the mapping.
2010, 26(1): 89-104
doi: 10.3934/dcds.2010.26.89
+[Abstract](1958)
+[PDF](240.0KB)
Abstract:
We study the differentiability of the solution of the Dirichlet problem associated to the system
We study the differentiability of the solution of the Dirichlet problem associated to the system
$A(u) \equiv - D_i (A_{ij}(x) D_j u) = \mu $
$u \in W^{1,1}_0$(Ω$, \IR^N)$
where Ω is an open bounded subset of $\IR^n$ $(n \geq 2)$ with $C^1$-boundary, $A$ is an elliptic operator with C 0, α-coefficients ($\alpha \in ]0,1]$) and $\mu$ is a signed Radon measure with finite total variation, satisfying a suitable density condition.
2010, 26(1): 105-134
doi: 10.3934/dcds.2010.26.105
+[Abstract](2048)
+[PDF](339.9KB)
Abstract:
We consider examples of discrete nonlinear Schrödinger equations in $\Z$ admitting ground states which are orbitally but not asymptotically stable in l $^2(\Z )$. The ground states contain internal modes which decouple from the continuous modes. The absence of leaking of energy from discrete to continues modes leads to an almost conservation and perpetual oscillation of the discrete modes. This is quite different from what is known for nonlinear Schrödinger equations in $\R ^d$. To achieve our goal we prove a Siegel normal form theorem, prove dispersive estimates for the linearized operators and prove some nonlinear estimates.
We consider examples of discrete nonlinear Schrödinger equations in $\Z$ admitting ground states which are orbitally but not asymptotically stable in l $^2(\Z )$. The ground states contain internal modes which decouple from the continuous modes. The absence of leaking of energy from discrete to continues modes leads to an almost conservation and perpetual oscillation of the discrete modes. This is quite different from what is known for nonlinear Schrödinger equations in $\R ^d$. To achieve our goal we prove a Siegel normal form theorem, prove dispersive estimates for the linearized operators and prove some nonlinear estimates.
2010, 26(1): 135-149
doi: 10.3934/dcds.2010.26.135
+[Abstract](2485)
+[PDF](223.3KB)
Abstract:
This paper is concerned with the Klein-Gordon-Maxwell system in a bounded space domain. We discuss the existence of standing waves $\psi=u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nabla\phi(x)$. We assume homogeneous Dirichlet boundary conditions on $u$ and non homogeneous Neumann boundary conditions on $\phi$. In the "linear" case we prove the existence of a nontrivial solution when the coupling constant is sufficiently small. Moreover we show that a suitable nonlinear perturbation in the matter equation gives rise to infinitely many solutions. These problems have a variational structure so that we can apply global variational methods.
This paper is concerned with the Klein-Gordon-Maxwell system in a bounded space domain. We discuss the existence of standing waves $\psi=u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nabla\phi(x)$. We assume homogeneous Dirichlet boundary conditions on $u$ and non homogeneous Neumann boundary conditions on $\phi$. In the "linear" case we prove the existence of a nontrivial solution when the coupling constant is sufficiently small. Moreover we show that a suitable nonlinear perturbation in the matter equation gives rise to infinitely many solutions. These problems have a variational structure so that we can apply global variational methods.
2010, 26(1): 151-196
doi: 10.3934/dcds.2010.26.151
+[Abstract](1943)
+[PDF](563.3KB)
Abstract:
We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy.
The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.
We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy.
The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to $(1+1)$-dimensional completely integrable soliton equations of differential-difference type.
2010, 26(1): 197-223
doi: 10.3934/dcds.2010.26.197
+[Abstract](2408)
+[PDF](286.8KB)
Abstract:
We study the traveling wave front solutions for a two-dimensional periodic lattice dynamical system with monostable nonlinearity. We first show that there is a minimal speed such that a traveling wave solution exists if and only if its speed is above this minimal speed. Then we prove that any wave profile is strictly monotone. Finally, we derive the convergence of discretized minimal speed to the continuous minimal speed.
We study the traveling wave front solutions for a two-dimensional periodic lattice dynamical system with monostable nonlinearity. We first show that there is a minimal speed such that a traveling wave solution exists if and only if its speed is above this minimal speed. Then we prove that any wave profile is strictly monotone. Finally, we derive the convergence of discretized minimal speed to the continuous minimal speed.
2010, 26(1): 225-249
doi: 10.3934/dcds.2010.26.225
+[Abstract](2038)
+[PDF](1427.0KB)
Abstract:
We consider a class $ \mathcal{Q}(M) \,$ consisting of smooth quartic differential forms which are defined on an oriented two-manifold $ M $, to each of which we associate a pair of transversal nets with common singularities. These quartic forms are related to geometric objects such as curvature lines, asymptotic lines of surfaces immersed in $\R^4.$ Local problems around the rank-2 singular points of the elements of $ \mathcal{Q}(M) \,$, such as stability, normal forms, finite determinacy, versal unfoldings, are studied in [2]. Here we make a similar study for a rank-1 singular point that is analogous to the saddle-node singularity of vector fields.
We consider a class $ \mathcal{Q}(M) \,$ consisting of smooth quartic differential forms which are defined on an oriented two-manifold $ M $, to each of which we associate a pair of transversal nets with common singularities. These quartic forms are related to geometric objects such as curvature lines, asymptotic lines of surfaces immersed in $\R^4.$ Local problems around the rank-2 singular points of the elements of $ \mathcal{Q}(M) \,$, such as stability, normal forms, finite determinacy, versal unfoldings, are studied in [2]. Here we make a similar study for a rank-1 singular point that is analogous to the saddle-node singularity of vector fields.
2010, 26(1): 251-263
doi: 10.3934/dcds.2010.26.251
+[Abstract](1941)
+[PDF](206.4KB)
Abstract:
We study the existence of transitive exchange transformations with flips defined on the unit circle $S^1$. We provide a complete answer to the question of whether there exists a transitive exchange transformation of $S^1$ defined on $n$ subintervals and having $f$ flips.
We study the existence of transitive exchange transformations with flips defined on the unit circle $S^1$. We provide a complete answer to the question of whether there exists a transitive exchange transformation of $S^1$ defined on $n$ subintervals and having $f$ flips.
2010, 26(1): 265-290
doi: 10.3934/dcds.2010.26.265
+[Abstract](2662)
+[PDF](329.5KB)
Abstract:
This paper studies the traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. The existence, uniqueness, asymptotics as well as the stability of the wave solutions are investigated. The traveling wave solutions, existed for a continuance of wave speeds, do not approach the equilibria exponentially with speed larger than the critical one. While with the critical speed, the wave solutions approach to one equilibrium exponentially fast and to the other equilibrium algebraically. This is in sharp contrast with the asymptotic behaviors of the wave solutions of the classical KPP and $m-th$ order Fisher equations. A delicate construction of super- and sub-solution shows that the wave solution with critical speed is globally asymptotically stable. A simpler alternative existence proof by LaSalle's Wazewski principle is also provided in the last section.
This paper studies the traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. The existence, uniqueness, asymptotics as well as the stability of the wave solutions are investigated. The traveling wave solutions, existed for a continuance of wave speeds, do not approach the equilibria exponentially with speed larger than the critical one. While with the critical speed, the wave solutions approach to one equilibrium exponentially fast and to the other equilibrium algebraically. This is in sharp contrast with the asymptotic behaviors of the wave solutions of the classical KPP and $m-th$ order Fisher equations. A delicate construction of super- and sub-solution shows that the wave solution with critical speed is globally asymptotically stable. A simpler alternative existence proof by LaSalle's Wazewski principle is also provided in the last section.
2010, 26(1): 291-302
doi: 10.3934/dcds.2010.26.291
+[Abstract](1752)
+[PDF](197.2KB)
Abstract:
Given $(X,T)$ and $(Y,S)$ mixing subshifts of finite type such that $(Y,S)$ is a factor of $(X,T)$ with factor map $\pi$:$\ X\to Y$, and positive Hölder continuous functions $\varphi$:$\ X\to \mathbb{R}$ and $\psi$:$\ Y\to \mathbb{R}$, we prove that the maximum of
Given $(X,T)$ and $(Y,S)$ mixing subshifts of finite type such that $(Y,S)$ is a factor of $(X,T)$ with factor map $\pi$:$\ X\to Y$, and positive Hölder continuous functions $\varphi$:$\ X\to \mathbb{R}$ and $\psi$:$\ Y\to \mathbb{R}$, we prove that the maximum of
$\frac{h_{\mu\circ \pi^{-1}}(S)}{\int \psi\circ\pi\d\mu}+ \frac{h_\mu(T)-h_{\mu\circ \pi^{-1}}(S)}{\int \varphi\d\mu}$
over all $T$-invariant Borel probability measures $\mu$ on $X$ is attained on the subset of ergodic measures. Here $h_\mu(T)$ stands for the metric entropy of $T$ with respect to $\mu$. As an application, we prove the existence of an ergodic invariant measure with full dimension for a class of transformations treated in [11], and also for the transformations treated in [17], where the author considers nonlinear skew-product perturbations of general Sierpinski carpets. In order to do so we establish a variational principle for the topological pressure of certain noncompact sets.
2010, 26(1): 303-312
doi: 10.3934/dcds.2010.26.303
+[Abstract](2390)
+[PDF](145.0KB)
Abstract:
For a class of one-dimensional reaction-diffusion equations, we establish the existence of generalized fronts, as recently defined by Berestycki and Hamel. We also prove uniform nondegeneracy estimates, such as a lower bound on the time derivative on some level sets, as well as a lower bound on the spatial derivative.
For a class of one-dimensional reaction-diffusion equations, we establish the existence of generalized fronts, as recently defined by Berestycki and Hamel. We also prove uniform nondegeneracy estimates, such as a lower bound on the time derivative on some level sets, as well as a lower bound on the spatial derivative.
2010, 26(1): 313-331
doi: 10.3934/dcds.2010.26.313
+[Abstract](2253)
+[PDF](224.6KB)
Abstract:
We study the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It was shown in our previous paper that in some parameter range, the problem has a time-local solution with prescribed moving singularities. Our concern in this paper is the existence of a time-global solution. By using a perturbed Haraux-Weissler equation, it is shown that there exists a forward self-similar solution with a moving singularity. Using this result, we also obtain a sufficient condition for the global existence of solutions with a moving singularity.
We study the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It was shown in our previous paper that in some parameter range, the problem has a time-local solution with prescribed moving singularities. Our concern in this paper is the existence of a time-global solution. By using a perturbed Haraux-Weissler equation, it is shown that there exists a forward self-similar solution with a moving singularity. Using this result, we also obtain a sufficient condition for the global existence of solutions with a moving singularity.
2010, 26(1): 333-346
doi: 10.3934/dcds.2010.26.333
+[Abstract](1924)
+[PDF](217.8KB)
Abstract:
In this paper we present an extendible, block gluing $\z^3$ shift of finite type $\w^{el}$ in which the topological entropy equals the $L$-projectional entropy for a two-dimensional sublattice $L$<$\z^3$, even so $\w^{el}$ is not a full $\z$ extension of $\w^{el}_L$. In particular this example shows that Theorem 4.1 of [4] does not generalize to $r$-dimensional sublattices $L$ for $r>1$.
Nevertheless we are able to reprove and extend the result about one-dimensional sublattices for general $\z^d$ shifts - instead of shifts of finite type - under the same mixing assumption as in [4] and by posing a stronger mixing condition we also obtain the corresponding statement for higher-dimensional sublattices.
In this paper we present an extendible, block gluing $\z^3$ shift of finite type $\w^{el}$ in which the topological entropy equals the $L$-projectional entropy for a two-dimensional sublattice $L$<$\z^3$, even so $\w^{el}$ is not a full $\z$ extension of $\w^{el}_L$. In particular this example shows that Theorem 4.1 of [4] does not generalize to $r$-dimensional sublattices $L$ for $r>1$.
Nevertheless we are able to reprove and extend the result about one-dimensional sublattices for general $\z^d$ shifts - instead of shifts of finite type - under the same mixing assumption as in [4] and by posing a stronger mixing condition we also obtain the corresponding statement for higher-dimensional sublattices.
2010, 26(1): 347-363
doi: 10.3934/dcds.2010.26.347
+[Abstract](1736)
+[PDF](256.9KB)
Abstract:
Recently, several numerical invariants have been introduced to characterize the distortion induced by automorphisms of a free group. We unify these by interpreting them in terms of an entropy function of a kind familiar in thermodynamic ergodic theory. We draw an analogy between this approach and the Manhattan curve associated to a pair of hyperbolic surfaces.
Recently, several numerical invariants have been introduced to characterize the distortion induced by automorphisms of a free group. We unify these by interpreting them in terms of an entropy function of a kind familiar in thermodynamic ergodic theory. We draw an analogy between this approach and the Manhattan curve associated to a pair of hyperbolic surfaces.
2010, 26(1): 365-378
doi: 10.3934/dcds.2010.26.365
+[Abstract](2042)
+[PDF](216.1KB)
Abstract:
Group extensions of Gaussian $\mathbb{G}$-actions with absolutely continuous spectrum in the orthocomplement of the functions depending on the first coordinate are constructed for $\mathbb{G}$ equal to $\mathbb{Z}^{d}$, $d\in\mathbb{N}\cup\{\infty\}$ or $\mathbb{Q}$.
Group extensions of Gaussian $\mathbb{G}$-actions with absolutely continuous spectrum in the orthocomplement of the functions depending on the first coordinate are constructed for $\mathbb{G}$ equal to $\mathbb{Z}^{d}$, $d\in\mathbb{N}\cup\{\infty\}$ or $\mathbb{Q}$.
2010, 26(1): 379-396
doi: 10.3934/dcds.2010.26.379
+[Abstract](2142)
+[PDF](250.3KB)
Abstract:
We study a simplified system of the original Ericksen-Leslie equations for the flow of nematic liquid crystals. This is a coupled non-parabolic dissipative dynamic system. We show the convergence of global classical solutions to single steady states as time goes to infinity by using the Łojasiewicz-Simon approach. Moreover, we provide an estimate on the convergence rate.
We study a simplified system of the original Ericksen-Leslie equations for the flow of nematic liquid crystals. This is a coupled non-parabolic dissipative dynamic system. We show the convergence of global classical solutions to single steady states as time goes to infinity by using the Łojasiewicz-Simon approach. Moreover, we provide an estimate on the convergence rate.
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