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Discrete & Continuous Dynamical Systems - A
February 2010 , Volume 27 , Issue 1
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2010, 27(1): 1-24
doi: 10.3934/dcds.2010.27.1
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Abstract:
We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,....$ which can be written as $I_j=\frac{1}{2}|F_j|^2$, where $F_j:H\rightarrow \R^2$, $F_j(0)=0$ for $j=1,2,....$ We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\rightarrow H$, such that $dF(0)=id$, $(F-id)$ is a $\kappa$-smoothing map ($\kappa\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps F ’j such that $F_j-$F ’j$=O(|u|^2)$ and each 1/2|F ’j|$^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism F’$: H\rightarrow H$, the germ (F’-id) is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector (1/2|F’j|$^2,j\ge1)$. Next we show that the theorem with $\kappa=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form 'identity plus a 1-smoothing analytic map'.
We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,....$ which can be written as $I_j=\frac{1}{2}|F_j|^2$, where $F_j:H\rightarrow \R^2$, $F_j(0)=0$ for $j=1,2,....$ We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\rightarrow H$, such that $dF(0)=id$, $(F-id)$ is a $\kappa$-smoothing map ($\kappa\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps F ’j such that $F_j-$F ’j$=O(|u|^2)$ and each 1/2|F ’j|$^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism F’$: H\rightarrow H$, the germ (F’-id) is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector (1/2|F’j|$^2,j\ge1)$. Next we show that the theorem with $\kappa=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form 'identity plus a 1-smoothing analytic map'.
2010, 27(1): 25-52
doi: 10.3934/dcds.2010.27.25
+[Abstract](2841)
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Abstract:
Despite their misleading label, rare events in stochastic systems are central to many applied phenomena. In this paper, we concentrate on one such situation - phase separation through homogeneous nucleation in binary alloys as described by the stochastic partial differential equation model due to Cahn, Hilliard, and Cook. We show that in the limit of small noise intensity, nucleation can be explained by the stochastically driven exit from the domain of attraction of an asymptotically stable homogeneous equilibrium state for the associated deterministic model. Furthermore, we provide insight into the subsequent nucleation dynamics via the structure of the attractor of the model in the absence of noise.
Despite their misleading label, rare events in stochastic systems are central to many applied phenomena. In this paper, we concentrate on one such situation - phase separation through homogeneous nucleation in binary alloys as described by the stochastic partial differential equation model due to Cahn, Hilliard, and Cook. We show that in the limit of small noise intensity, nucleation can be explained by the stochastically driven exit from the domain of attraction of an asymptotically stable homogeneous equilibrium state for the associated deterministic model. Furthermore, we provide insight into the subsequent nucleation dynamics via the structure of the attractor of the model in the absence of noise.
2010, 27(1): 53-73
doi: 10.3934/dcds.2010.27.53
+[Abstract](2165)
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Abstract:
We prove the disjointness of almost all interval exchange transformations from ELF systems (systems of probabilistic origin) for a countable subset of permutations including the symmetric permutations
We prove the disjointness of almost all interval exchange transformations from ELF systems (systems of probabilistic origin) for a countable subset of permutations including the symmetric permutations
$ 1\ 2\ \ldots \ m-1 \ m $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ m-1 \ldots \ 2\ 1 $
for m=3,5,7.
Some disjointness properties of special flows built over interval exchange transformations and under piecewise constant roof function are investigated as well.
2010, 27(1): 75-116
doi: 10.3934/dcds.2010.27.75
+[Abstract](2588)
+[PDF](378.1KB)
Abstract:
The existence of at least two homoclinic orbits is proved by A. Ambrosetti and V. Coti Zelati (Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89 (1993), 177-194) for autonomous Lagrangian systems
The existence of at least two homoclinic orbits is proved by A. Ambrosetti and V. Coti Zelati (Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89 (1993), 177-194) for autonomous Lagrangian systems
$\ddot{q}+V'(q)=0, ~q\in C^2(\R,\R^m),~m\geq 2 $
where $V:\R^m\rightarrow\R$ is a function of the form
$ V(q)=-\frac{|q|^2}{2}+W(q) $
with $W\in C^2(\R^m,\R)$ superquadratic, satisfying a "pinching''
hypothesis and an hypothesis on its second derivative.
The present work deals with potentials of the form $W(q,\dot{q})$
that weakly depend on $\dot{q}$. In this case an homoclinic orbit
corresponds to a classical solution to the equation
$\ddot{q}-q+W_1(q,\dot{q})-\frac{d}{dt}W_2(q,\dot{q})=0,$
where $W_i=\partial_i W$ for $i=1,2$.
2010, 27(1): 117-132
doi: 10.3934/dcds.2010.27.117
+[Abstract](1916)
+[PDF](232.1KB)
Abstract:
We study a non-relativistic charged quantum particle moving in a bounded open set $\Omega\subset\R^3$ with smooth boundary under the action of a zero-range potential. In the electrostatic case the standing wave solutions take the form $\psi(t,x)=u(x)e^{-i\omega t}$ where $u$ formally satisfies $-\Delta u+\alpha\varphi u-\beta\delta_{x_0} u=\omega u$ and the electric potential $\varphi$ is given by $-\Delta\varphi = u^2$. We introduce the definition of ground state. We show the existence of such solutions for each $\beta>0$ and the compactness as $\beta\to 0$.
We study a non-relativistic charged quantum particle moving in a bounded open set $\Omega\subset\R^3$ with smooth boundary under the action of a zero-range potential. In the electrostatic case the standing wave solutions take the form $\psi(t,x)=u(x)e^{-i\omega t}$ where $u$ formally satisfies $-\Delta u+\alpha\varphi u-\beta\delta_{x_0} u=\omega u$ and the electric potential $\varphi$ is given by $-\Delta\varphi = u^2$. We introduce the definition of ground state. We show the existence of such solutions for each $\beta>0$ and the compactness as $\beta\to 0$.
2010, 27(1): 133-169
doi: 10.3934/dcds.2010.27.133
+[Abstract](1826)
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Abstract:
We construct stationary solutions to the non-barotropic, compressible Euler and Navier-Stokes equations in several space dimensions with spherical or cylindrical symmetry. The equation of state is assumed to satisfy standard monotonicity and convexity assumptions. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in an exterior domain. On the other hand, stationary smooth solutions in an interior domain necessarily become sonic and cannot be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres (or cylinders).
Next we construct smooth solutions wε to the Navier-Stokes system converging to the previously constructed Euler shocks in the small viscosity limit ε → 0. The viscous solutions are obtained by a new technique for constructing solutions to a class of two-point boundary problems with a fast transition region. The construction is explicit in the sense that it produces high order expansions in powers of ε for wε, and the coefficients in the expansion satisfy simple, explicit ODEs, which are linear except in the case of the leading term. The solutions to the Euler equations described above provide the slowly varying contribution to the leading term in the expansion.
The approach developed here is applicable to a variety of singular perturbation problems, including the construction of heteroclinic orbits with fast transitions. For example, a variant of our method is used in [W] to give a new construction of detonation profiles for the reactive Navier-Stokes equations.
We construct stationary solutions to the non-barotropic, compressible Euler and Navier-Stokes equations in several space dimensions with spherical or cylindrical symmetry. The equation of state is assumed to satisfy standard monotonicity and convexity assumptions. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in an exterior domain. On the other hand, stationary smooth solutions in an interior domain necessarily become sonic and cannot be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres (or cylinders).
Next we construct smooth solutions wε to the Navier-Stokes system converging to the previously constructed Euler shocks in the small viscosity limit ε → 0. The viscous solutions are obtained by a new technique for constructing solutions to a class of two-point boundary problems with a fast transition region. The construction is explicit in the sense that it produces high order expansions in powers of ε for wε, and the coefficients in the expansion satisfy simple, explicit ODEs, which are linear except in the case of the leading term. The solutions to the Euler equations described above provide the slowly varying contribution to the leading term in the expansion.
The approach developed here is applicable to a variety of singular perturbation problems, including the construction of heteroclinic orbits with fast transitions. For example, a variant of our method is used in [W] to give a new construction of detonation profiles for the reactive Navier-Stokes equations.
2010, 27(1): 171-183
doi: 10.3934/dcds.2010.27.171
+[Abstract](2222)
+[PDF](217.8KB)
Abstract:
We study the Navier-Stokes system with initial data belonging to sum of two weak-$L^{p}$ spaces, which contains the sum of homogeneous function with different degrees. The domain $\Omega$ can be either an exterior domain, the half-space, the whole space or a bounded domain with dimension $n\geq 2$. We obtain the existence of local mild solutions in the same class of initial data and moreover we show results about uniqueness, regularity and continuous dependence of solutions with respect to the initial data. To obtain our results we prove a new Hölder-type inequality on the sum of Lorentz spaces.
We study the Navier-Stokes system with initial data belonging to sum of two weak-$L^{p}$ spaces, which contains the sum of homogeneous function with different degrees. The domain $\Omega$ can be either an exterior domain, the half-space, the whole space or a bounded domain with dimension $n\geq 2$. We obtain the existence of local mild solutions in the same class of initial data and moreover we show results about uniqueness, regularity and continuous dependence of solutions with respect to the initial data. To obtain our results we prove a new Hölder-type inequality on the sum of Lorentz spaces.
2010, 27(1): 185-204
doi: 10.3934/dcds.2010.27.185
+[Abstract](2221)
+[PDF](278.6KB)
Abstract:
Let $\tau _r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for typical $x,x_0$
Let $\tau _r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for typical $x,x_0$
liminfr → 0 $ \frac{\log \tau _r(x,x_0)}{-\log r} = \infty.$
By considering a suitable reparametrization of the flow generated by
a suspension of this translation, using a previous construction by
Fayad, we show the existence of a mixing system on three torus
having the same properties. The speed of mixing of this example must
be subpolynomial, because we also show that: in a system having
polynomial decay of correlations, the limsupr → 0 of the
above ratio of logarithms (which is also called the upper hitting
time indicator) is bounded from above by a function of the local
dimension and the speed of correlation decay.
More generally, this shows that reparametrizations of torus
translations having a Liouville component cannot be polynomially
mixing.
2010, 27(1): 205-216
doi: 10.3934/dcds.2010.27.205
+[Abstract](2488)
+[PDF](215.2KB)
Abstract:
Let $f$ be a diffeomorphism of a closed $n$-dimensional $C^\infty$ manifold, and $p$ be a hyperbolic saddle periodic point of $f$. In this paper, we introduce the notion of $C^1$-stably weakly shadowing for a closed $f$-invariant set, and prove that for the homoclinic class $H_f(p)$ of $p$, if $f_{|H_f(p)}$ is $C^1$-stably weakly shadowing, then $H_f(p)$ admits a dominated splitting. Especially, on a 3-dimensional manifold, the splitting on $H_f(p)$ is partially hyperbolic, and if in addition, $f$ is far from homoclinic tangency, then $H_f(p)$ is strongly partially hyperbolic.
Let $f$ be a diffeomorphism of a closed $n$-dimensional $C^\infty$ manifold, and $p$ be a hyperbolic saddle periodic point of $f$. In this paper, we introduce the notion of $C^1$-stably weakly shadowing for a closed $f$-invariant set, and prove that for the homoclinic class $H_f(p)$ of $p$, if $f_{|H_f(p)}$ is $C^1$-stably weakly shadowing, then $H_f(p)$ admits a dominated splitting. Especially, on a 3-dimensional manifold, the splitting on $H_f(p)$ is partially hyperbolic, and if in addition, $f$ is far from homoclinic tangency, then $H_f(p)$ is strongly partially hyperbolic.
2010, 27(1): 217-229
doi: 10.3934/dcds.2010.27.217
+[Abstract](2237)
+[PDF](185.3KB)
Abstract:
We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples.
We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples.
2010, 27(1): 231-236
doi: 10.3934/dcds.2010.27.231
+[Abstract](2301)
+[PDF](125.6KB)
Abstract:
Spatial analyticity properties of Koch-Tataru solutions of the Navier-Stokes equations are obtained directly from the equations. Time decay rates of higher order derivatives follow as a simple consequence.
Spatial analyticity properties of Koch-Tataru solutions of the Navier-Stokes equations are obtained directly from the equations. Time decay rates of higher order derivatives follow as a simple consequence.
2010, 27(1): 237-263
doi: 10.3934/dcds.2010.27.237
+[Abstract](2065)
+[PDF](287.2KB)
Abstract:
In two space dimension, we show that the steady state the solution of fluid/particle system may tend to after a long time is completely determined by the initial total momentum. Based on this observation, we prove the global-in-time existence of the classical solutions for arbitrary initial data to the system that couples the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. By linearized method and Littlewood-Paley analysis, the exponential rate of the convergence toward steady state is obtained under some specific assumptions.
In two space dimension, we show that the steady state the solution of fluid/particle system may tend to after a long time is completely determined by the initial total momentum. Based on this observation, we prove the global-in-time existence of the classical solutions for arbitrary initial data to the system that couples the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. By linearized method and Littlewood-Paley analysis, the exponential rate of the convergence toward steady state is obtained under some specific assumptions.
2010, 27(1): 265-284
doi: 10.3934/dcds.2010.27.265
+[Abstract](2184)
+[PDF](677.5KB)
Abstract:
In this paper, we discuss a discrete version of the Turing continuous model of morphogenesis. We describe some dynamical properties of the asymptotic behaviors for trajectories escaping to infinity and those which remain bounded, and find various types of invariant sets of trajectories in this system. Finally, some numerical results of asymptotic behaviors of trajectories are presented.
In this paper, we discuss a discrete version of the Turing continuous model of morphogenesis. We describe some dynamical properties of the asymptotic behaviors for trajectories escaping to infinity and those which remain bounded, and find various types of invariant sets of trajectories in this system. Finally, some numerical results of asymptotic behaviors of trajectories are presented.
2010, 27(1): 285-300
doi: 10.3934/dcds.2010.27.285
+[Abstract](2039)
+[PDF](344.0KB)
Abstract:
In this paper, we study a two-parameter family $\{\varphi_{\mu,\nu}\}$ of three-dimensional diffeomorphisms which have a bifurcation induced by simultaneous generation of a heterodimensional cycle and a heterodimensional tangency associated to two saddle points. We show that such a codimension-$2$ bifurcation generates a quadratic homoclinic tangency associated to one of the saddle continuations which unfolds generically with respect to some one-parameter subfamily of $\{\varphi_{\mu,\nu}\}$. Moreover, from this result together with some well-known facts, we detect some nonhyperbolic phenomena (i.e., the existence of nonhyperbolic strange attractors and the $C^{2}$ robust tangencies) arbitrarily close to the codimension-$2$ bifurcation.
In this paper, we study a two-parameter family $\{\varphi_{\mu,\nu}\}$ of three-dimensional diffeomorphisms which have a bifurcation induced by simultaneous generation of a heterodimensional cycle and a heterodimensional tangency associated to two saddle points. We show that such a codimension-$2$ bifurcation generates a quadratic homoclinic tangency associated to one of the saddle continuations which unfolds generically with respect to some one-parameter subfamily of $\{\varphi_{\mu,\nu}\}$. Moreover, from this result together with some well-known facts, we detect some nonhyperbolic phenomena (i.e., the existence of nonhyperbolic strange attractors and the $C^{2}$ robust tangencies) arbitrarily close to the codimension-$2$ bifurcation.
2010, 27(1): 301-323
doi: 10.3934/dcds.2010.27.301
+[Abstract](2414)
+[PDF](272.4KB)
Abstract:
We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops $L_x^\infty$-norm blowup in finite time.
We consider a nonlocal aggregation equation with nonlinear diffusion which arises from the study of biological aggregation dynamics. As a degenerate parabolic problem, we prove the well-posedness, continuation criteria and smoothness of local solutions. For compactly supported nonnegative smooth initial data we prove that the gradient of the solution develops $L_x^\infty$-norm blowup in finite time.
2010, 27(1): 325-335
doi: 10.3934/dcds.2010.27.325
+[Abstract](1949)
+[PDF](183.3KB)
Abstract:
This paper is concerned with the bifurcation of limit cycles from a class of one-parameter family of quadratic reversible system under quadratic perturbations. The exact upper bound of the number of limit cycles is given.
This paper is concerned with the bifurcation of limit cycles from a class of one-parameter family of quadratic reversible system under quadratic perturbations. The exact upper bound of the number of limit cycles is given.
2010, 27(1): 337-355
doi: 10.3934/dcds.2010.27.337
+[Abstract](2310)
+[PDF](257.8KB)
Abstract:
In this paper, we consider the minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. We prove that if the Hamiltonian function $H\in C^2(\R^{2n}, \R)$ is super-quadratic and convex, for every number $\tau>0$, there exists at least one $\tau$-periodic brake orbit $(\tau,x)$ with minimal period $\tau$ or $\tau/2$ provided $H(Nx)=H(x)$.
In this paper, we consider the minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. We prove that if the Hamiltonian function $H\in C^2(\R^{2n}, \R)$ is super-quadratic and convex, for every number $\tau>0$, there exists at least one $\tau$-periodic brake orbit $(\tau,x)$ with minimal period $\tau$ or $\tau/2$ provided $H(Nx)=H(x)$.
2010, 27(1): 357-368
doi: 10.3934/dcds.2010.27.357
+[Abstract](1592)
+[PDF](179.0KB)
Abstract:
We establish the uniqueness of subsonic potential flows in a three-dimensional finite duct, a semi-infinite duct and an infinite duct with quadrate sections, as well as flows in half space and whole space. Moreover, some extremum principles for the related elliptic equations are proved under suitable assumptions in unbounded domains.
We establish the uniqueness of subsonic potential flows in a three-dimensional finite duct, a semi-infinite duct and an infinite duct with quadrate sections, as well as flows in half space and whole space. Moreover, some extremum principles for the related elliptic equations are proved under suitable assumptions in unbounded domains.
2010, 27(1): 369-382
doi: 10.3934/dcds.2010.27.369
+[Abstract](2093)
+[PDF](7833.2KB)
Abstract:
We study certain discontinuous maps by means of a coding map defined on a special partition of the phase space which is such that the points of discontinuity of the map, $\mathcal{D}$, all belong to the union of the boundaries of elements in the partition.
For maps acting locally as homeomorphisms in a compact space, we prove that, if the set of points whose trajectory comes arbitrarily close to the set of discontinuities is closed and not the full space then all points not in that set are rationally coded, i.e., their codings eventually settle on a repeated block of symbols.
In particular, for piecewise isometries, which are discontinuous maps acting locally as isometries, we give a topological description of the equivalence classes of the coding map in terms of the connected components generated by the closure of the preimages of $\mathcal{D}$.
We study certain discontinuous maps by means of a coding map defined on a special partition of the phase space which is such that the points of discontinuity of the map, $\mathcal{D}$, all belong to the union of the boundaries of elements in the partition.
For maps acting locally as homeomorphisms in a compact space, we prove that, if the set of points whose trajectory comes arbitrarily close to the set of discontinuities is closed and not the full space then all points not in that set are rationally coded, i.e., their codings eventually settle on a repeated block of symbols.
In particular, for piecewise isometries, which are discontinuous maps acting locally as isometries, we give a topological description of the equivalence classes of the coding map in terms of the connected components generated by the closure of the preimages of $\mathcal{D}$.
2010, 27(1): 383-388
doi: 10.3934/dcds.2010.27.383
+[Abstract](2321)
+[PDF](132.4KB)
Abstract:
Given a real-valued continuous function $f$ defined on the phase space of a dynamical system, an invariant measure is said to be maximizing if it maximises the integral of $f$ over the set of all invariant measures. Extending results of Bousch, Jenkinson and Brémont, we show that the ergodic maximizing measures of functions belonging to a residual subset of the continuous functions may be characterised as those measures which belong to a residual subset of the ergodic measures.
Given a real-valued continuous function $f$ defined on the phase space of a dynamical system, an invariant measure is said to be maximizing if it maximises the integral of $f$ over the set of all invariant measures. Extending results of Bousch, Jenkinson and Brémont, we show that the ergodic maximizing measures of functions belonging to a residual subset of the continuous functions may be characterised as those measures which belong to a residual subset of the ergodic measures.
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