ISSN:

1078-0947

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1553-5231

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## Discrete & Continuous Dynamical Systems - A

April 2013 , Volume 33 , Issue 4

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2013, 33(4): 1231-1246
doi: 10.3934/dcds.2013.33.1231

*+*[Abstract](2499)*+*[PDF](396.3KB)**Abstract:**

We present a novel class of higher order energies motivated by the study of network formation in binary mixtures of functionalized polymers and solvent. For a broad class of Lagrangians, we introduce their

*functionalized*form, which is a higher order energy balancing the square of the variational derivative against the original energy. We show that the functionalized energies have global minimizers over several natural spaces of admissible functions. The critical points of the functionalized Lagrangian contain those of the original Lagrangian, however we demonstrate that for a sufficient strength of the functionalization all the critical points of the original Lagrangian are saddle points of the functionalized Lagrangian, and the global minima is a new structure.

2013, 33(4): 1247-1273
doi: 10.3934/dcds.2013.33.1247

*+*[Abstract](2603)*+*[PDF](522.7KB)**Abstract:**

Let $Q\rightarrow X$ be a principal bundle having as structural group $G$ a reductive Lie group in the Harish-Chandra class that includes the case when $G$ is semi-simple with finite center. A semiflow $\phi _{k}$ of endomorphisms of $Q$ induces a semiflow $\psi _{k}$ on the associated bundle $\mathbb{E}=Q\times _{G}\mathbb{F}$, where $\mathbb{F}$ is the maximal flag bundle of $G$. The $A$-component of the Iwasawa decomposition $G=KAN$ yields an additive vector valued cocycle $\mathsf{a}\left( k,\xi \right) $, $\xi \in \mathbb{E}$, over $\psi _{k}$ with values in the Lie algebra $\mathfrak{a}$ of $A$. We prove the Multiplicative Ergodic Theorem of Oseledets for this cocycle: If $\nu $ is a probability measure invariant by the semiflow on $X$ then the $\mathfrak{a}$-Lyapunov exponent $\lambda \left( \xi \right) =\lim \frac{1}{k}\mathsf{a}\left( k,\xi \right) $ exists for every $\xi $ on the fibers above a set of full $\nu $-measure. The level sets of $\lambda \left( \cdot \right) $ on the fibers are described in algebraic terms. When $\phi _{k}$ is a flow the description of the level sets is sharpened. We relate the cocycle $\mathsf{a}\left( k,\xi \right) $ with the Lyapunov exponents of a linear flow on a vector bundle and other growth rates.

2013, 33(4): 1275-1291
doi: 10.3934/dcds.2013.33.1275

*+*[Abstract](1932)*+*[PDF](3531.5KB)**Abstract:**

This paper is devoted to the study and approximation of systems of ordinary differential equations based on an analysis of a certain error functional associated, in a natural way, with the original problem. We prove that in seeking to minimize the error by using standard descent schemes, the procedure can never get stuck in local minima, but will always and steadily decrease the error until getting to the original solution. One main step in the procedure relies on a very particular linearization of the problem: in some sense, it is like a globally convergent Newton type method. Although our objective here is not to perform a rigorous numerical study of the method, we illustrate its potential for approximation by considering some stiff systems of equations. The performance is astonishingly very good due to the fact that we can use very robust methods to approximate linear stiff problems like implicit collocation schemes. We also include a couple of typical test models for the Lorentz system and the Kepler problem, again confirming a very good performance. We believe that this approach can be used in a systematic way to examine other situations and other types of equations due to its flexibility and its simplicity.

2013, 33(4): 1293-1296
doi: 10.3934/dcds.2013.33.1293

*+*[Abstract](1642)*+*[PDF](232.6KB)**Abstract:**

We show that a well known lemma concerning conditions equivalent to topological transitivity is false when posed in a setting that is too general. We also explore some ways of remedying this problem.

2013, 33(4): 1297-1311
doi: 10.3934/dcds.2013.33.1297

*+*[Abstract](1983)*+*[PDF](381.5KB)**Abstract:**

We study the relation between the notions of exponential dichotomy and admissibility for a nonautonomous dynamics with discrete time. More precisely, we consider $\mathbb{Z}$-cocycles defined by a sequence of linear operators in a Banach space, and we give criteria for the existence of an exponential dichotomy in terms of the admissibility of the pairs $(\ell^p,\ell^q)$ of spaces of sequences, with $p\le q$ and $(p,q)\ne(1,\infty)$. We extend the existing results in several directions. Namely, we consider the general case of nonuniform exponential dichotomies; we consider $\mathbb{Z}$-cocycles and not only $\mathbb{N}$-cocycles; and we consider exponential dichotomies that need not be invertible in the stable direction. We also exhibit a collection of admissible pairs of spaces of sequences for any nonuniform exponential dichotomy.

2013, 33(4): 1313-1332
doi: 10.3934/dcds.2013.33.1313

*+*[Abstract](2415)*+*[PDF](884.7KB)**Abstract:**

In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the $\alpha$-continued fraction transformations $T_\alpha$ and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.

2013, 33(4): 1333-1349
doi: 10.3934/dcds.2013.33.1333

*+*[Abstract](2198)*+*[PDF](367.7KB)**Abstract:**

We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.

2013, 33(4): 1351-1363
doi: 10.3934/dcds.2013.33.1351

*+*[Abstract](2397)*+*[PDF](362.6KB)**Abstract:**

We show, when $G$ is a nilpotent or reductive Lie group, that the entropy of any surjective endomorphism coincides with the entropy of its restriction to the toral component of the center of $G$. In particular, if $G$ is a semi-simple Lie group, the entropy of any surjective endomorphism always vanishes. Since every compact group is reductive, the formula for the entropy of a endomorphism of a compact group reduces to the formula for the entropy of an endomorphism of a torus. We also characterize the recurrent set of conjugations of linear semi-simple Lie groups.

2013, 33(4): 1365-1374
doi: 10.3934/dcds.2013.33.1365

*+*[Abstract](2043)*+*[PDF](337.9KB)**Abstract:**

We establish some results on the existence of pullback attractors for non--autonomous delay differential equations with multiple delays. In particular, we generalise some recent works on the existence of pullback attractors for delay differential equations.

2013, 33(4): 1375-1388
doi: 10.3934/dcds.2013.33.1375

*+*[Abstract](1755)*+*[PDF](370.4KB)**Abstract:**

For a sequence of subadditive potentials, a method of choosing state points with negative growth rates for an ergodic dynamical system was given in [5]. This paper first generalizes this result to the non-ergodic dynamics, and then proves that under some mild additional hypothesis, one can choose points with negative growth rates from a positive Lebesgue measure set, even if the system does not preserve any measure that is absolutely continuous with respect to Lebesgue measure.

2013, 33(4): 1389-1405
doi: 10.3934/dcds.2013.33.1389

*+*[Abstract](2372)*+*[PDF](469.4KB)**Abstract:**

The global well-posedness on the Cauchy problem of nonlinear Schrödinger equations (NLS) is studied for a class of critical nonlinearity

*below*$L^2$ in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index $s_c$ is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.

2013, 33(4): 1407-1429
doi: 10.3934/dcds.2013.33.1407

*+*[Abstract](2485)*+*[PDF](466.4KB)**Abstract:**

Let $(\mathcal{M}, \tilde{g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the problem $$ \varepsilon^2 Δ_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0 in \mathcal{M}, $$ where $\varepsilon >0$ is a small parameter and $V$ is a positive, smooth function in $\mathcal{M}$. Let $ \mathcal{K}\subset \mathcal{M}$ be an $(N-1)$-dimensional smooth submanifold that divides $\mathcal{M}$ into two disjoint components $\mathcal{M}_{\pm}$. We assume $\mathcal{K}$ is stationary and non-degenerate relative to the weighted area functional $\int_{\mathcal{K}}V^{\frac{1}{2}}$. We prove that there exist two transition layer solutions $u_\varepsilon^{(1)}, u_\varepsilon^{(2)}$ when $\varepsilon$ is sufficiently small. The first layer solution $u_\varepsilon^{(1)}$ approaches $-1$ in $\mathcal{M}_{-}$ and $+1$ in $\mathcal{M}_{+}$ as $\varepsilon$ tends to 0, while the other solution $u_\varepsilon^{(2)}$ exhibits a transition layer in the opposite direction.

2013, 33(4): 1431-1450
doi: 10.3934/dcds.2013.33.1431

*+*[Abstract](2081)*+*[PDF](465.2KB)**Abstract:**

The evolutions of states is described corresponding to the Glauber dynamics of an infinite system of interacting particles in continuum. The description is conducted on both micro- and mesoscopic levels. The microscopic description is based on solving linear equations for correlation functions by means of an Ovsjannikov-type technique, which yields the evolution in a scale of Banach spaces. The mesoscopic description is performed by means of the Vlasov scaling, which yields a linear infinite chain of equations obtained from those for the correlation function. Its main peculiarity is that, for the initial correlation function of the inhomogeneous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type.

2013, 33(4): 1451-1476
doi: 10.3934/dcds.2013.33.1451

*+*[Abstract](2387)*+*[PDF](526.7KB)**Abstract:**

Let $f$ be a diffeomorphism of a manifold preserving a hyperbolic Borel probability measure $ μ $ having transverse intersections for almost every pair of stable and unstable manifolds. A lower bound on the Hausdorff dimension of generic sets is given in terms of the Lyapunov exponents and the metric entropy. Furthermore we obtain a lower bound for the large deviation rate.

2013, 33(4): 1477-1498
doi: 10.3934/dcds.2013.33.1477

*+*[Abstract](1915)*+*[PDF](324.8KB)**Abstract:**

For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_\lambda$ provides an algorithm to expand any positive real number in $\lambda$-continued fraction. We prove the conjugacy between $T_\lambda$ and some $\beta$-shift, $\beta>1$. Some properties of the map $\lambda\mapsto\beta(\lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.

2013, 33(4): 1499-1511
doi: 10.3934/dcds.2013.33.1499

*+*[Abstract](2173)*+*[PDF](370.7KB)**Abstract:**

Considered here is a system of smectic liquid crystals possessing polarizations described by the Oseen-Frank and Chen-Lubensky energies. We establish partial regularity of minimizers for the governing energy functional using the idea of $(c,\beta)$-almost minimizer introduced in [9].

2013, 33(4): 1513-1544
doi: 10.3934/dcds.2013.33.1513

*+*[Abstract](2080)*+*[PDF](627.3KB)**Abstract:**

Our goal in this paper is to investigate how the geometric shape of a swimmer affects the forces acting upon it in a 3-$D$ incompressible fluid, such as governed by the non-stationary Stokes or Navier-Stokes equations. Namely, we are interested in the following question:

*How will the swimmer's internal forces (i.e., not moving the center of swimmer's mass when it is not inside a fluid) ``transform'' their actions when the swimmer is placed into a fluid (thus, possibly, creating its self-propelling motion)?*We focus on the case when the swimmer's body consists of either small parallelepipeds or balls. Such problems are of interest in biology and engineering application dealing with propulsion systems in fluids.

2013, 33(4): 1545-1562
doi: 10.3934/dcds.2013.33.1545

*+*[Abstract](2147)*+*[PDF](426.6KB)**Abstract:**

We investigate a class of non-hyperbolic diffeomorphisms defined on the product space. By using the Pesin theory combined with the general theory of differentiable dynamical systems, we prove that there are exactly two SRB attractors, and their basins cover a full measure subset of the ambient manifold. Furthermore, we prove that the basins of SRB attractors have the strange intermingled phenomenon, i.e. they are measure-theoretically dense in each other. The intermingled phenomena have been observed in many physical systems by numerical experiments, and considered to be important to some fundamental problems in physical, biology and computer science etc. Finally, we describe a concrete example for application.

2013, 33(4): 1563-1581
doi: 10.3934/dcds.2013.33.1563

*+*[Abstract](1812)*+*[PDF](401.6KB)**Abstract:**

In this paper, we study the existence and multiplicity of periodic solutions of Liénard equations $$ x''+f(x)x'+V'(x)+g(x)=p(t), $$ where $V$ is a $2\pi/n$-isochronous potential. When $F(F(x)=\int_0^xf(s)ds)$ and $g$ are bounded, we provide new sufficient conditions to ensure the existence of periodic solutions of this equations. Moreover, we prove the multiplicity of periodic solutions of the given equations under certain bounded conditions by using topological degree method. When $F, g$ satisfy a certain class of unbounded conditions, we also give sufficient conditions to ensure the existence of $2\pi$-periodic solutions of the given equations.

2013, 33(4): 1583-1601
doi: 10.3934/dcds.2013.33.1583

*+*[Abstract](1557)*+*[PDF](395.9KB)**Abstract:**

We study the energy decay to the wave equation with a dissipative boundary condition on partially rectangular domains. We give a polynomial order energy decay under the assumption that the damping term may vanish on the rectangular part. A resolvent estimate for the correspondent stationary problem is proved.

2013, 33(4): 1603-1614
doi: 10.3934/dcds.2013.33.1603

*+*[Abstract](1661)*+*[PDF](407.7KB)**Abstract:**

This article provides sufficient conditions for the existence of periodic solutions with nonconstant sign in a family of polynomial, non-auto-nomous, first-order differential equations that arise as a generalization of the Abel equation of the second kind.

2013, 33(4): 1615-1631
doi: 10.3934/dcds.2013.33.1615

*+*[Abstract](2107)*+*[PDF](1768.6KB)**Abstract:**

We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1,12], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.

2013, 33(4): 1633-1644
doi: 10.3934/dcds.2013.33.1633

*+*[Abstract](2461)*+*[PDF](440.0KB)**Abstract:**

We develop a geometrical method to detect the presence of chaotic dynamics in delay differential equations. An application to the classical Lotka-Volterra model with delay is given.

2013, 33(4): 1645-1655
doi: 10.3934/dcds.2013.33.1645

*+*[Abstract](2229)*+*[PDF](354.5KB)**Abstract:**

We show that the generalized Yang-Mills system with Hamiltonian $H=\frac12(y_1^2+y_2^2)+\frac12(ax_1^2+bx_2^2)+\frac14cx_1^4+\frac14dx_2^4+\frac12ex_1^2x_2^2$ is meromorphically integrable in Liouvillian sense(i.e., the existence of an additional meromorphic first integral) if and only if (A) $e=0$, or (B) $c=d=e$, or (C) $a=b, e=3c=3d$, or (D) $b=4a, e=3c, d=8c$, or (E) $b=4a, e=6c, d=16c$, or (F) $b=4a, e=3d, c=8d$, or (G) $b=4a, e=6d, c=16d$. Therefore, we get a complete classification of the Yang-Mills Hamiltonian system in sense of integrability and non-integrability.

2013, 33(4): 1657-1697
doi: 10.3934/dcds.2013.33.1657

*+*[Abstract](1815)*+*[PDF](639.2KB)**Abstract:**

We discuss some properties of a forward-backward parabolic problem that arises in models of phase transition in which two stable phases are separated by an interface. Here we consider a formulation of the problem that comes from a Sobolev approximation of it. In particular we prove uniqueness of the previous problem extending to nonlinear diffusion function a result obtained in [21] in the piecewise linear case. Moreover, we analyze the third order partial differential problem that approximates the forward-backward parabolic one. In particular, for some classes of initial data, we obtain a priori estimates that generalize that proved in [22]. Using these results we study the singular limit of the Sobolev approximation, as a consequence we obtain existence of the forward-backward problem for a class of initial data.

2013, 33(4): 1699-1712
doi: 10.3934/dcds.2013.33.1699

*+*[Abstract](2380)*+*[PDF](413.2KB)**Abstract:**

This paper is concerned with the problem of well-posedness for a modified two-component Camassa-Holm system in Besov spaces with the critical index $s=\frac 3 2$.

2013, 33(4): 1713-1739
doi: 10.3934/dcds.2013.33.1713

*+*[Abstract](3042)*+*[PDF](437.1KB)**Abstract:**

This paper is devoted to the continuation of solutions to the generalized Camassa-Holm equation beyond wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear system. This formulation allows one to continue the solution after collision time, giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global conservative or dissipative solutions, which depend continuously on the initial data.

2019 Impact Factor: 1.338

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