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Discrete & Continuous Dynamical Systems - A

2013 , Volume 33 , Issue 8

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The reversibility problem for quasi-homogeneous dynamical systems
Antonio Algaba, Estanislao Gamero and Cristóbal García
2013, 33(8): 3225-3236 doi: 10.3934/dcds.2013.33.3225 +[Abstract](88) +[PDF](361.8KB)
In this paper, we obtain necessary and sufficient conditions for the reversibility of a quasi-homogeneous $n$-dimensional system. As a consequence, we get necessary conditions for an arbitrary system to be orbital-reversible. Moreover, we give sufficient conditions for orbital-reversibility in terms of the existence of Lie symmetries of the vector field. The results obtained are conveniently adapted to the case of planar systems, where we give sufficient conditions for a degenerate planar vector field to have a center at the origin. We apply the results to some case studies. Namely, we consider a family of planar vector fields, where we determine centers which are not orbital-reversible. We also study some tridimensional systems, where nonlinear involutions are determined in the reversible situations.
Maximizing entropy of cycles on trees
Lluís Alsedà, David Juher, Deborah M. King and Francesc Mañosas
2013, 33(8): 3237-3276 doi: 10.3934/dcds.2013.33.3237 +[Abstract](94) +[PDF](651.0KB)
In this paper we give a partial characterization of the periodic tree patterns of maximum entropy for a given period. More precisely, we prove that each periodic pattern with maximal entropy is irreducible (has no block structures) and simplicial (any vertex belongs to the periodic orbit). Moreover, we also prove that it is maximodal in the sense that every point of the periodic orbit is a "turning point".
Toeplitz kneading sequences and adding machines
Lori Alvin
2013, 33(8): 3277-3287 doi: 10.3934/dcds.2013.33.3277 +[Abstract](86) +[PDF](367.6KB)
In this paper we provide a characterization for a shift maximal sequence of 1's and 0's to be the kneading sequence for a unimodal map $f$ with $f|_{\omega(c)}$ topologically conjugate to an adding machine, where $c$ is the turning point of $f$. We show that the unimodal map $f$ has an embedded adding machine if and only if $\mathcal{K}(f)$ is a one-sided, non-periodic Toeplitz sequence with the finite time containment property. We then show the existence of unimodal maps with Toeplitz kneading sequences that do not have the finite time containment property.
Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations
Youcef Amirat and Kamel Hamdache
2013, 33(8): 3289-3320 doi: 10.3934/dcds.2013.33.3289 +[Abstract](77) +[PDF](530.4KB)
In this paper we study the equations of flow and heat transfer in a magnetic fluid with internal rotations, when the fluid is subjected to the action of an external magnetic field. The system of equations is a combination of the Navier-Stokes equations, the magnetization relaxation equation of Bloch type, the magnetostatic equations and the temperature equation. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions and establish a blow-up criterium for strong solutions. We then prove the global-in-time existence of strong solutions, under smallness assumptions on the initial data and the external magnetic field.
No entire function with real multipliers in class $\mathcal{S}$
Agnieszka Badeńska
2013, 33(8): 3321-3327 doi: 10.3934/dcds.2013.33.3321 +[Abstract](117) +[PDF](309.4KB)
We prove that there is no entire transcendental function in class $\mathcal{S}$ with real multipliers of all repelling periodic orbits.
On the control of non holonomic systems by active constraints
Alberto Bressan, Ke Han and Franco Rampazzo
2013, 33(8): 3329-3353 doi: 10.3934/dcds.2013.33.3329 +[Abstract](87) +[PDF](540.3KB)
The paper is concerned with mechanical systems which are controlled by implementing a number of time-dependent, frictionless holonomic constraints. The main novelty is due to the presence of additional non-holonomic constraints. We develop a general framework to analyze these problems, deriving the equations of motion and studying the continuity properties of the "control-to-trajectory" maps. Various geometric characterizations are provided, in order that the equations be affine w.r.t. the time derivative of the control. In this case the system is fit for jumps, and the evolution is well defined also in connection with discontinuous control functions. The classical Roller Racer provides an example where the non-affine dependence of the equations on the derivative of the control is due only to the non-holonomic constraint. This is a case where the presence of quadratic terms in the equations can be used for controllability purposes.
An alternative approach to generalised BV and the application to expanding interval maps
Oliver Butterley
2013, 33(8): 3355-3363 doi: 10.3934/dcds.2013.33.3355 +[Abstract](94) +[PDF](398.2KB)
We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where the weighting used in the transfer operator is not better than piecewise Hölder continuous and the partition on which the map is continuous may possess a countable number of elements. For such weighted transfer operators we give upper bounds for both the spectral radius and for the essential spectral radius.
On the mixing properties of piecewise expanding maps under composition with permutations
Nigel P. Byott, Mark Holland and Yiwei Zhang
2013, 33(8): 3365-3390 doi: 10.3934/dcds.2013.33.3365 +[Abstract](86) +[PDF](520.4KB)
We consider the effect on the mixing properties of a piecewise smooth interval map $f$ when its domain is divided into $N$ equal subintervals and $f$ is composed with a permutation of these. The case of the stretch-and-fold map $f(x)=mx \bmod 1$ for integers $m \geq 2$ is examined in detail. We give a combinatorial description of those permutations $\sigma$ for which $\sigma \circ f$ is still (topologically) mixing, and show that the proportion of such permutations tends to $1$ as $N \to \infty$. We then investigate the mixing rate of $\sigma \circ f$ (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that composition with a permutation cannot improve the mixing rate of $f$, but typically makes it worse. Under some mild assumptions on $m$ and $N$, we obtain a precise value for the worst mixing rate as $\sigma$ ranges through all permutations; this can be made arbitrarily close to $1$ as $N → ∞$ (with $m$ fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small $m$ and $N$, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps $f$ for which composition with permutations produces different behaviour than that obtained from the stretch-and-fold map.
Optimal partial regularity results for nonlinear elliptic systems in Carnot groups
Shuhong Chen and Zhong Tan
2013, 33(8): 3391-3405 doi: 10.3934/dcds.2013.33.3391 +[Abstract](75) +[PDF](153.2KB)
In this paper, we consider partial regularity for weak solutions of second-order nonlinear elliptic systems in Carnot groups. By the method of A-harmonic approximation, we establish optimal interior partial regularity of weak solutions to systems under controllable growth conditions with sub-quadratic growth in Carnot groups.
On the Cauchy problem for the two-component Dullin-Gottwald-Holm system
Yong Chen, Hongjun Gao and Yue Liu
2013, 33(8): 3407-3441 doi: 10.3934/dcds.2013.33.3407 +[Abstract](88) +[PDF](581.3KB)
Considered herein is the initial-value problem for a two-component Dullin-Gottwald-Holm system. The local well-posedness in the Sobolev space $H^{s}(\mathbb{R})$ with $s>3/2$ is established by using the bi-linear estimate technique to the approximate solutions. Then the wave-breaking criteria and global solutions are determined in $H^s(\mathbb{R}), s > 3/2.$ Finally, existence of the solitary-wave solutions is demonstrated.
Spreading speeds of $N$-season spatially periodic integro-difference models
Weiwei Ding, Xing Liang and Bin Xu
2013, 33(8): 3443-3472 doi: 10.3934/dcds.2013.33.3443 +[Abstract](125) +[PDF](535.2KB)
In this paper, the spreading speeds of $N$-season spatially periodic integro-difference models are investigated. The variational formula of the spreading speeds is given via the principal eigenvalues of the respective positive linear operators. The effects of the spatial and temporal distribution of the intrinsic growth rates on the spreading speeds are considered.
Geometric inequalities and symmetry results for elliptic systems
Serena Dipierro
2013, 33(8): 3473-3496 doi: 10.3934/dcds.2013.33.3473 +[Abstract](90) +[PDF](448.9KB)
We obtain some Poincaré type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form \begin{eqnarray*} \left\{ \begin{array}{ll} div\left( a\left( |\nabla u|\right) \nabla u\right) = F_1(u, v), \\ div\left( b\left( |\nabla v|\right) \nabla v\right) = F_2(u, v), \end{array} \right. \end{eqnarray*} where $F ∈ C^{1,1}_{loc}(\mathbb{R}^2)$.
    Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in $\mathbb{R}^2$.
Uniformity in the Wiener-Wintner theorem for nilsequences
Tanja Eisner and Pavel Zorin-Kranich
2013, 33(8): 3497-3516 doi: 10.3934/dcds.2013.33.3497 +[Abstract](82) +[PDF](492.9KB)
We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.
On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations
Daoyuan Fang and Ruizhao Zi
2013, 33(8): 3517-3541 doi: 10.3934/dcds.2013.33.3517 +[Abstract](86) +[PDF](523.6KB)
This paper is devoted to the study of the inhomogeneous hyperdissipative Navier-Stokes equations on the whole space $\mathbb{R}^N,N\geq3$. Compared with the classical inhomogeneous Navier-Stokes, the dissipative term $- Δ u$ here is replaced by $D^2u$, where $D$ is a Fourier multiplier whose symbol is $m(\xi)=|\xi|^{\frac{N+2}{4}}$. For arbitrary small positive constants $ε$ and $δ$, global well-posedness is showed for the data $(\rho_0, u_0)$ such that $(ρ_{0} - 1, u_0)∈ H^{\frac{N}{2}+ε} × H^{δ}$ with $\inf_{x\in \mathbb{R}^N}\rho_0>0$. To our best knowledge, this is the first result on the inhomogeneous hyperdissipative Navier-Stokes equations, and it can also be viewed as the high-dimensional generalization of the 2D result for classical inhomogeneous Navier-Stokes equations given by Danchin [Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), 353--386.]
Discretization of dynamical systems with first integrals
Michal Fečkan and Michal Pospíšil
2013, 33(8): 3543-3554 doi: 10.3934/dcds.2013.33.3543 +[Abstract](81) +[PDF](350.4KB)
We find conditions under which the first integral of an ordinary differential equation becomes a discrete Lyapunov function for its numerical discretization. This result is applied for precluding periodic and bounded orbits under discretization for several cases.
On "Arnold's theorem" on the stability of the solar system
Jacques Féjoz
2013, 33(8): 3555-3565 doi: 10.3934/dcds.2013.33.3555 +[Abstract](92) +[PDF](349.7KB)
Arnold's theorem on the planetary problem states that, assuming that the masses of $n$ planets are small enough, there exists in the phase space a set of initial conditions of positive Lebesgue measure, leading to quasiperiodic motions with $3n-1$ frequencies. Arnold's initial proof is complete only for the plane $2$-planet problem. Arnold had missed a resonance later discovered by Herman. The first complete proof, by Herman-Féjoz, relies on the weak non-degeneracy condition of Arnold-Pyartli. A second proof, by Chierchia-Pinzari, is closer to Arnold's initial idea and shows the strong non-degeneracy of the problem after suitable reduction by (part of) the symmetry of rotation. We review and compare these proofs. In an appendix, we define the Poincaré coordinates and prove their symplectic nature through the shortest possible computation.
Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations
Armengol Gasull, Hector Giacomini and Joan Torregrosa
2013, 33(8): 3567-3582 doi: 10.3934/dcds.2013.33.3567 +[Abstract](102) +[PDF](449.3KB)
It is well-known that the existence of traveling wave solutions for reaction-diffusion partial differential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar differential equations. We introduce a method for finding explicit upper and lower bounds of these heteroclinic orbits. In particular, for the classical Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to locate these solutions analytically and with very high accuracy. These results allow one to construct analytical approximate expressions for the traveling wave solutions with a rigorous control of the errors for arbitrary values of the independent variables. These explicit expressions are very simple and tractable for practical purposes. They are constructed with exponential and rational functions.
On the moments of solutions to linear parabolic equations involving the biharmonic operator
Filippo Gazzola
2013, 33(8): 3583-3597 doi: 10.3934/dcds.2013.33.3583 +[Abstract](86) +[PDF](432.1KB)
We consider the solutions to Cauchy problems for the parabolic equation $u_\tau +\Delta^2u=0$ in $\mathbb{R}_+\times\mathbb{R}^n$, with fast decay initial data. We study the behavior of their moments. This enables us to give a more precise description of the sign-changing behavior of solutions corresponding to positive initial data.
Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities
Gabriele Grillo, Matteo Muratori and Maria Michaela Porzio
2013, 33(8): 3599-3640 doi: 10.3934/dcds.2013.33.3599 +[Abstract](83) +[PDF](741.4KB)
We study weighted porous media equations on domains $\Omega\subseteq{\mathbb R}^N$, either with Dirichlet or with Neumann homogeneous boundary conditions when $\Omega\not={\mathbb R}^N$. Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, $L^{q_0}$-$L^\varrho$ smoothing effects ($1\leq q_0<\varrho<\infty$) are discussed for short time, in connection with the validity of a Poincaré inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. In fact, we prove full equivalence between certain $L^{q_0}$-$L^\varrho$ smoothing effects and suitable weighted Poincaré-type inequalities. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case $\Omega={\mathbb R}^N$ when the corresponding weight makes its measure finite, so that solutions converge to their weighted mean value instead than to zero. Examples are given in terms of wide classes of weights.
Partial hyperbolicity on 3-dimensional nilmanifolds
Andy Hammerlindl
2013, 33(8): 3641-3669 doi: 10.3934/dcds.2013.33.3641 +[Abstract](109) +[PDF](787.5KB)
Every partially hyperbolic diffeomorphism on a 3-dimensional nilmanifold is leaf conjugate to a nilmanifold automorphism. Moreover, if the nilmanifold is not the 3-torus, the center foliation is an invariant circle bundle.
Branch interactions and long-term dynamics for the diblock copolymer model in one dimension
Ian Johnson, Evelyn Sander and Thomas Wanner
2013, 33(8): 3671-3705 doi: 10.3934/dcds.2013.33.3671 +[Abstract](61) +[PDF](1825.5KB)
Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki [26]. Their model yields a Cahn-Hilliard-like equation, where a nonlocal term is added to the standard Cahn-Hilliard energy. We study the long-term dynamics of this model on one-dimensional domains through a combination of bifurcation theoretic results and numerical simulations. Our results shed light on how the complicated bifurcation behavior of the diblock copolymer model is related to the better known bifurcation structure of the Cahn-Hilliard equation. In addition, we demonstrate that this knowledge can be used to predict the long-term dynamics of solutions originating close to the homogeneous equilibrium. In particular, we show that the periodicity of the long-term limit of such solutions can be predicted by tracking certain secondary bifurcation points in the bifurcation diagram, and that the long-term limit is in general not given by the global energy minimizer.
Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations
Yong Jung Kim, Wei-Ming Ni and Masaharu Taniguchi
2013, 33(8): 3707-3718 doi: 10.3934/dcds.2013.33.3707 +[Abstract](86) +[PDF](388.7KB)
Assume a single reaction-diffusion equation has zero as an asymptotically stable stationary point. Then we prove that there exist no localized travelling waves with non-zero speed. If $[\liminf_{|x|\to\infty}u(x),\limsup_{|x|\to\infty}u(x)]$ is included in an open interval of zero that does not include other stationary points, then the speed has to be zero or the travelling profile $u$ has to be identically zero.
Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards
Michel L. Lapidus and Robert G. Niemeyer
2013, 33(8): 3719-3740 doi: 10.3934/dcds.2013.33.3719 +[Abstract](69) +[PDF](1019.7KB)
The Koch snowflake $KS$ is a nowhere differentiable curve. The billiard table $Ω (KS)$ with boundary $KS$ is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes $Ω (KS)$ such an interesting, yet difficult, table to analyze.
    In this paper, we approach this problem by approximating (from the inside) $Ω (KS)$ by well-defined (prefractal) rational polygonal billiard tables $Ω (KS_{n})$. We first show that the flat surface $S(KS_{n})$ determined from the rational billiard $Ω (KS_{n})$ is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [6], allows us to define a sequence of compatible orbits of prefractal billiards $Ω (KS_{n})$. Using a particular addressing system, we define a hybrid orbit of a prefractal billiard $Ω (KS_{n})$ and show that every dense orbit of a prefractal billiard $Ω (KS_{n})$ is a dense hybrid orbit of $Ω (KS_{n})$. This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits.
    We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of $Ω(KS)$. In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. We conjecture that such a path is indeed the subset of what will eventually be an orbit of the Koch snowflake fractal billiard, once an appropriate `fractal law of reflection' is determined.
    Finally, we close with a discussion of several open problems and potential directions for further research. We discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining 'fractal flat surfaces' naturally associated with the billiard flows.
A footnote on expanding maps
Carlangelo Liverani
2013, 33(8): 3741-3751 doi: 10.3934/dcds.2013.33.3741 +[Abstract](98) +[PDF](411.3KB)
I introduce Banach spaces on which it is possible to precisely characterize the spectrum of the transfer operator associated to a piecewise expanding map with Hölder weight.
K-groups of the full group actions on one-sided topological Markov shifts
Kengo Matsumoto
2013, 33(8): 3753-3765 doi: 10.3934/dcds.2013.33.3753 +[Abstract](85) +[PDF](390.5KB)
Let $(X_A,\sigma_A)$ be the one-sided topological Markov shift for an $N\times N$ irreducible matrix $A$ with entries in $\{ 0,1 \}$. We will first show that the continuous full group $\Gamma_A$ of $(X_A,\sigma_A)$ is non amenable as a discrete group and contains all finite groups and free groups. We will second introduce the K-group $K^0(X_A,\Gamma_A)$ for the action of $\Gamma_A$ on $X_A$, and show that the group $K^0(X_A,\Gamma_A)$ with the position of the constant $1$ function is a complete invariant for the topological conjugacy class of the action of $\Gamma_A$ on $X_A$ under some conditions.
$\varepsilon$-neighborhoods of orbits and formal classification of parabolic diffeomorphisms
Maja Resman
2013, 33(8): 3767-3790 doi: 10.3934/dcds.2013.33.3767 +[Abstract](92) +[PDF](642.6KB)
In this article we study the dynamics generated by germs of parabolic diffeomorphisms $f:(\mathbb{C},0)\rightarrow (\mathbb{C},0)$ tangent to the identity. We show how formal classification of a given parabolic diffeomorphism can be deduced from the asymptotic development of what we call directed area of the $\varepsilon$-neighborhood of any orbit near the origin. Relevant coefficients and constants in the development have a geometric meaning. They present fractal properties of the orbit, namely its box dimension, Minkowski content and what we call residual content.
A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density
Anthony Suen
2013, 33(8): 3791-3805 doi: 10.3934/dcds.2013.33.3791 +[Abstract](103) +[PDF](209.3KB)
We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in $\mathbb{R}^3$. We establish a blow-up criterion for the local strong solutions in terms of the density and magnetic field. Namely, if the density is away from vacuum ($\rho= 0$) and the concentration of mass ($\rho=\infty$) and if the magnetic field is bounded above in terms of $L^\infty$-norm, then a local strong solution can be continued globally in time.
Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero
Juntao Sun, Jifeng Chu and Zhaosheng Feng
2013, 33(8): 3807-3824 doi: 10.3934/dcds.2013.33.3807 +[Abstract](209) +[PDF](406.6KB)
In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first order periodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we establish some new criteria to guarantee that Hamiltonian systems, with asymptotically quadratic terms and spectrum point zero, have at least one and infinitely many homoclinic orbits under certain conditions.
Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies
Nicola Soave and Susanna Terracini
2013, 33(8): 3825-3829 doi: 10.3934/dcds.2013.33.3825 +[Abstract](98) +[PDF](293.9KB)
This paper aims at completing and clarifying a delicate step in the proof of Theorem 5.3 of our paper [1], where it was used the differentiability of a function $F$, which a priori can appear not necessarily differentiable.
Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms
Genni Fragnelli, Paolo Nistri and Duccio Papini
2013, 33(8): 3831-3834 doi: 10.3934/dcds.2013.33.3831 +[Abstract](97) +[PDF](268.6KB)
We correct a flaw in the proof of [1, Lemma 2.3].

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