# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

September 2016 , Volume 36 , Issue 9

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2016, 36(9): 4619-4635 doi: 10.3934/dcds.2016001 +[Abstract](1897) +[PDF](468.6KB)
Abstract:
We consider a minimal compact lamination by hyperbolic surfaces. We prove that if no leaf is simply connected, then the horocycle flow on its unitary tangent bundle is minimal.
2016, 36(9): 4637-4664 doi: 10.3934/dcds.2016002 +[Abstract](2067) +[PDF](1863.4KB)
Abstract:
The present work deals with numerical methods for computing slow stable invariant manifolds as well as their invariant stable and unstable normal bundles. The slow manifolds studied here are sub-manifolds of the stable manifold of a hyperbolic equilibrium point. Our approach is based on studying certain partial differential equations equations whose solutions parameterize the invariant manifolds/bundles. Formal solutions of the partial differential equations are obtained via power series arguments, and truncating the formal series provides an explicit polynomial representation for the desired chart maps. The coefficients of the formal series are given by recursion relations which are amenable to computer calculations. The parameterizations conjugate the dynamics on the invariant manifolds and bundles to a prescribed linear dynamical systems. Hence in addition to providing accurate representation of the invariant manifolds and bundles our methods describe the dynamics on these objects as well. Example computations are given for vector fields which arise as Galerkin projections of a partial differential equation. As an application we illustrate the use of the parameterized slow manifolds and their linear bundles in the computation of heteroclinic orbits.
2016, 36(9): 4665-4702 doi: 10.3934/dcds.2016003 +[Abstract](1503) +[PDF](701.2KB)
Abstract:
Polynomials from the closure of the principal hyperbolic domain of the cubic connectedness locus have some specific properties, which were studied in a recent paper by the authors. The family of (affine conjugacy classes of) all polynomials with these properties is called the Main Cubioid. In this paper, we describe a combinatorial counterpart of the Main Cubioid --- the set of invariant laminations that can be associated to polynomials from the Main Cubioid.
2016, 36(9): 4703-4721 doi: 10.3934/dcds.2016004 +[Abstract](2039) +[PDF](468.8KB)
Abstract:
In this paper we consider positive supersolutions of the elliptic equation $-\triangle u = f(u) |\nabla u|^q$, posed in exterior domains of $\mathbb{R}^N$ ($N\ge 2$), where $f$ is continuous in $[0,+\infty)$ and positive in $(0,+\infty)$ and $q>0$. We classify supersolutions $u$ into four types depending on the function $m(R)=\inf_{|x|=R} u(x)$ for large $R$, and give necessary and sufficient conditions in order to have supersolutions of each of these types. As a consequence, we also obtain Liouville theorems for supersolutions depending on the values of $N$, $q$ and on some integrability properties on $f$ at zero or infinity. We also describe these questions when the equation is posed in the whole $\mathbb{R}^N$.
2016, 36(9): 4723-4738 doi: 10.3934/dcds.2016005 +[Abstract](1460) +[PDF](482.1KB)
Abstract:
We first show that the subgroup of the abelian real group $\mathbb{R}$ generated by the coordinates of a point in $x\in\mathbb{R}^n$ completely classifies the GL$(n,\mathbb{Z})$-orbit of $x$. This yields a short proof of J.S. Dani's theorem: the GL$(n,\mathbb{Z})$-orbit of $x\in\mathbb{R}^n$ is dense iff $x_i/x_j\in \mathbb{R}\setminus \mathbb{Q}$ for some $i,j=1,\dots,n$. We then classify GL$(n,\mathbb{Z})$-orbits of rational affine subspaces $F$ of $\mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope associated to $F$ yields a complete classifier of the GL$(n,\mathbb{Z})$-orbit of $F$.
2016, 36(9): 4739-4759 doi: 10.3934/dcds.2016006 +[Abstract](1848) +[PDF](513.3KB)
Abstract:
Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting on a compact Riemanian manifold $M$ without boundary. We state and prove several sufficient conditions for the topological entropy of $f$ to be positive. The conditions deal with the dynamical behaviour of the (non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue measure is $\delta$-recurrent then the entropy of $f$ is positive. We give counterexamples showing that these sufficient conditions are not necessary. Finally, in the case of partially hyperbolic diffeomorphisms, we give a positive lower bound for the entropy relating it with the dimension of the unstable and stable sub-bundles.
2016, 36(9): 4761-4811 doi: 10.3934/dcds.2016007 +[Abstract](1565) +[PDF](661.0KB)
Abstract:
In [13], Chen-Ma-Salani established the strict convexity of spacetime level sets of solutions to heat equation in convex rings, using the constant rank theorem and a deformation method. In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, that is, establish the corresponding microscopic spacetime convexity principles for spacetime level sets. In fact, the results hold for fully nonlinear parabolic equations under a general structural condition, including the $p$-Laplacian parabolic equations ($p >1$) and some mean curvature type parabolic equations.
2016, 36(9): 4813-4837 doi: 10.3934/dcds.2016008 +[Abstract](1426) +[PDF](1176.9KB)
Abstract:
The paper is concerned with the discretization and solution of DAEs of index $1$ and subject to a highly oscillatory forcing term. Separate asymptotic expansions in inverse powers of the oscillatory parameter are constructed to approximate the differential and algebraic variables of the DAEs. The series are truncated to enable practical implementation. Numerical experiments are provided to illustrate the effectiveness of the method.
2016, 36(9): 4839-4870 doi: 10.3934/dcds.2016009 +[Abstract](1949) +[PDF](590.3KB)
Abstract:
In this paper, we study the asymptotic behavior of solution to the initial boundary value problem for the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line $\mathbb{R}_{+}:=(0,\infty).$ Our idea mainly comes from [10] which describes the large time behavior of solution for the non-isentropic Navier-Stokes equations in a half line. The electric field brings us some additional troubles compared with Navier-Stokes equations in the absence of the electric field. We obtain the convergence rate of global solution towards corresponding stationary solution. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proofs are given by a weighted energy method.
2016, 36(9): 4871-4893 doi: 10.3934/dcds.2016010 +[Abstract](1493) +[PDF](603.3KB)
Abstract:
Let $x$ and $y$ be points in a billiard table $M=M(\sigma)$ in $\mathbb{\mathbb{R}}^{2}$ that is bounded by a curve $\sigma$. We assume $\sigma\in\Sigma_{r}$ with $r\geq2$, where $\Sigma_{r}$ is the set of simple closed $C^{r}$ curves in $\mathbb{R}^{2}$ with positive curvature. A subset $B$ of $M\setminus\{x,y\}$ is called a blocking set for the pair $(x,y)$ if every billiard path in $M$ from $x$ to $y$ passes through a point in $B$. If a finite blocking set exists, the pair $(x,y)$ is called secure in $M;$ if not, it is called insecure. We show that for $\sigma$ in a dense $G_{\delta}$ subset of $\Sigma_{r}$ with the $C^{r}$ topology, there exists a dense $G_{\delta}$ subset $\mathcal{\mathcal{R}=R}(\sigma)$ of $M(\sigma)\times M(\sigma)$ such that $(x,y)$ is insecure in $M(\sigma)$ for each $(x,y)\in\mathcal{R}$. In this sense, for the generic Birkhoff billiard, the generic pair of interior points is insecure. This is related to a theorem of S. Tabachnikov [24] that $(x,y)$ is insecure for all sufficiently close points $x$ and $y$ on a strictly convex arc on the boundary of a smooth table.
2016, 36(9): 4895-4914 doi: 10.3934/dcds.2016011 +[Abstract](2306) +[PDF](461.6KB)
Abstract:
We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for stochastic partial differential equations with BMO$_x$ coefficients.
2016, 36(9): 4915-4923 doi: 10.3934/dcds.2016012 +[Abstract](1587) +[PDF](363.3KB)
Abstract:
In this paper we establish some regularity criteria for the three-dimensional Boussinesq system with the temperature-dependent viscosity and thermal diffusivity in a bounded domain.
2016, 36(9): 4925-4943 doi: 10.3934/dcds.2016013 +[Abstract](1735) +[PDF](398.5KB)
Abstract:
By variational methods, this paper considers 4k-periodic solutions of a kind of differential delay systems with $2k-1$ lags. Our results reveal the fact that the number of $4k-$periodic orbits depends only upon the eigenvalues of both matrices $A_{\infty}$ and $A_0$. The conditions are more definite and easier to be examined. Moreover, two examples are given to illustrate the applications of the results.
2016, 36(9): 4945-4962 doi: 10.3934/dcds.2016014 +[Abstract](1439) +[PDF](874.7KB)
Abstract:
The model for disordered actomyosin bundles recently derived in [6] includes the effects of cross-linking of parallel and anti-parallel actin filaments, their polymerization and depolymerization, and, most importantly, the interaction with the motor protein myosin, which leads to sliding of anti-parallel filaments relative to each other. The model relies on the assumption that actin filaments are short compared to the length of the bundle. It is a two-phase model which treats actin filaments of both orientations separately. It consists of quasi-stationary force balances determining the local velocities of the filament families and of transport equation for the filaments. Two types of initial-boundary value problems are considered, where either the bundle length or the total force on the bundle are prescribed. In the latter case, the bundle length is determined as a free boundary. Local in time existence and uniqueness results are proven. For the problem with given bundle length, a global solution exists for short enough bundles. For small prescribed force, a formal approximation can be computed explicitly, and the bundle length tends to a limiting value.
2016, 36(9): 4963-4996 doi: 10.3934/dcds.2016015 +[Abstract](1453) +[PDF](5391.5KB)
Abstract:
Brenier and Grenier [SIAM J. Numer. Anal., 1998] proved that sticky particle dynamics with a large number of particles allow to approximate the entropy solution to scalar one-dimensional conservation laws with monotonic initial data. In [arXiv:1501.01498], we introduced a multitype version of this dynamics and proved that the associated empirical cumulative distribution functions converge to the viscosity solution, in the sense of Bianchini and Bressan [Ann. of Math. (2), 2005], of one-dimensional diagonal hyperbolic systems with monotonic initial data of arbitrary finite variation. In the present paper, we analyse the $L^1$ error of this approximation procedure, by splitting it into the discretisation error of the initial data and the non-entropicity error induced by the evolution of the particle system. We prove that the error at time $t$ is bounded from above by a term of order $(1+t)/n$, where $n$ denotes the number of particles, and give an example showing that this rate is optimal. We last analyse the additional error introduced when replacing the multitype sticky particle dynamics by an iterative scheme based on the typewise sticky particle dynamics, and illustrate the convergence of this scheme by numerical simulations.
2016, 36(9): 4997-5010 doi: 10.3934/dcds.2016016 +[Abstract](1904) +[PDF](402.2KB)
Abstract:
We study a dynamics of solutions to a system of reaction-diffusion equations modeling a biological pattern formation. This model has activator-inhibitor type nonlinearities and we show that it has solutions blowing up in a finite time. More precisely, in the case of absence of a diffusion of an activator, we show that there are solutions which blow up in a finite time at one point, only. This result holds true for the whole range of nonlinearity exponents in the considered activator-inhibitor system. Next, we consider a range of nonlinearities, where some space-homogeneous solutions blow up in a finite time and we show an analogous result for space-inhomogeneous solutions.
2016, 36(9): 5011-5024 doi: 10.3934/dcds.2016017 +[Abstract](1474) +[PDF](446.6KB)
Abstract:
In this paper, topological conjugacy for two-sided non-hyperbolic and non-autonomous discrete dynamical systems is studied. It is shown that if the system has covering relations with weak Lyapunov condition determined by a transition matrix, there exists a sequence of compact invariant sets restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by the transition matrix. Moreover, if the systems have covering relations with exponential dichotomy and small Lipschitz perturbations, then there is a constructive verification proof of the weak Lyapunov condition, and so topological dynamics of these systems are fully understood by symbolic representations. In addition, the tolerance of Lipschitz perturbation can be characterised by the dichotomy tuple . Here, the weak Lyapunov condition is adapted from [12,24,15] and the exponential dichotomy is from [2].
2016, 36(9): 5025-5046 doi: 10.3934/dcds.2016018 +[Abstract](2359) +[PDF](502.5KB)
Abstract:
In this paper, we consider a fully parabolic chemotaxis system \begin{eqnarray*}\label{1} \left\{ \begin{array}{llll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+u-\mu u^r,\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-v+u,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain $\Omega\subset R^n(n=2,3)$, where $\chi>0, \mu>0$ and $r\geq 2$.
For the dimensions $n=2$ and $n=3$, we establish results on the global existence and boundedness of classical solutions to the corresponding initial-boundary problem, provided that $\chi$, $\mu$ and $r$ satisfy some explicit conditions. Apart from this, we also show that if $\frac{\mu^{\frac{1}{r-1}}}{\chi}>20$ and $r\geq 2$ and $r\in \mathbb{N}$ the solution of the system approaches the steady state $\left(\mu^{-\frac{1}{r-1}}, \mu^{-\frac{1}{r-1}}\right)$ as time tends to infinity.
2016, 36(9): 5047-5066 doi: 10.3934/dcds.2016019 +[Abstract](1952) +[PDF](446.8KB)
Abstract:
In this paper we mainly investigate the Cauchy problem of a three-component Camassa-Holm system. By using Littlewood-Paley theory and transport equations theory, we establish the local well-posedness of the system in the critical Besov space. Moreover, we obtain some weighted $L^p$ estimates of strong solutions to the system. By taking suitable weighted functions, we can get the persistence properties of strong solutions on exponential, algebraic and logarithmic decay rates, respectively.
2016, 36(9): 5067-5096 doi: 10.3934/dcds.2016020 +[Abstract](1448) +[PDF](717.7KB)
Abstract:
The correlation integral and determinism are quantitative characteristics of a dynamical system based on the recurrence of orbits. For strongly non-chaotic interval maps, the determinism equals $1$ for every small enough threshold. This means that trajectories of such systems are perfectly predictable in the infinite horizon. In this paper we study the correlation integral and determinism for the family of $2^\infty$ non-chaotic maps, first considered by Delahaye in 1980. The determinism in a finite horizon equals $1$. However, the behaviour of the determinism in the infinite horizon is counter-intuitive. Sharp bounds on the determinism are provided.
2016, 36(9): 5097-5118 doi: 10.3934/dcds.2016021 +[Abstract](1743) +[PDF](437.6KB)
Abstract:
We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensates and study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one or two spatial dimensions, respectively, are rigorously obtained as special cases of an averaged three dimensional limit model. In the particular case where the rotation axis is not parallel to the strongly confining direction the resulting limiting model(s) include a negative, and thus, purely repulsive quadratic potential, which is not present in the original equation and which can be seen as an effective centrifugal force counteracting the confinement.
2016, 36(9): 5119-5129 doi: 10.3934/dcds.2016022 +[Abstract](1552) +[PDF](697.0KB)
Abstract:
Based on mathematical analysis, this paper proves the existence of homoclinic orbits in a new class of 3-dimensional piecewise affine systems, and gives an example to illustrate the effectiveness of the method.
2016, 36(9): 5131-5162 doi: 10.3934/dcds.2016023 +[Abstract](1564) +[PDF](527.5KB)
Abstract:
In the planar $N$-center problem, given a non-trivial free homotopy class of the configuration space satisfying certain conditions, we show that there is at least one collision-free $T$-periodic solution for any positive $T.$ The direct method of calculus of variations is used and the main difficulty is to show that minimizers under certain topological constraints are free of collision.
2016, 36(9): 5163-5181 doi: 10.3934/dcds.2016024 +[Abstract](2207) +[PDF](440.3KB)
Abstract:
This paper is devoted to study the abstract functional differential equation (FDE) of the following form $$\dot{u}(t)=Au(t)+\Phi u_t,$$ where $A$ generates a $C$-regularized semigroup, which is the generalization of $C_0$-semigroup and can be applied to deal with many important differential operators that the $C_0$-semigroup can not be used to. We first show that the $C$-well-posedness of a FDE is equivalent to the $\mathscr{C}$-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator $\mathscr{C}$ is related with the operator $C$ and will be defined in the following text. Then, by making use of a perturbation result of $C$-regularized semigroup, a sufficient condition is provided for the $C$-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.
2016, 36(9): 5183-5199 doi: 10.3934/dcds.2016025 +[Abstract](1681) +[PDF](430.1KB)
Abstract:
We consider a space-inhomogeneous KPP equation with a nonlocal diffusion and an almost-periodic nonlinearity. By employing and adapting the theory of homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.
2016, 36(9): 5201-5221 doi: 10.3934/dcds.2016026 +[Abstract](1918) +[PDF](523.6KB)
Abstract:
In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well-posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploiting the balance law and some new estimates, we prove the existence and uniqueness of global weak solutions for the C-H equation with a forcing in $H^1(\mathbb{R})$.

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