# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

January 2015 , Volume 35 , Issue 1

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2015, 35(1): 1-23 doi: 10.3934/dcds.2015.35.1 +[Abstract](1833) +[PDF](958.8KB)
Abstract:
We study the rhomboidal symmetric-mass $1$, $m$, $1$, $m$ four-body problem in the four-degrees-of-freedom setting, where $0 < m \leq 1$. Under suitable changes of variables, isolated binary collisions at the origin are regularizable. Analytic existence of the orbit in the four-degrees-of-freedom setting is established. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is analytically reduced to computing three entries of a $4 \times 4$ matrix related to the monodromy matrix. Additionally, it is shown that the four-degrees-of-freedom setting has a two-degrees-of-freedom invariant set, and linear stability results in the subset comes for free'' from the calculation in the full space. The final numerical analysis shows that the four-degrees-of-freedom orbit is linearly unstable except for a very small interval about $m = 0.4$, whereas the two-degrees-of-freedom orbit is linearly stable for all but very small values of $m$.
2015, 35(1): 25-42 doi: 10.3934/dcds.2015.35.25 +[Abstract](2584) +[PDF](439.5KB)
Abstract:
The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities $u$ and $v= 2\arctan u_x$ along each characteristic, it is proved that the Cauchy problem with general initial data $u_0\in H^1(\mathbb{R})$ has a unique solution, globally in time.
2015, 35(1): 43-58 doi: 10.3934/dcds.2015.35.43 +[Abstract](1826) +[PDF](415.2KB)
Abstract:
For a dissipative Anosov diffeomorphism $f$ of the 2-torus, we give examples of $f$-invariant measures $\mu$ such that the group of $\mu$-preserving diffeomorphisms is virtually cyclic.
2015, 35(1): 59-72 doi: 10.3934/dcds.2015.35.59 +[Abstract](2669) +[PDF](367.5KB)
Abstract:
We deal with the existence of invariant cones in a family of three-dimensional non-observable piecewise linear systems with two zones of linearity. We find a subfamily of systems with one invariant cone foliated by periodic orbits. After that, we perturb the system by making it observable and non-homogeneous. Then, the periodic orbits that remain after the perturbation are analyzed.
2015, 35(1): 73-98 doi: 10.3934/dcds.2015.35.73 +[Abstract](1705) +[PDF](1270.2KB)
Abstract:
We introduce a class of set evolution equations that can be used to describe population's movements as well as various instances of individual-population interactions. Optimal control/management problems can be formalized and tackled in this framework. A rigorous analytical structure is established and the basic well posedness results are obtained. Several examples show possible applications and their numerical integrations display possible qualitative behaviors of solutions.
2015, 35(1): 99-116 doi: 10.3934/dcds.2015.35.99 +[Abstract](2016) +[PDF](276.1KB)
Abstract:
In this paper, we establish a unilateral global bifurcation result from interval for a class of $p$-Laplacian problems. By applying above result, we study the spectrum of a class of half-quasilinear problems. Moreover, we also investigate the existence of nodal solutions for a class of half-quasilinear eigenvalue problems.
2015, 35(1): 117-137 doi: 10.3934/dcds.2015.35.117 +[Abstract](2070) +[PDF](471.2KB)
Abstract:
We study the existence, the decay and the blow-up of solutions to the Cauchy problem for the multi-dimensional generalized sixth-order Boussinesq equation: $$u_{tt} - \Delta u - \Delta^{2} u- \mu \Delta ^{3} u = \Delta f(u),\; t>0, \; x \in {\mathbb{R}^{n}}, n \geq 1,$$ where $f(u)= \gamma |u|^{p-1}u, \; \gamma \in \mathbb{R}, \; p \geq 2, \; \mu > 1/4$. We find two global existence results for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$ On the other hand we show that if $\mu= 1/3$ and $p>13/2$, then the solution with small initial data decays in time. A blow up in finite time result is also obtained for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$
2015, 35(1): 139-154 doi: 10.3934/dcds.2015.35.139 +[Abstract](1877) +[PDF](1490.6KB)
Abstract:
This paper is devoted to multiplicity results of solutions to nonlocal elliptic equations modeling gravitating systems. By considering the case of Fermi--Dirac statistics as a singular perturbation of Maxwell--Boltzmann statistics, we are able to produce multiplicity results. Our method is based on cumulated mass densities and a logarithmic change of coordinates that allow us to describe the set of all solutions by a non-autonomous perturbation of an autonomous dynamical system. This has interesting consequences in terms of bifurcation diagrams, which are illustrated by some numerical computations. More specifically, we study a model based on the Fermi function as well as a simplified one for which estimates are easier to establish. The main difficulty comes from the fact that the mass enters in the equation as a parameter which makes the whole problem non-local.
Jingbo Dou and
2015, 35(1): 155-171 doi: 10.3934/dcds.2015.35.155 +[Abstract](2235) +[PDF](438.2KB)
Abstract:
In this paper we establish a Liouville type theorem for an integral system on the upper half space $\mathbb{R}_+^{n}$ \begin{equation*} \begin{cases} u(y)=\int_{\mathbb{R}^{n}_+}\frac{f(v(x))}{|x-y|^{n-\alpha}}dx,&\quad y\in\partial\mathbb{R}^{n}_+,\\ v(x)=\int_{\partial\mathbb{R}^{n}_+}\frac{g(u(y))}{|x-y|^{n-\alpha}}dy,&\quad x\in\mathbb{R}_+^{n}. \end{cases} \end{equation*} This integral system arises from the Euler-Lagrange equation corresponding to Hardy-Littlewood-Sobolev inequality on the upper half space. Under natural structure conditions on $f$ and $g$, we classify positive solutions to the above system basing on the method of moving sphere in integral forms and the Hardy-Littlewood-Sobolev inequality on the upper half space.
2015, 35(1): 173-204 doi: 10.3934/dcds.2015.35.173 +[Abstract](1772) +[PDF](427.8KB)
Abstract:
In this paper, we establish an estimate for the solutions of a small-divisor equation with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the derivative nonlinear Schrödinger equations subject to small Hamiltonian perturbations.
2015, 35(1): 205-224 doi: 10.3934/dcds.2015.35.205 +[Abstract](2276) +[PDF](503.3KB)
Abstract:
We obtain a parametric (and an orbital) normal form for any non-degenerate perturbation of the generalized saddle-node case of Bogdanov--Takens singularity. Explicit formulas are derived and greatly simplified for an efficient implementation in any computer algebra system. A Maple program is prepared for an automatic parametric normal form computation. A section is devoted to present some practical formulas which avoid technical details of the paper.
2015, 35(1): 225-246 doi: 10.3934/dcds.2015.35.225 +[Abstract](2336) +[PDF](753.5KB)
Abstract:
We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption, as well as the behavior of Bose-Einstein condensates. For small $L^2$ norm (low power), the solution executes beating oscillations between the two wells. There is a power threshold at which a symmetry breaking bifurcation occurs. The set of guided mode solutions splits into two families of solutions. One type of solution is concentrated in either well of the potential, but not both. Solutions in the second family undergo tunneling oscillations between the two wells. A finite dimensional reduction (system of ODEs) derived in [17] is expected to well-approximate the PDE dynamics on long time scales. In particular, we revisit this reduction, find a class of exact solutions and shadow them in the (NLS/GP) system by applying the approach of [17].
2015, 35(1): 247-281 doi: 10.3934/dcds.2015.35.247 +[Abstract](2004) +[PDF](654.6KB)
Abstract:
We show how any labeled convex polygon associated to a compact semi-toric system, as defined by Vũ ngọc, determines Karshon's labeled directed graph which classifies the underlying Hamiltonian $S^1$-space up to isomorphism. Then we characterize adaptable compact semi-toric systems, i.e. those whose underlying Hamiltonian $S^1$-action can be extended to an effective Hamiltonian $\mathbb{T}^2$-action, as those which have at least one associated convex polygon which satisfies the Delzant condition.
2015, 35(1): 283-299 doi: 10.3934/dcds.2015.35.283 +[Abstract](2268) +[PDF](458.7KB)
Abstract:
We study the solutions $u\in C^\infty(\mathbb{R}^{2m})$ of the problem $$\label{P0} (-\Delta)^mu=\bar Qe^{2mu}, \text{ where }\bar Q=\pm (2m-1)!, \quad V :=\int_{\mathbb{R}^{2m}}e^{2mu}dx <\infty,(1)$$ particularly when $m>1$. Problem (1) corresponds to finding conformal metrics $g_u:=e^{2u}|dx|^2$ on $\mathbb{R}^{2m}$ with constant $Q$-curvature $\bar Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $|x|\to \infty$ can be simultaneously prescribed, under certain restrictions. When $\bar Q= (2m-1)!$ we need to assume $V < vol(S^{2m})$, but surprisingly for $\bar Q=-(2m-1)!$ the volume $V$ can be chosen arbitrarily.
2015, 35(1): 301-322 doi: 10.3934/dcds.2015.35.301 +[Abstract](2584) +[PDF](419.2KB)
Abstract:
Whether or not classical solutions of the 3D incompressible MHD equations with full dissipation and magnetic diffusion can develop finite-time singularities is a long standing open problem of fluid dynamics and PDE theory. In this paper, we investigate the Cauchy problem for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. We get a unique global smooth solution under the assumption that $u_\theta$ and $b_r$ are trivial. In absence of some viscosities, there is no smoothing effect on the derivatives of that direction. However, we take full advantage of the structures of MHD system to make up this shortcoming.
2015, 35(1): 323-339 doi: 10.3934/dcds.2015.35.323 +[Abstract](2139) +[PDF](470.5KB)
Abstract:
We investigate a class of nonlocal conservation laws with the nonlinear advection coupling both local and nonlocal mechanism, which arises in several applications such as the collective motion of cells and traffic flows. It is proved that the $C^1$ solution regularity of this class of conservation laws will persist at least for a short time. This persistency may continue as long as the solution gradient remains bounded. Based on this result, we further identify sub-thresholds for finite time shock formation in traffic flow models with Arrhenius look-ahead dynamics.
2015, 35(1): 341-352 doi: 10.3934/dcds.2015.35.341 +[Abstract](2314) +[PDF](386.3KB)
Abstract:
In this note we show that, for a class of partially hyperbolic $C^r$ ($r \geq 1$) diffeomorphisms, (1) Shub's entropy conjecture holds true; (2) SRB measures exist as zero-noise limits.
2015, 35(1): 353-365 doi: 10.3934/dcds.2015.35.353 +[Abstract](2113) +[PDF](434.8KB)
Abstract:
In this article we intend to contribute in the understanding of the ergodic properties of the set of robustly transitive local diffeomorphisms on a compact manifold without boundary. We prove that $C^1$ generic robustly transitive local diffeomorphisms have a residual subset of points with dense pre-orbits. Moreover, $C^1$ generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbits, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results hold for an important class of robustly transitive maps.
2015, 35(1): 367-397 doi: 10.3934/dcds.2015.35.367 +[Abstract](2081) +[PDF](569.3KB)
Abstract:
In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems, and we study the properties of these systems, including their regularity and reversibility, from the perspective of symplectic and Poisson geometry. Next, we use this framework---along with a generalized notion of generating function due to Śniatycki and Tulczyjew [18]---to develop a theory of discrete constrained Lagrangian mechanics. This allows for systems with arbitrary constraints, including those which are non-integrable (in an appropriate discrete, variational sense). In addition to characterizing the dynamics of these constrained systems, we also develop a theory of reduction and Noether symmetries, and study the relationship between the dynamics and variational principles. Finally, we apply this theory to discretize several concrete examples of constrained systems in mechanics and optimal control.
2015, 35(1): 399-409 doi: 10.3934/dcds.2015.35.399 +[Abstract](1904) +[PDF](336.0KB)
Abstract:
We prove Liouville-type theorems for a parabolic system in dimension $N=1$ and for radial solutions in all dimensions under an optimal Sobolev growth restriction on the nonlinearities. This seems to be the first example of a Liouville-type theorem in the whole Sobolev subcritical range for a parabolic system (even for radial solutions). Moreover, this also seems to be the first application of the Gidas-Spruck technique to a parabolic system.
2015, 35(1): 411-426 doi: 10.3934/dcds.2015.35.411 +[Abstract](1665) +[PDF](429.7KB)
Abstract:
We consider the $\Gamma$-limit of a highly oscillatory Riemannian metric length functional as its period tends to 0. The metric coefficient takes values in either $\{1,\infty\}$ or $\{1,\beta \epsilon^{-p}\}$ where $\beta,\epsilon > 0$ and $p \in (0,\infty)$. We find that for a large class of metrics, in particular those metrics whose surface of discontinuity forms a differentiable manifold, the $\Gamma$-limit exists, as in the case of a uniformly bounded sequence of metrics. However, the existence of the $\Gamma$-limit for the corresponding boundary value problem depends on the value of $p$. Specifically, we show that the power $p=1$ is critical in that the $\Gamma$-limit exists for $p < 1$, whereas it ceases to exist for $p \geq 1$. The results here have applications in both nonlinear optics and the effective description of a Hamiltonian particle in a discontinuous potential.
2015, 35(1): 427-440 doi: 10.3934/dcds.2015.35.427 +[Abstract](2768) +[PDF](398.0KB)
Abstract:
In this paper, we obtain the existence of infinitely many solutions for the following Schrödinger-Poisson system \begin{equation*} \begin{cases} -\Delta u+a(x)u+ \phi u=k(x)|u|^{q-2}u- h(x)|u|^{p-2}u,\quad &x\in \mathbb{R}^3,\\ -\Delta \phi=u^2,\ \lim_{|x|\to +\infty}\phi(x)=0, &x\in \mathbb{R}^3, \end{cases} \end{equation*} where $1 < q < 2 < p < +\infty$, $a(x)$, $k(x)$ and $h(x)$ are measurable functions satisfying suitable assumptions.
2015, 35(1): 441-466 doi: 10.3934/dcds.2015.35.441 +[Abstract](2693) +[PDF](510.5KB)
Abstract:
The regularity of the pullback random attractor for a stochastic FitzHugh-Nagumo system on $\mathbb R^n$ driven by deterministic non-autonomous forcing is proved. More precisely, the pullback random attractor is shown to be compact in $H^1(\mathbb R^n)\times L^2(\mathbb R^n)$ and attract all tempered sets of $L^2(\mathbb R^n)\times L^2(\mathbb R^n)$ in the topology of $H^1(\mathbb R^n)\times L^2(\mathbb R^n)$. The proof is based on tail estimates technique, eigenvalues of the Laplace operator in bounded domains and some new estimates of solutions.
2015, 35(1): 467-482 doi: 10.3934/dcds.2015.35.467 +[Abstract](2247) +[PDF](403.6KB)
Abstract:
In this paper we analyze the behavior of small amplitude solutions of the variant of the classical Kap system given by $\partial_t u = \partial_x v - 2 \partial_x(v^3), \quad \partial_t v = \partial_x u - \frac 1 3 \partial_{xxx} u.$ It is proved that the above equation admits small-amplitude solutions that are quasiperiodic in time and that correspond to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof relies on the Hamiltonian formulation of the problem, the study of its Birkhoff normal form and an infinite dimensional KAM theorem. This is the abstract of your paper and it should not exceed.
2015, 35(1): 483-512 doi: 10.3934/dcds.2015.35.483 +[Abstract](1930) +[PDF](355.6KB)
Abstract:
In this paper we study the singularly perturbed parabolic system of competing species. This problem exhibit a phase separation" phenomena when the interaction between different species is very strong. We are concerned with the case where different species may have different diffusion rates. We identify its singular limit with the heat flow (i.e. gradient flow) of harmonic maps into a metric space with non-positive curvature, by establishing the system of differential inequalities satisfied by this heat flow and uniqueness of the solution to the corresponding initial-boundary value problem.
2015, 35(1): 513-536 doi: 10.3934/dcds.2015.35.513 +[Abstract](2215) +[PDF](567.6KB)
Abstract:
In this paper, the compressible Navier-Stokes-Korteweg equations with a potential external force is considered in $\mathbb{R}^3$. Under the smallness assumption on both the external force and the initial perturbation of the stationary solution in some Sobolev spaces, we establish the existence theory of global solutions to the stationary profile. What's more, when the initial perturbation is bounded in $L^p$-norm with $1\leq p<2$, the optimal time decay rates of the solution in $L^q$-norm with $2\leq q\leq 6$ and its first order derivative in $L^2$-norm are shown. On the other hand, when the $\dot{H}^{-s}$ norm $(s\in(0,\frac{3}{2}])$ of the perturbation is finite, we obtain the optimal time decay rates of the solution and its first order derivative in $L^2$-norm.
2015, 35(1): 537-553 doi: 10.3934/dcds.2015.35.537 +[Abstract](2862) +[PDF](479.4KB)
Abstract:
The subject of the paper is scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.
2015, 35(1): 555-582 doi: 10.3934/dcds.2015.35.555 +[Abstract](2206) +[PDF](655.6KB)
Abstract:
In this paper, we are concerned with a model arising from electro-hydrodynamics, which is a coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms. The local well-posedness and global well-posedness with small initial data to the 3-D Cauchy problem of this system are established in the critical Besov space $\dot{B}^{-1+\frac{3}{p}}_{p,1}(\mathbb{R}^{3})\times(\dot{B}^{-2+\frac{3}{q}}_{q,1}(\mathbb{R}^{3}))^{2}$ with suitable choices of $p, q$. Especially, we prove that there exist two positive constants $c_{0}, C_{0}$ depending on the coefficients of system except $\mu$ such that if \begin{equation*} \big(\|u_{0}^{h}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}+(\mu+1)\|(v_{0},w_{0})\|_{\dot{B}^{-2+\frac{3}{q}}_{q,1}} \big) \exp\Big\{\frac{C_{0}}{\mu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}^{2}+1)\Big\}\leq c_{0}\mu, \end{equation*} then the above local solution can be extended to the global one. This result implies the global well-posedness of this system with large initial vertical velocity component.
2015, 35(1): 583-591 doi: 10.3934/dcds.2015.35.583 +[Abstract](2103) +[PDF](325.6KB)
Abstract:
We consider the Gierer-Meinhardt system (1.1), shown below, on a bounded smooth domain $\Omega\subset\mathbb{R}^n$ ($n\ge1$) with a homogeneous Neumann boundary condition. For suitable exponents $a$, $b$, $c$ and $d$, we establish certain sufficient conditions for global existence. Theorem 1.1 here, combined with Theorem 1.2 of [6], implies a classical phenomenon on the effect of the initial data on global existence and finite time blow-up. This work is a continuation of our earlier result [6] for the Gierer-Meinhardt system.
The Gierer-Meinhardt system was introduced in [1] to model activator-inhibitor systems in pattern formation in ecological systems.
2015, 35(1): 593-594 doi: 10.3934/dcds.2015.35.593 +[Abstract](2034) +[PDF](216.8KB)
Abstract:
We correct a flaw in the proof of Proposition 6.3 in [1].

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