# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

August 2017 , Volume 37 , Issue 8

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2017, 37(8): 4159-4190 doi: 10.3934/dcds.2017177 +[Abstract](1663) +[HTML](11) +[PDF](580.3KB)
Abstract:

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to 1. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem.

2017, 37(8): 4191-4211 doi: 10.3934/dcds.2017178 +[Abstract](1716) +[HTML](7) +[PDF](480.4KB)
Abstract:

Surfaces admitting flows all whose orbits are dense are called minimal. Minimal orientable surfaces were characterized by J.C. Benière in 1998, leaving open the nonorientable case. This paper fills this gap providing a characterization of minimal nonorientable surfaces of finite genus. We also construct an example of a minimal nonorientable surface with infinite genus and conjecture that any nonorientable surface without combinatorial boundary is minimal.

2017, 37(8): 4213-4230 doi: 10.3934/dcds.2017179 +[Abstract](2530) +[HTML](24) +[PDF](467.2KB)
Abstract:

In this paper, we study the following quasilinear elliptic equation with critical Sobolev exponent:

which models the self-channeling of a high-power ultra short laser in matter, where N ≥ 3; 2 < p < 2* = $\frac{{2N}}{{N -2}}$ and V (x) is a given positive potential. Combining the change of variables and an abstract result developed by Jeanjean in [14], we obtain the existence of positive ground state solutions for the given problem.

2017, 37(8): 4231-4238 doi: 10.3934/dcds.2017180 +[Abstract](1642) +[HTML](9) +[PDF](342.9KB)
Abstract:

We prove that separated net arising from certain higher rank $\mathbb R.k$ actions on homogeneous spaces is bi-Lipschitz equivalent to a lattice.

2017, 37(8): 4239-4247 doi: 10.3934/dcds.2017181 +[Abstract](1724) +[HTML](11) +[PDF](287.4KB)
Abstract:

We introduce a new type of recursion operator suitable to generate a class of nonlocal symmetries for those second-order evolution equations in $1+1$ dimension which allow the complete integration of their time-independent versions. We show that this class of evolution equations is $C$-integrable (linearizable by a point transformation). We also discuss some applications.

2017, 37(8): 4249-4275 doi: 10.3934/dcds.2017182 +[Abstract](1596) +[HTML](16) +[PDF](2359.8KB)
Abstract:

Geometric integrators for nonholonomic systems were introduced by Cortés and Martínez in [4] by proposing a discrete Lagrange-D'Alembert principle. Their approach is based on the definition of a discrete Lagrangian $L_d$ and a discrete constraint space $D_d$. There is no recipe to construct these objects and the performance of the integrator is sensitive to their choice.

Cortés and Martínez [4] claim that choosing $L_d$ and $D_d$ in a consistent manner with respect to a finite difference map is necessary to guarantee an approximation of the continuous flow within a desired order of accuracy. Although this statement is given without proof, similar versions of it have appeared recently in the literature.

We evaluate the importance of the consistency condition by comparing the performance of two different geometric integrators for the nonholonomic Suslov problem, only one of which corresponds to a consistent choice of $L_d$ and $D_d$. We prove that both integrators produce approximations of the same order, and, moreover, that the non-consistent discretisation outperforms the other in numerical experiments and in terms of energy preservation. Our results indicate that the consistency of a discretisation might not be the most relevant feature to consider in the construction of nonholonomic geometric integrators.

2017, 37(8): 4277-4308 doi: 10.3934/dcds.2017183 +[Abstract](2059) +[HTML](22) +[PDF](528.0KB)
Abstract:

We consider a diffuse interface model for tumor growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation, which models a tumor growing in the presence of a nutrient species and surrounded by healthy tissue. The well-posedness of the system equipped with Neumann boundary conditions was found to require regular potentials with quadratic growth. In this work, Dirichlet boundary conditions are considered, and we establish the well-posedness of the system for regular potentials with higher polynomial growth and also for singular potentials. New difficulties are encountered due to the higher polynomial growth, but for regular potentials, we retain the continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms as established in the previous work. Furthermore, we deduce the well-posedness of a variant of the model with quasi-static nutrient by rigorously passing to the limit where the ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is small.

2017, 37(8): 4309-4328 doi: 10.3934/dcds.2017184 +[Abstract](2031) +[HTML](10) +[PDF](460.1KB)
Abstract:

We prove that standing-waves which are solutions to the non-linear Schrödinger equation in dimension one, and whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term satisfies a Euler differential inequality. When the non-linear term is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.

2017, 37(8): 4329-4346 doi: 10.3934/dcds.2017185 +[Abstract](1660) +[HTML](16) +[PDF](473.8KB)
Abstract:

In this paper, we show the existence of solutions for the coupled Dirac system

where $M$ is an $n$-dimensional compact Riemannian spin manifold, $D$ is the Dirac operator on $M$, and $H:\Sigma M\oplus \Sigma M\to \mathbb{R}$ is a real valued superquadratic function of class $C^1$ in the fiber direction with subcritical growth rates. Our proof relies on a generalized linking theorem applied to a strongly indefinite functional on a product space of suitable fractional Sobolev spaces. Furthermore, we consider the $\mathbb{Z}_2$-invariant $H$ that includes a nonlinearity of the form

where $f(x)$ and $g(x)$ are strictly positive continuous functions on $M$ and $p, q>1$ satisfy

In this case we obtain infinitely many solutions of the coupled Dirac system by using a generalized fountain theorem.

2017, 37(8): 4347-4378 doi: 10.3934/dcds.2017186 +[Abstract](1646) +[HTML](7) +[PDF](1471.3KB)
Abstract:

We consider a dynamical system to have memory if it remembers the current state as well as the state before that. The dynamics is defined as follows: $x_{n+1}=T_{\alpha }(x_{n-1}, x_{n})=\tau (\alpha \cdot x_{n}+(1-\alpha)\cdot x_{n-1}),$ where $\tau$ is a one-dimensional map on $I=[0, 1]$ and $0 < \alpha < 1$ determines how much memory is being used. $T_{\alpha }$ does not define a dynamical system since it maps $U=I\times I$ into $I$. In this note we let $\tau$ be the symmetric tent map. We shall prove that for $0 < \alpha < 0.46,$ the orbits of $\{x_{n}\}$ are described statistically by an absolutely continuous invariant measure (acim) in two dimensions. As $\alpha$ approaches $0.5$ from below, that is, as we approach a balance between the memory state $x_{n-1}$ and the present state $x_{n}$, the support of the acims become thinner until at $\alpha =0.5$, all points have period 3 or eventually possess period 3. For $% 0.5 < \alpha < 0.75$, we have a global attractor: for all starting points in $U$ except $(0, 0)$, the orbits are attracted to the fixed point $(2/3, 2/3).$ At $%\alpha=0.75,$ we have slightly more complicated periodic behavior.

2017, 37(8): 4379-4390 doi: 10.3934/dcds.2017187 +[Abstract](1410) +[HTML](11) +[PDF](373.7KB)
Abstract:

The Livšic Theorem for Hölder continuous cocycles with values in Banach rings is proved. We consider a transitive homeomorphism ${\sigma :X\to X}$ that satisfies the Anosov Closing Lemma and a Hölder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. The map $a(x)$ is a coboundary with a Hölder continuous transition function if and only if $a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)$ is the identity for each periodic point $p=\sigma^n p$.

2017, 37(8): 4391-4398 doi: 10.3934/dcds.2017188 +[Abstract](1530) +[HTML](15) +[PDF](297.0KB)
Abstract:

In this paper, we present an explicit and exact solution of the nonlinear governing equations including Coriolis and centripetal terms for internal azimuthal waves with a uniform current in the $\beta$-plane approximation near the equator. This solution is described in the Lagrangian framework. The unidirectional azimuthal internal trapped are symmetric about the equator and propagate eastward above the thermocline and beneath the near-surface layer.

2017, 37(8): 4399-4437 doi: 10.3934/dcds.2017189 +[Abstract](1439) +[HTML](12) +[PDF](796.5KB)
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We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a $\mathbb{Z}_2$ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.

2017, 37(8): 4439-4460 doi: 10.3934/dcds.2017190 +[Abstract](1972) +[HTML](14) +[PDF](745.1KB)
Abstract:

The main object of the paper is finding a unique solution to Riemann problem for generalized Chaplygin gas model. That is a model of the dark energy in Universe introduced in the last decade. It permits an infinite mass concentration so one has to consider solutions containing the Dirac delta function. Although it was easy to construct solution to any Riemann problem, the usual admissibility conditions, overcompressiveness, do not exclude unwanted delta-type waves when a classical solution exists. We are using Shadow Wave approach in order to solve that uniqueness problem since they are well adopted for using Lax entropy–entropy flux conditions and there is a rich family of convex entropies.

2017, 37(8): 4461-4487 doi: 10.3934/dcds.2017191 +[Abstract](1563) +[HTML](11) +[PDF](588.4KB)
Abstract:

We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of Banach scale Lie algebras (this notion is defined in the article). This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses mould calculus (recalled in the article) and properties of the solution of a universal mould equation studied in a preceding paper.

2017, 37(8): 4489-4505 doi: 10.3934/dcds.2017192 +[Abstract](1828) +[HTML](11) +[PDF](453.1KB)
Abstract:

This paper is concerned with the stationary Gierer-Meinhardt system with singularity:

where $-\infty < p < 1$, $-1 < s$, and $q, r, d_1, d_2$ are positive constants, $a_1, \, a_2$ are nonnegative constants, $\rho_1, \, \rho_2$ are smooth nonnegative functions and $\Omega\subset \mathbb{R}^d\, (d\geq1)$ is a bounded smooth domain. New sufficient conditions, some of which are necessary, on the existence of classical solutions are established. A uniqueness result of solutions in any space dimension is also derived. Previous results are substantially improved; moreover, a much simpler mathematical approach with potential application in other problems is developed.

2017, 37(8): 4507-4542 doi: 10.3934/dcds.2017193 +[Abstract](1868) +[HTML](18) +[PDF](482.6KB)
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This paper is devoted to studying a non-autonomous stochastic linear evolution equation in Banach spaces of martingale type 2. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to stochastic diffusion equations.

2017, 37(8): 4543-4563 doi: 10.3934/dcds.2017194 +[Abstract](2206) +[HTML](34) +[PDF](472.8KB)
Abstract:

Let $(X, T)$ be a topological dynamical system, and $\mu$ be a $T$-invariant Borel probability measure on $X$. Let $\mathcal{F}$ be a family of subsets of $\mathbb{Z}_+$. We introduce notions of $\mathcal{F}$-sensitivity for $\mu$ and block $\mathcal{F}$-sensitivity for $\mu$.

Let $\mathcal{F}_t$ (resp. $\mathcal{F}_{ip}$) be the families consisting of thick sets (resp. IP-sets). The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either $\mathcal{F}_{t}$-sensitive for $\mu$ or an almost one-to-one extension of its maximal equicontinous factor. (2) a minimal system is either block $\mathcal{F}_{t}$-sensitive for $\mu$ or a proximal extension of its maximal equicontinous factor. (3) a minimal system is either block $\mathcal{F}_{ip}$-sensitive for $\mu$ or an almost one-to-one extension of its $\infty$-step nilfactor.

We also introduce the notion of topological $l$-sensitivity, and construct a minimal system which is $l$-sensitive but not $(l+1)$-sensitive for $l\in\mathbb{N}$.

2017, 37(8): 4565-4583 doi: 10.3934/dcds.2017195 +[Abstract](2224) +[HTML](14) +[PDF](470.3KB)
Abstract:

In this paper, we study the following Hamiltonian elliptic system with gradient term and inverse square potential

for $x\in\mathbb{R}^{N}$, where $N\geq3$, $\mu\in\mathbb{R}$, and $V(x)$, $\vec{b}(x)$ and $H(x, u, v)$ are $1$-periodic in $x$. Under suitable conditions, we prove that the system possesses a ground state solution via variational methods for sufficiently small $\mu\geq0$. Moreover, we provide the comparison of the energy of ground state solutions for the case $\mu>0$ and $\mu=0$. Finally, we also give the convergence property of ground state solutions as $\mu\to0^+$.

2017, 37(8): 4585-4586 doi: 10.3934/dcds.2017196 +[Abstract](1863) +[HTML](15) +[PDF](177.1KB)
Abstract:

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