# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

July 2020 , Volume 40 , Issue 7

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2020, 40(7): 4073-4092 doi: 10.3934/dcds.2020172 +[Abstract](583) +[HTML](114) +[PDF](572.54KB)
Abstract:

In this work we are going to consider the classical Hénon-Devaney map given by

We are going to construct conjugacy to a subshift of finite type, providing a global understanding of the map's behavior.We extend the coding to a more general class of maps that can be seen as a map in a square with a fixed discontinuity.

2020, 40(7): 4093-4112 doi: 10.3934/dcds.2020173 +[Abstract](411) +[HTML](109) +[PDF](427.18KB)
Abstract:

The principal aim of this paper is to construct an explicit sequence of weighted divergence free vector fields which accelerates the rate of convergence of planar Ornstein-Uhlenbeck diffusion to its equilibrium state. The rate of convergence is expressed in terms of the spectral gap of the diffusion generator. We construct an explicit sequence of vector fields which pushes the spectral gap to infinity. The acceleration of the diffusion results from the strong oscillation of the flow lines generated by the vector field.

2020, 40(7): 4113-4130 doi: 10.3934/dcds.2020174 +[Abstract](411) +[HTML](105) +[PDF](383.58KB)
Abstract:

We prove existence of solitary-wave solutions to the equation

for weak assumptions on the dispersion \begin{document}$L$\end{document} and the nonlinearity \begin{document}$n$\end{document}. The symbol \begin{document}$m$\end{document} of the Fourier multiplier \begin{document}$L$\end{document} is allowed to be of low positive order (\begin{document}$s > 0$\end{document}), while \begin{document}$n$\end{document} need only be locally Lipschitz and asymptotically homogeneous at zero. We shall discover such solutions in Sobolev spaces contained in \begin{document}$H^{1+s}$\end{document}.

2020, 40(7): 4131-4162 doi: 10.3934/dcds.2020175 +[Abstract](452) +[HTML](116) +[PDF](478.09KB)
Abstract:

We consider the Kawahara model and two fourth order semi-linear Schrödinger equations in any spatial dimension. We construct the corresponding normalized ground states, which we rigorously show to be spectrally stable.

For the Kawahara model, our results provide a significant extension in parameter space of the current rigorous results. In fact, our results establish (modulo an additional technical assumption, which should be satisfied at least generically), spectral stability for all normalized waves constructed therein - in all dimensions, for all acceptable values of the parameters. This, combined with the results of [5], provides orbital stability, for all normalized waves enjoying the non-degeneracy property. The validity of the non-degeneracy property for generic waves remains an intriguing open question.

At the same time, we verify and clarify recent numerical simulations of the spectral stability of these solitons. For the fourth order NLS models, we improve upon recent results on spectral stability of very special, explicit solutions in the one dimensional case. Our multidimensional results for fourth order anisotropic NLS seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second derivatives vs. pure Laplacian) has implications on the range of existence and stability of the normalized waves.

2020, 40(7): 4163-4177 doi: 10.3934/dcds.2020176 +[Abstract](496) +[HTML](117) +[PDF](407.22KB)
Abstract:

It has been shown by Le Jan that, given a memoryless-noise random dynamical system together with an ergodic distribution for the associated Markov transition probabilities, if the support of the ergodic distribution admits locally asymptotically stable trajectories, then there is a random attracting set consisting of finitely many points, whose basin of forward-time attraction includes a random full measure open set. In this paper, we present necessary and sufficient conditions for this attracting set to be a singleton. Our result does not require the state space to be compact, but holds on general Lusin metric spaces (in both discrete and continuous time).

2020, 40(7): 4179-4196 doi: 10.3934/dcds.2020177 +[Abstract](278) +[HTML](87) +[PDF](466.82KB)
Abstract:

In this paper, we will construct a new type of non-landing exponential rays, each of whose accumulation sets is bounded, disjoint from the ray and homeomorphic to the closed topologist's sine curve.

2020, 40(7): 4197-4229 doi: 10.3934/dcds.2020178 +[Abstract](281) +[HTML](79) +[PDF](502.89KB)
Abstract:

The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order.

2020, 40(7): 4231-4258 doi: 10.3934/dcds.2020179 +[Abstract](311) +[HTML](84) +[PDF](466.71KB)
Abstract:

We relate together different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the \begin{document}$L^2$\end{document} norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Kuznetsov equation in a half-space with periodic in time initial and boundary data) are obtained.

2020, 40(7): 4259-4286 doi: 10.3934/dcds.2020180 +[Abstract](365) +[HTML](88) +[PDF](1063.61KB)
Abstract:

The dynamics of symbolic systems, such as multidimensional subshifts of finite type or cellular automata, are known to be closely related to computability theory. In particular, the appropriate tools to describe and classify topological entropy for this kind of systems turned out to be algorithmic. Part of the great importance of these symbolic systems relies on the role they have played in understanding more general systems over non-symbolic spaces. The aim of this article is to investigate topological entropy from a computability point of view in this more general, not necessarily symbolic setting. In analogy to effective subshifts, we consider computable maps over effective compact sets in general metric spaces, and study the computability properties of their topological entropies. We show that even in this general setting, the entropy is always a \begin{document}$\Sigma_2$\end{document}-computable number. We then study how various dynamical and analytical constrains affect this upper bound, and prove that it can be lowered in different ways depending on the constraint considered. In particular, we obtain that all \begin{document}$\Sigma_2$\end{document}-computable numbers can already be realized within the class of surjective computable maps over \begin{document}$\{0,1\}^{{\mathbb N}}$\end{document}, but that this bound decreases to \begin{document}$\Pi_{1}$\end{document}(or upper)-computable numbers when restricted to expansive maps. On the other hand, if we change the geometry of the ambient space from the symbolic \begin{document}$\{0,1\}^{{\mathbb N}}$\end{document} to the unit interval \begin{document}$[0,1]$\end{document}, then we find a quite different situation – we show that the possible entropies of computable systems over \begin{document}$[0,1]$\end{document} are exactly the \begin{document}$\Sigma_{1}$\end{document}(or lower)-computable numbers and that this characterization switches down to precisely the computable numbers when we restrict the class of system to the quadratic family.

2020, 40(7): 4287-4305 doi: 10.3934/dcds.2020181 +[Abstract](358) +[HTML](72) +[PDF](387.61KB)
Abstract:

The paper addresses the weak-strong uniqueness property and singular limit for the compressible Primitive Equations (PE). We show that a weak solution coincides with the strong solution emanating from the same initial data. On the other hand, we prove compressible PE will approach to the incompressible inviscid PE equations in the regime of low Mach number and large Reynolds number in the case of well-prepared initial data. To the best of the authors' knowledge, this is the first work to bridge the link between the compressible PE with incompressible inviscid PE.

2020, 40(7): 4307-4340 doi: 10.3934/dcds.2020182 +[Abstract](309) +[HTML](97) +[PDF](434.43KB)
Abstract:

We solve Cauchy problems for some \begin{document}$\mu$\end{document}-Camassa-Holm integro-partial differential equations in the analytic category. The equations to be considered are \begin{document}$\mu$\end{document}CH of Khesin-Lenells-Misiołek, \begin{document}$\mu$\end{document}DP of Lenells-Misiołek-Tiğlay, the higher-order \begin{document}$\mu$\end{document}CH of Wang-Li-Qiao and the non-quasilinear version of Qu-Fu-Liu. We prove the unique local solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Moreover, we show the existence of a unique global-in-time analytic solution for \begin{document}$\mu$\end{document}CH, \begin{document}$\mu$\end{document}DP and the higher-order \begin{document}$\mu$\end{document}CH. The present work is the first result of such a global nature for these equations.

2020, 40(7): 4341-4378 doi: 10.3934/dcds.2020183 +[Abstract](269) +[HTML](78) +[PDF](762.27KB)
Abstract:

It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincaré disk model. Our tables have \begin{document}$m\ge3$\end{document} ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group \begin{document}${\rm{PSU}}^\pm_{1,1} \mathbb{Z}[i]$\end{document}. The resulting billiard map \begin{document}$\widetilde B$\end{document} acts on the de Sitter space \begin{document}$x_1^2+x_2^2-x_3^2 = 1$\end{document}, and has a natural factor \begin{document}$B$\end{document} on the unit circle, the pythagorean triples appearing as the \begin{document}$B$\end{document}-preimages of fixed points. We compute the invariant densities of these maps, and prove the Lagrange and Galois theorems: A complex number of unit modulus has a preperiodic (purely periodic) \begin{document}$B$\end{document}-orbit precisely when it is quadratic (and isolated from its conjugate by a billiard wall) over \begin{document}$\mathbb{Q}(i)$\end{document}.

Each \begin{document}$B$\end{document} as above is a \begin{document}$(m-1)$\end{document}-to-\begin{document}$1$\end{document} orientation-reversing covering map of the circle, a property shared by the group character \begin{document}$T(z) = z^{-(m-1)}$\end{document}. We prove that there exists a homeomorphism \begin{document}$\Phi$\end{document}, unique up to postcomposition with elements in a dihedral group, that conjugates \begin{document}$B$\end{document} with \begin{document}$T$\end{document}; in particular \begin{document}$\Phi$\end{document} -whose prototype is the classical Minkowski question mark function- establishes a bijection between the set of points of degree \begin{document}$\le2$\end{document} over \begin{document}$\mathbb{Q}(i)$\end{document} and the torsion subgroup of the circle. We provide an explicit formula for \begin{document}$\Phi$\end{document}, and prove that \begin{document}$\Phi$\end{document} is singular and Hölder continuous with exponent \begin{document}$\log(m-1)$\end{document} divided by the maximal periodic mean free path in the associated billiard table.

2020, 40(7): 4379-4425 doi: 10.3934/dcds.2020184 +[Abstract](314) +[HTML](79) +[PDF](580.21KB)
Abstract:

In this article, we consider the Schrödinger flow of maps from two dimensional hyperbolic space \begin{document}${{\mathbb{H}}}^2$\end{document} to sphere \begin{document}${{\mathbb{S}}}^2$\end{document}. First, we prove the local existence and uniqueness of Schrödinger flow for initial data \begin{document}$u_0\in\mathbf{H}^3$\end{document} using an approximation scheme and parallel transport introduced by McGahagan [32]. Second, using the Coulomb gauge, we reduce the study of the equivariant Schrödinger flow to that of a system of coupled Schrödinger equations with potentials. Then we prove the global existence of equivariant Schrödinger flow for small initial data \begin{document}$u_0\in\mathbf{H}^1$\end{document} by Strichartz estimates and perturbation method.

2020, 40(7): 4427-4451 doi: 10.3934/dcds.2020185 +[Abstract](329) +[HTML](79) +[PDF](431.42KB)
Abstract:

We study a periodic-parabolic Droop model of two species competing for a single-limited nutrient in an unstirred chemostat, where the nutrient is added to the culture vessel by way of periodic forcing function in time. For the single species model, we establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear periodic eigenvalue problem. In particular, when diffusion rate is sufficiently small or large, the sign can be determined. We then show that for the competition model, when diffusion rates for both species are small, there exists a coexistence periodic solution.

2020, 40(7): 4453-4477 doi: 10.3934/dcds.2020186 +[Abstract](314) +[HTML](76) +[PDF](541.93KB)
Abstract:

We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is \begin{document}$\mathbb Z$\end{document}. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approximate the hard square constant on \begin{document}$\mathbb Z^2$\end{document}, we show that the entropy of the hard square tree shift on the regular \begin{document}$k$\end{document}-tree increases with \begin{document}$k$\end{document}, in contrast to the case of \begin{document}$\mathbb Z^k$\end{document}. We prove that the strip entropy approximations increase strictly to the entropy of the golden mean tree shift for \begin{document}$k = 2,\dots,8$\end{document} and propose that this holds for all \begin{document}$k \geq 2$\end{document}. We study the dynamics of the map of the simplex that advances the vector of ratios of symbol counts as the width of the approximating strip is increased, providing a fairly complete description for the golden mean subshift on the \begin{document}$k$\end{document}-tree for all \begin{document}$k$\end{document}. This map provides an efficient numerical method for approximating the entropies of tree shifts defined by nearest neighbor restrictions. Finally, we show that counting configurations over certain other patterns besides the natural finite subtrees yields the same value of entropy for tree SFT's.

2020, 40(7): 4479-4496 doi: 10.3934/dcds.2020187 +[Abstract](278) +[HTML](72) +[PDF](392.99KB)
Abstract:

In this paper we study the incompressible non-inertial Qian-Sheng model, which describes the hydrodynamics of nematic liquid crystals without inertial effect in the \begin{document}$Q$\end{document}-tensor framework. Under some proper assumptions on the viscous coefficients, we prove the local well-posedness with large initial data and the global existence with small size of the initial data in the classical solutions regime.

2020, 40(7): 4497-4518 doi: 10.3934/dcds.2020188 +[Abstract](299) +[HTML](82) +[PDF](393.9KB)
Abstract:

In this paper, nearly integrable system under almost periodic perturbations is studied

where \begin{document}$x\in\mathbb{T}^n, \, y\in\mathbb{R}^n$\end{document}, \begin{document}$\omega_0\in\mathbb{R}^n$\end{document} is the frequency vector, and the perturbations \begin{document}$f, g$\end{document} are real analytic almost periodic functions in \begin{document}$t$\end{document} with the infinite frequency \begin{document}$\omega = (\cdots, \omega_\lambda, \cdots)_{\lambda\in\mathbb{Z}}$\end{document}. We also assume that the above system is reversible with respect to the involution \begin{document}$\mathcal{M}_0:(x, y)\rightarrow (-x, y)$\end{document}. By KAM iterative method, we prove the existence of invariant tori for the above reversible system. As an application, we discuss the existence of almost periodic solutions and the boundedness of all solutions for a second-order nonlinear differential equation.

2020, 40(7): 4519-4531 doi: 10.3934/dcds.2020189 +[Abstract](327) +[HTML](80) +[PDF](320.52KB)
Abstract:

We consider the setting for the disappearance of uniform hyperbolicity as in Bjerklöv and Saprykina (2008 Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp \begin{document}$\frac12$\end{document}-Hölder continuous.

In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum is sharp \begin{document}$\frac12$\end{document}-Hölder continuous below the lowest energy of the spectrum, in the large coupling regime.

2020, 40(7): 4533-4564 doi: 10.3934/dcds.2020190 +[Abstract](323) +[HTML](87) +[PDF](444.68KB)
Abstract:

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

2020, 40(7): 4565-4576 doi: 10.3934/dcds.2020191 +[Abstract](324) +[HTML](86) +[PDF](303.97KB)
Abstract:

This paper is concerned with the Cauchy problem for a two- component cubic Camassa-Holm system with peakons, which is a natural multi-component extension of the single Fokas-Olver-Rosenau-Qiao equation. By sufficiently exploiting the fine structure of the system, we derive two useful conservation laws which turns out an exponential increase estimate for the \begin{document}$L^\infty$\end{document}-norm of the strong solution within its lifespan. As a result, two new blow-up solutions with certain initial profiles are established.

2020, 40(7): 4577-4578 doi: 10.3934/dcds.2020192 +[Abstract](271) +[HTML](79) +[PDF](211.45KB)
Abstract:

We correct an error which has occurred in the proof of Lemma 4.1 in the paper "The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension" [Discrete Contin. Dyn. Syst., 36 (2016), 5743-5761].

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