# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete and Continuous Dynamical Systems

February 2022 , Volume 42 , Issue 2

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2022, 42(2): 505-535 doi: 10.3934/dcds.2021126 +[Abstract](590) +[HTML](236) +[PDF](477.63KB)
Abstract:

This paper is devoted to the study of a stochastic epidemiological model which is a variant of the SIR model to which we add an extra factor in the transition rate from susceptible to infected accounting for the inflow of infection due to immigration or environmental sources of infection. This factor yields the formation of new clusters of infections, without having to specify a priori and explicitly their date and place of appearance.

We establish an exact deterministic description for such stochastic processes on 1D lattices (finite lines, semi-infinite lines, infinite lines) by showing that the probability of infection at a given point in space and time can be obtained as the solution of a deterministic ODE system on the lattice. Our results allow stochastic initial conditions and arbitrary spatio-temporal heterogeneities on the parameters.

We then apply our results to some concrete situations and obtain useful qualitative results and explicit formulae on the macroscopic dynamics and also the local temporal behavior of each individual. In particular, we provide a fine analysis of some aspects of cluster formation through the study of patient-zero problems and the effects of time-varying point sources.

Finally, we show that the space-discrete model gives rise to new space-continuous models, which are either ODEs or PDEs, depending on the rescaling regime assumed on the parameters.

2022, 42(2): 537-553 doi: 10.3934/dcds.2021127 +[Abstract](700) +[HTML](238) +[PDF](380.35KB)
Abstract:

In this paper, we establish the \begin{document}$W^{1,p}$\end{document} estimates for solutions of second order elliptic problems with drift terms in bounded Lipschitz domains by using a real variable method. For scalar equations, we prove that the \begin{document}$W^{1,p}$\end{document} estimates hold for \begin{document}$\frac{3}{2}-\varepsilon<p<3+\varepsilon$\end{document} for \begin{document}$d\geq3$\end{document}, and the range for \begin{document}$p$\end{document} is sharp. For elliptic systems, we prove that the \begin{document}$W^{1,p}$\end{document} estimates hold for \begin{document}$\frac{2d}{d+1}-\varepsilon<p<\frac{2d}{d-1}+\varepsilon$\end{document} under the assumption that the Lipschitz constant of the domain is small.

2022, 42(2): 555-595 doi: 10.3934/dcds.2021128 +[Abstract](622) +[HTML](235) +[PDF](490.18KB)
Abstract:

In this paper, we continue to develop Aubry-Mather and weak KAM theories for contact Hamiltonian systems \begin{document}$H(x,u,p)$\end{document} with certain dependence on the contact variable \begin{document}$u$\end{document}. For the Lipschitz dependence case, we obtain some properties of the Mañé set. For the non-decreasing case, we provide some information on the Aubry set, such as the comparison property, graph property and a partially ordered relation for the collection of all projected Aubry sets with respect to backward weak KAM solutions. Moreover, we find a new flow-invariant set \begin{document}$\tilde{\mathcal{S}}_s$\end{document} consists of strongly static orbits, which coincides with the Aubry set \begin{document}$\tilde{\mathcal{A}}$\end{document} in classical Hamiltonian systems. Nevertheless, a class of examples are constructed to show \begin{document}$\tilde{\mathcal{S}}_s\subsetneqq\tilde{\mathcal{A}}$\end{document} in the contact case. As their applications, we find some new phenomena appear even if the strictly increasing dependence of \begin{document}$H$\end{document} on \begin{document}$u$\end{document} fails at only one point, and we show that there is a difference for the vanishing discount problem from the negative direction between the minimal viscosity solution and non-minimal ones.

2022, 42(2): 597-604 doi: 10.3934/dcds.2021129 +[Abstract](512) +[HTML](199) +[PDF](154.76KB)
Abstract:

Any \begin{document}$C^d$\end{document} conservative map \begin{document}$f$\end{document} of the \begin{document}$d$\end{document}-dimensional unit ball \begin{document}${\mathbb B}^d$\end{document}, \begin{document}$d\geq 2$\end{document}, can be realized by renormalized iteration of a \begin{document}$C^d$\end{document} perturbation of identity: there exists a conservative diffeomorphism of \begin{document}${\mathbb B}^d$\end{document}, arbitrarily close to identity in the \begin{document}$C^d$\end{document} topology, that has a periodic disc on which the return dynamics after a \begin{document}$C^d$\end{document} change of coordinates is exactly \begin{document}$f$\end{document}.

2022, 42(2): 605-621 doi: 10.3934/dcds.2021130 +[Abstract](517) +[HTML](202) +[PDF](385.57KB)
Abstract:

Motivated by radiation hydrodynamics, we analyse a \begin{document}$2\times2$\end{document} system consisting of a one-dimensional viscous conservation law with strictly convex flux –the viscous Burgers' equation being a paradigmatic example– coupled with an elliptic equation, named viscous Hamer-type system. In the regime of small viscosity and for large shocks, namely when the profile of the corresponding underlying inviscid model undergoes a discontinuity –usually called sub-shock– it is proved the existence of a smooth propagating front, regularising the jump of the corresponding inviscid equation. The proof is based on Geometric Singular Perturbation Theory (GSPT) as introduced in the pioneering work of Fenichel [5] and subsequently developed by Szmolyan [21]. In addition, the case of small shocks and large viscosity is also addressed via a standard bifurcation argument.

2022, 42(2): 623-641 doi: 10.3934/dcds.2021131 +[Abstract](578) +[HTML](217) +[PDF](427.46KB)
Abstract:

We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map \begin{document}$f$\end{document} has positive entropy if and only if some iterate \begin{document}$f^k$\end{document} has a periodic orbit with three aligned points consecutive in time, that is, a triplet \begin{document}$(a,b,c)$\end{document} such that \begin{document}$f^k(a) = b$\end{document}, \begin{document}$f^k(b) = c$\end{document} and \begin{document}$b$\end{document} belongs to the interior of the unique interval connecting \begin{document}$a$\end{document} and \begin{document}$c$\end{document} (a forward triplet of \begin{document}$f^k$\end{document}). We also prove a new criterion of entropy zero for simplicial \begin{document}$n$\end{document}-periodic patterns \begin{document}$P$\end{document} based on the non existence of forward triplets of \begin{document}$f^k$\end{document} for any \begin{document}$1\le k<n$\end{document} inside \begin{document}$P$\end{document}. Finally, we study the set \begin{document}$\mathcal{X}_n$\end{document} of all \begin{document}$n$\end{document}-periodic patterns \begin{document}$P$\end{document} that have a forward triplet inside \begin{document}$P$\end{document}. For any \begin{document}$n$\end{document}, we define a pattern that attains the minimum entropy in \begin{document}$\mathcal{X}_n$\end{document} and prove that this entropy is the unique real root in \begin{document}$(1,\infty)$\end{document} of the polynomial \begin{document}$x^n-2x-1$\end{document}.

2022, 42(2): 643-677 doi: 10.3934/dcds.2021132 +[Abstract](653) +[HTML](219) +[PDF](476.01KB)
Abstract:

In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any \begin{document}$n$\end{document}-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure \begin{document}$p$\end{document} and the definition of weak solutions. Detailed arguments of converting the equations on the time-varying domains into the corresponding equations on the cylindrical domains are presented. Finally, we show the well-posedness of weak solutions and the existence of a compact pullback attractor for the 2D MHD equations.

2022, 42(2): 679-706 doi: 10.3934/dcds.2021133 +[Abstract](569) +[HTML](199) +[PDF](498.39KB)
Abstract:

We associate to a perturbation \begin{document}$(f_t)$\end{document} of a (stably mixing) piecewise expanding unimodal map \begin{document}$f_0$\end{document} a two-variable fractional susceptibility function \begin{document}$\Psi_\phi(\eta, z)$\end{document}, depending also on a bounded observable \begin{document}$\phi$\end{document}. For fixed \begin{document}$\eta \in (0,1)$\end{document}, we show that the function \begin{document}$\Psi_\phi(\eta, z)$\end{document} is holomorphic in a disc \begin{document}$D_\eta\subset \mathbb{C}$\end{document} centered at zero of radius \begin{document}$>1$\end{document}, and that \begin{document}$\Psi_\phi(\eta, 1)$\end{document} is the Marchaud fractional derivative of order \begin{document}$\eta$\end{document} of the function \begin{document}$t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t$\end{document}, at \begin{document}$t = 0$\end{document}, where \begin{document}$\mu_t$\end{document} is the unique absolutely continuous invariant probability measure of \begin{document}$f_t$\end{document}. In addition, we show that \begin{document}$\Psi_\phi(\eta, z)$\end{document} admits a holomorphic extension to the domain \begin{document}$\{\, (\eta, z) \in \mathbb{C}^2\mid 0<\Re \eta <1, \, z \in D_\eta \,\}$\end{document}. Finally, if the perturbation \begin{document}$(f_t)$\end{document} is horizontal, we prove that \begin{document}$\lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0}$\end{document}.

2022, 42(2): 707-735 doi: 10.3934/dcds.2021134 +[Abstract](526) +[HTML](202) +[PDF](459.91KB)
Abstract:

We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death(COD) state from the exponential \begin{document}$\ell^1$\end{document}-stability and the existence of the equilibrium solution.

2022, 42(2): 737-757 doi: 10.3934/dcds.2021135 +[Abstract](565) +[HTML](196) +[PDF](439.6KB)
Abstract:

We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger \begin{document}$W^{1,2}$\end{document} convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.

2022, 42(2): 759-779 doi: 10.3934/dcds.2021136 +[Abstract](651) +[HTML](248) +[PDF](432.03KB)
Abstract:

In this work we consider a two-species predator-prey chemotaxis model

in a bounded domain with smooth boundary. We prove that if (1.7)-(1.13) hold, the model (\begin{document}$\ast$\end{document}) admits a global boundedness of classical solutions in any physically meaningful dimension. Moreover, we show that the global classical solutions \begin{document}$(u,v,w)$\end{document} exponentially converges to constant stable steady state \begin{document}$(u_\ast,v_\ast,w_\ast)$\end{document}. Inspired by [5], we employ the special structure of (\begin{document}$\ast$\end{document}) and carefully balance the triple cross diffusion. Indeed, we introduced some functions and combined them in a way that allowed us to cancel the bad items.

2022, 42(2): 781-813 doi: 10.3934/dcds.2021137 +[Abstract](606) +[HTML](211) +[PDF](497.01KB)
Abstract:

This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space \begin{document}$\mathbb{R}^N,\ N \geq 2.$\end{document} The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.

2022, 42(2): 815-839 doi: 10.3934/dcds.2021138 +[Abstract](553) +[HTML](206) +[PDF](635.2KB)
Abstract:

We prove quantitative statistical stability results for a large class of small \begin{document}$C^{0}$\end{document} perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.

2022, 42(2): 841-862 doi: 10.3934/dcds.2021139 +[Abstract](480) +[HTML](196) +[PDF](376.13KB)
Abstract:

We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.

2022, 42(2): 863-884 doi: 10.3934/dcds.2021140 +[Abstract](522) +[HTML](202) +[PDF](1605.86KB)
Abstract:

The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model.

2022, 42(2): 885-902 doi: 10.3934/dcds.2021141 +[Abstract](631) +[HTML](240) +[PDF](325.16KB)
Abstract:

This paper is devoted to understanding the global stability of perturbations near a background magnetic field of the 2D magnetohydrodynamic (MHD) equations with partial dissipation. We establish the global stability for the solutions of the nonlinear MHD system by the bootstrap argument.

2022, 42(2): 903-929 doi: 10.3934/dcds.2021142 +[Abstract](511) +[HTML](195) +[PDF](583.28KB)
Abstract:

We study the approximation of the nonlocal-interaction equation restricted to a compact manifold \begin{document}${\mathcal{M}}$\end{document} embedded in \begin{document}${\mathbb{R}}^d$\end{document}, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on \begin{document}${\mathcal{M}}$\end{document} can be approximated by the classical nonlocal-interaction equation on \begin{document}${\mathbb{R}}^d$\end{document} by adding an external potential which strongly attracts to \begin{document}${\mathcal{M}}$\end{document}. The proof relies on the Sandier–Serfaty approach [23,24] to the \begin{document}$\Gamma$\end{document}-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on \begin{document}${\mathcal{M}}$\end{document}, which was shown [10]. We also provide an another approximation to the interaction equation on \begin{document}${\mathcal{M}}$\end{document}, based on iterating approximately solving an interaction equation on \begin{document}${\mathbb{R}}^d$\end{document} and projecting to \begin{document}${\mathcal{M}}$\end{document}. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.

2022, 42(2): 931-988 doi: 10.3934/dcds.2021143 +[Abstract](513) +[HTML](172) +[PDF](849.41KB)
Abstract:

In this text I study the asymptotics of the complexity function of minimal multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [12] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the \begin{document}$\Delta_2$\end{document}-computable numbers in \begin{document}$[0, d]$\end{document}. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all \begin{document}$d \ge 2$\end{document} the possible values for entropy dimensions of \begin{document}$d$\end{document}-dimensional SFT are the \begin{document}$\Delta_2$\end{document}-computable numbers in \begin{document}$[0, d-1]$\end{document}. In the present text I prove formally this result for \begin{document}$d = 3$\end{document}. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case \begin{document}$d = 2$\end{document} requires some substantial changes to be made in order to adapt the construction that are not discussed here.

2022, 42(2): 989-1010 doi: 10.3934/dcds.2021144 +[Abstract](485) +[HTML](146) +[PDF](632.17KB)
Abstract:

It is known that, for many transcendental entire functions in the Eremenko-Lyubich class \begin{document}$\mathcal{B}$\end{document}, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are criniferous. In this paper, we extend this result to a new class of maps in \begin{document}$\mathcal{B}$\end{document}. Furthermore, we show that if a map belongs to this class, then its Julia set contains a Cantor bouquet; in other words, it is a subset of \begin{document}$\mathbb{C}$\end{document} ambiently homeomorphic to a straight brush.

2022, 42(2): 1011-1037 doi: 10.3934/dcds.2021145 +[Abstract](478) +[HTML](173) +[PDF](561.49KB)
Abstract:

In this paper we show that the solution of an obstacle problem for linearly elastic elliptic membrane shells enjoys higher differentiability properties in the interior of the domain where it is defined.

2020 Impact Factor: 1.392
5 Year Impact Factor: 1.610
2020 CiteScore: 2.2