# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

April 2020 , Volume 40 , Issue 4

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2020, 40(4): 1989-2009 doi: 10.3934/dcds.2020102 +[Abstract](763) +[HTML](91) +[PDF](874.73KB)
Abstract:

We consider skew tent maps \begin{document}$T_{ { \alpha }, { \beta }}(x)$\end{document} such that \begin{document}$( \alpha, \beta)\in[0, 1]^{2}$\end{document} is the turning point of \begin{document}$T _{\alpha, \beta}$\end{document}, that is, \begin{document}$T_{ { \alpha }, { \beta }} = \frac{ { \beta }}{ { \alpha }}x$\end{document} for \begin{document}$0\leq x \leq { \alpha }$\end{document} and \begin{document}$T_{ { \alpha }, { \beta }}(x) = \frac{ { \beta }}{1- { \alpha }}(1-x)$\end{document} for \begin{document}${ \alpha }<x\leq 1$\end{document}. We denote by \begin{document}$\underline{M} = K( \alpha, \beta)$\end{document} the kneading sequence of \begin{document}$T _{\alpha, \beta}$\end{document}, by \begin{document}$h( \alpha, \beta)$\end{document} its topological entropy and \begin{document}$\Lambda = \Lambda_{\alpha, \beta}$\end{document} denotes its Lyapunov exponent. For a given kneading squence \begin{document}$\underline{M}$\end{document} we consider isentropes (or equi-topological entropy, or equi-kneading curves), \begin{document}$( \alpha, \Psi_{ \underline{M}}( \alpha))$\end{document} such that \begin{document}$K( \alpha, \Psi_{ \underline{M}}( \alpha)) = \underline{M}$\end{document}. On these curves the topological entropy \begin{document}$h( \alpha, \Psi_{ \underline{M}}( \alpha))$\end{document} is constant.

We show that \begin{document}$\Psi_{ \underline{M}}'( \alpha)$\end{document} exists and the Lyapunov exponent \begin{document}$\Lambda_{\alpha, \beta}$\end{document} can be expressed by using the slope of the tangent to the isentrope. Since this latter can be computed by considering partial derivatives of an auxiliary function \begin{document}$\Theta_{ \underline{M}}$\end{document}, a series depending on the kneading sequence which converges at an exponential rate, this provides an efficient new method of finding the value of the Lyapunov exponent of these maps.

2020, 40(4): 2011-2016 doi: 10.3934/dcds.2020103 +[Abstract](586) +[HTML](81) +[PDF](285.43KB)
Abstract:

In hyperbolic dynamics, a well-known result is: every hyperbolic Lyapunov stable set, is attracting; it's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics. This question is still open, although some partial results have been presented. We will prove that all sectional-hyperbolic transitive Lyapunov stable set of codimension one of a vector field \begin{document}$X$\end{document} over a compact manifold, with unique singularity Lorenz-like, which is of boundary-type, is an attractor of \begin{document}$X$\end{document}.

2020, 40(4): 2017-2035 doi: 10.3934/dcds.2020104 +[Abstract](637) +[HTML](84) +[PDF](399.24KB)
Abstract:

In this paper, we consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in Euclidean \begin{document}$R^{n+1}$\end{document} with speed \begin{document}$u^\alpha\sigma_k^\beta$\end{document} firstly, where \begin{document}$u$\end{document} is support function of the hypersurface, \begin{document}$\alpha, \beta \in R^1$\end{document}, and \begin{document}$\beta>0$\end{document}, \begin{document}$\sigma_k$\end{document} is the \begin{document}$k$\end{document}-th symmetric polynomial of the principal curvature radii of the hypersurface, \begin{document}$k$\end{document} is an integer and \begin{document}$1\le k\le n$\end{document}. For \begin{document}$\alpha\le1-k\beta$\end{document}, \begin{document}$\beta>\frac{1}{k}$\end{document} we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for \begin{document}$\alpha\le1-k\beta$\end{document}, \begin{document}$\beta>\frac{1}{k}$\end{document}, we prove that the flow with the speed \begin{document}$fu^\alpha\sigma_k^\beta$\end{document} exists for all time and converges smoothly after normalisation to a soliton which is a solution of \begin{document}$fu^{\alpha-1}\sigma_k^{\beta} = c$\end{document} provided that \begin{document}$f$\end{document} is a smooth positive function on \begin{document}$S^n$\end{document} and satisfies that \begin{document}$(\nabla_i\nabla_jf^{\frac{1}{1+k\beta-\alpha}}+\delta_{ij}f^{\frac{1}{1+k\beta-\alpha}})$\end{document} is positive definite. When \begin{document}$\beta = 1$\end{document}, our argument provides a proof to the well-known \begin{document}$L_p$\end{document} Christoffel-Minkowski problem for the case \begin{document}$p\ge k+1$\end{document} where \begin{document}$p = 2-\alpha$\end{document}, which is identify with Ivaki's recent result. Especially, we obtain the same result for \begin{document}$k = n$\end{document} without any constraint on smooth positive function \begin{document}$f$\end{document}. Finally, we also give a counterexample for the two anisotropic expanding flows when \begin{document}$\alpha>1-k\beta$\end{document}.

2020, 40(4): 2037-2060 doi: 10.3934/dcds.2020105 +[Abstract](785) +[HTML](96) +[PDF](507.8KB)
Abstract:

We study the orientation flocking for the deterministic counterpart of a stochastic agent-based model introduced by Degond, Frouvelle and Merino-Aceituno in 2017, where the orientation is defined as a \begin{document}${\rm SO}(3)$\end{document} matrix. Their proposed model can be reduced to the other collective dynamics models such as the Lohe matrix model and the Viscek-type model as special cases. In this work, we study the emergent dynamics of the orientation flocking model in two frameworks. First, we present sufficient conditions leading to the orientation flocking when the natural frequency matrices are identical. To be precise, we prove that all orientation matrices asymptotically converge to the common one, and the spatial position diameter remains uniformly bounded. Second, we show the emergence of orientation-locked states for non-identical natural frequency matrices, that is, the difference of any two orientation matrices tends to the definite constant matrix. On the other hand, we establish the finite-in-time stability with respect to initial data of the proposed orientation flocking model. We also present the numerical results consistent with our rigorous analysis. Our work remains valid even for dimensions greater than three.

2020, 40(4): 2061-2087 doi: 10.3934/dcds.2020106 +[Abstract](623) +[HTML](121) +[PDF](389.8KB)
Abstract:

We prove the global well-posedness of the free interface problem for the two-phase incompressible Euler Equations with damping for the small initial data, where the effect of surface tension is included on the free surfaces. Moreover, the solution decays exponentially to the equilibrium.

2020, 40(4): 2089-2103 doi: 10.3934/dcds.2020107 +[Abstract](521) +[HTML](63) +[PDF](358.08KB)
Abstract:

In this paper, we study the quasi-shadowing property for partially hyperbolic flows. A partially hyperbolic flow \begin{document}$\varphi_{t}$\end{document} has the quasi-shadowing property if for any \begin{document}$(\delta,T)$\end{document}-pseudoorbit \begin{document}$g(t)$\end{document} of \begin{document}$\varphi_{t}$\end{document} there exist a sequence of points \begin{document}$\{y_{k}\}_{k\in\mathbb{Z}}$\end{document} and a reparametrization \begin{document}$\alpha$\end{document} such that \begin{document}$\varphi_{\alpha(t)-\alpha(kT)}(y_k)$\end{document} trace \begin{document}$g(t)$\end{document} in which \begin{document}$y_{k}$\end{document} is obtained from \begin{document}$\varphi_{\alpha(kT)-\alpha((k-1)T)}(y_{k-1})$\end{document} by a motion along the central direction. We prove that any partially hyperbolic flow \begin{document}$\varphi_{t}$\end{document} has the quasi-shadowing property. We also investigate the limit quasi-shadowing properties for flows. That is, a partially hyperbolic flow has the \begin{document}$\mathcal{L}^p$\end{document}, limit and asymptotic quasi-shadowing properties.

2020, 40(4): 2105-2134 doi: 10.3934/dcds.2020108 +[Abstract](611) +[HTML](147) +[PDF](560.54KB)
Abstract:

This paper investigates a reaction-advection-diffusion system that describes the evolution of population distributions of two competing species in an enclosed bounded habitat. Here the competition relationships are assumed to be of the Beddington–DeAngelis type. In particular, we consider a situation where first species disperses by a combination of random walk and directed movement along with the population distribution of the second species which disperse randomly within the habitat. We obtain a set of results regarding the qualitative properties of this advective competition system. First of all, we show that this system is globally well-posed and its solutions are classical and uniformly bounded in time. Then we study its steady states in a one-dimensional interval by examining the combined effects of diffusion and advection on the existence and stability of nonconstant positive steady states of the strongly coupled elliptic system. Our stability result of these nontrivial steady states provides a selection mechanism for stable wavemodes of the time-dependent system. Moreover, in the limit of diffusion rates, the steady states of this fully elliptic system can be approximated by nonconstant positive solutions of a shadow system that admits boundary spikes and layers. Furthermore, for the fully elliptic system, we construct solutions with a single boundary spike or an inverted boundary spike, i.e., the first species concentrates on a boundary point while the second species dominates the remaining habitat. These spatial structures model the spatial segregation phenomenon through interspecific competitions. Finally, we perform some numerical simulations to illustrate and support our theoretical findings.

2020, 40(4): 2135-2163 doi: 10.3934/dcds.2020109 +[Abstract](522) +[HTML](78) +[PDF](423.66KB)
Abstract:

We consider generalized models on coral broadcast spawning phenomena involving diffusion, advection, chemotaxis, and reactions when egg and sperm densities are different. We prove the global-in-time existence of the regular solutions of the models as well as their temporal decays in two and three dimensions. We also show that the total masses of egg and sperm density have positive lower bounds as time tends to infinity in three dimensions.

2020, 40(4): 2165-2187 doi: 10.3934/dcds.2020110 +[Abstract](643) +[HTML](83) +[PDF](401.37KB)
Abstract:

In this paper, we study saturable nonlinear Schrödinger equations with nonzero intensity function which makes the nonlinearity become not superlinear near zero. Using the Nehari manifold and the Lusternik-Schnirelman category, we prove the existence of multiple positive solutions for saturable nonlinear Schrödinger equations with nonzero intensity function which satisfies suitable conditions. The ideas contained here might be useful to obtain multiple positive solutions of the other non-homogeneous nonlinear elliptic equations.

2020, 40(4): 2189-2212 doi: 10.3934/dcds.2020111 +[Abstract](582) +[HTML](107) +[PDF](409.73KB)
Abstract:

In this paper, we investigate the asymptotic behavior of local solutions for the semilinear elliptic system \begin{document}$-\Delta \mathbf{u} = |\mathbf{u}|^{p-1}\mathbf{u}$\end{document} with boundary isolated singularity, where \begin{document}$1<p<\frac{n+2}{n-2}$\end{document}, \begin{document}$n\geq 2$\end{document} and \begin{document}$\mathbf{u}$\end{document} is a \begin{document}$C^2$\end{document} nonnegative vector-valued function defined on the half space. This work generalizes the correspondence results of Bidaut-Véron-Ponce-Véron on the scalar case, and Ghergu-Kim-Shahgholian on the internal singularity case.

2020, 40(4): 2213-2265 doi: 10.3934/dcds.2020112 +[Abstract](519) +[HTML](84) +[PDF](642.23KB)
Abstract:

In this paper, we are concerned with the global smooth solution problem for 3-D compressible isentropic Euler equations with vanishing density in an infinitely expanding ball. It is well-known that the classical solution of compressible Euler equations generally forms the shock as well as blows up in finite time due to the compression of gases. However, for the rarefactive gases, it is expected that the compressible Euler equations will admit global smooth solutions. We now focus on the movement of compressible gases in an infinitely expanding ball. Because of the conservation of mass, the fluid in the expanding ball becomes rarefied meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. We will confirm this interesting phenomenon from the mathematical point of view. Through constructing some anisotropy weighted Sobolev spaces, and by carrying out the new observations and involved analysis on the radial speed and angular speeds together with the divergence and rotations of velocity, the uniform weighted estimates on sound speed and velocity are established. From this, the pointwise time-decay estimate of sound speed is obtained, and the smooth gas fluids without vacuum are shown to exist globally.

2020, 40(4): 2267-2283 doi: 10.3934/dcds.2020113 +[Abstract](628) +[HTML](71) +[PDF](344.27KB)
Abstract:

In this paper, we introduce the concepts of rescaled expansiveness and the rescaled shadowing property for flows on metric spaces which are dynamical properties, and present a spectral decomposition theorem for flows. More precisely, we prove that if a flow is rescaling expansive and has the rescaled shadowing property on a locally compact metric space, then it admits the spectral decomposition. Moreover, we show that if a flow on locally compact metric space has the rescaled shadowing property then its restriction on nonwandering set also has the rescaled shadowing property.

2020, 40(4): 2285-2313 doi: 10.3934/dcds.2020114 +[Abstract](499) +[HTML](69) +[PDF](433.7KB)
Abstract:

We prove a forward Ergodic Closing Lemma for nonsingular \begin{document}$C^1$\end{document} endomorphisms, claiming that the set of eventually strongly closable points is a total probability set. The "forward" means that the closing perturbation is involved along a finite part of the forward orbit of a point in a total probability set, which is the same perturbation as in Mañé's Ergodic Closing Lemma for \begin{document}$C^1$\end{document} diffeomorphisms. As an application, Shub's Entropy Conjecture for nonsingular \begin{document}$C^1$\end{document} endomorphisms away from homoclinic tangencies is proved, extending the result for \begin{document}$C^1$\end{document} diffeomorphisms by Liao, Viana and Yang.

2020, 40(4): 2315-2333 doi: 10.3934/dcds.2020115 +[Abstract](510) +[HTML](70) +[PDF](412.91KB)
Abstract:

For transformations with regularly varying property, we identify a class of moduli of continuity related to the local behavior of the dynamics near a fixed point, and we prove that this class is not compatible with the existence of continuous sub-actions. The dynamical obstruction is given merely by a local property. As a natural complement, we also deal with the question of the existence of continuous sub-actions focusing on a particular dynamic setting. Applications of both results include interval maps that are expanding outside a neutral fixed point, as Manneville-Pomeau and Farey maps.

2020, 40(4): 2335-2346 doi: 10.3934/dcds.2020116 +[Abstract](486) +[HTML](95) +[PDF](347.16KB)
Abstract:

We consider a set of necessary conditions which are efficient heuristics for deciding when a set of Wang tiles cannot tile a group.

Piantadosi [19] gave a necessary and sufficient condition for the existence of a valid tiling of any free group. This condition is actually necessary for the existence of a valid tiling for an arbitrary finitely generated group.

We consider two other conditions: the first, also given by Piantadosi [19], is a necessary and sufficient condition to decide if a set of Wang tiles gives a strongly periodic tiling of the free group; the second, given by Chazottes et. al. [9], is a necessary condition to decide if a set of Wang tiles gives a tiling of \begin{document}$\mathbb Z^2$\end{document}.

We show that these last two conditions are equivalent. Joining and generalising approaches from both sides, we prove that they are necessary for having a valid tiling of any finitely generated amenable group, confirming a remark of Jeandel [14].

2020, 40(4): 2347-2365 doi: 10.3934/dcds.2020117 +[Abstract](493) +[HTML](76) +[PDF](381.54KB)
Abstract:

Recently, in connection with C*-algebra theory, the first author and Danilo Royer introduced ultragraph shift spaces. In this paper we define a family of metrics for the topology in such spaces, and use these metrics to study the existence of chaos in the shift. In particular we characterize all ultragraph shift spaces that have Li-Yorke chaos (an uncountable scrambled set), and prove that such property implies the existence of a perfect and scrambled set in the ultragraph shift space. Furthermore, this scrambled set can be chosen compact, which is not the case for a labelled edge shift (with the product topology) of an infinite graph.

2020, 40(4): 2367-2391 doi: 10.3934/dcds.2020118 +[Abstract](544) +[HTML](163) +[PDF](425.41KB)
Abstract:

Given a smooth bounded domain \begin{document}$\Omega \subset \mathbb {R}^n$\end{document} and consider the problem

where \begin{document}$p$\end{document} is subcritical exponent (\begin{document}$p > 1$\end{document} if \begin{document}$n = 2$\end{document} and \begin{document}$1 < p < \frac{n+2}{n-2}$\end{document} if \begin{document}$n \geq 3$\end{document}), \begin{document}$\sigma > 0$\end{document} is a large parameter and \begin{document}$\nu$\end{document} denotes the outward normal of \begin{document}$\partial\Omega$\end{document}. Let \begin{document}$\Gamma$\end{document} be an interior straighline intersecting orthogonally with \begin{document}$\partial\Omega$\end{document}. Assuming moreover that \begin{document}$\Gamma$\end{document} satisfies a non-degeneracy condition, we construct a new class of solutions which consist of large number of spikes concentrating on \begin{document}$\Gamma$\end{document}, showing as in [5,6] that higher dimensional concentration can exist without resonance condition.

2020, 40(4): 2393-2419 doi: 10.3934/dcds.2020119 +[Abstract](681) +[HTML](86) +[PDF](648.41KB)
Abstract:

This paper studies the existence of subharmonics of arbitrary order in a generalized class of non-autonomous predator-prey systems of Volterra type with periodic coefficients. When the model is non-degenerate it is shown that the Poincaré–Birkhoff twist theorem can be applied to get the existence of subharmonics of arbitrary order. However, in the degenerate models, whether or not the twist theorem can be applied to get subharmonics of a given order might depend on the particular nodal behavior of the several weight function-coefficients involved in the setting of the model. Finally, in order to analyze how the subharmonics might be lost as the model degenerates, the exact point-wise behavior of the \begin{document}$T$\end{document}-periodic solutions of a non-degenerate model is ascertained as a perturbation parameter makes it degenerate.

2020, 40(4): 2421-2439 doi: 10.3934/dcds.2020120 +[Abstract](525) +[HTML](67) +[PDF](394.38KB)
Abstract:

This paper is concerned with the 1-dimensional quintic nonlinear Schrödinger equations with real valued \begin{document}$C^{\infty}$\end{document}-smooth given potential

subject to Dirichlet boundary conditions. By means of normal form theory and an infinite-dimensional Kolmogorov-Arnold-Moser (KAM, for short) theorem, it is proved that the above equation admits a family of elliptic tori where lies small amplitude quasi-periodic solutions with two frequencies of high modes.

2020, 40(4): 2441-2474 doi: 10.3934/dcds.2020121 +[Abstract](782) +[HTML](79) +[PDF](497.1KB)
Abstract:

We study sensitivity, topological equicontinuity and even continuity in dynamical systems. In doing so we provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but no equicontinuity point. We define what it means for a system to be eventually sensitive; we give a dichotomy for transitive dynamical systems in relation to eventual sensitivity. Along the way we define a property called splitting and discuss its relation to some existing notions of chaos. The approach we take is topological rather than metric.

2020, 40(4): 2475-2493 doi: 10.3934/dcds.2020122 +[Abstract](606) +[HTML](122) +[PDF](367.91KB)
Abstract:

In this paper, we investigate a generalized two-component rotational b-family system arising in the rotating fluid with the effect of the Coriolis force. First, we study the persistence properties of the system in weighted \begin{document}$L^p$\end{document}-spaces, for a large class of moderate weights. Secondly, in order to overcome the difficulty arising from higher order nonlinearity and no conservation law, we take the advantage of the specially intrinsic structure of the system and make use of commutator estimate, and then derive two blow-up results for the strong solutions to the system.

2020, 40(4): 2495-2514 doi: 10.3934/dcds.2020123 +[Abstract](508) +[HTML](73) +[PDF](415.14KB)
Abstract:

Given an isoparametric function \begin{document}$f$\end{document} on the \begin{document}$n$\end{document}-dimensional round sphere, we consider functions of the form \begin{document}$u = w\circ f$\end{document} to reduce the semilinear elliptic problem

with \begin{document}$\lambda>0$\end{document} and \begin{document}$1<p$\end{document}, into a singular ODE in \begin{document}$[0,\pi]$\end{document} of the form \begin{document}$w" + \frac{h(r)}{\sin r} w' + \frac{\lambda}{\ell^2}\left(\vert w\vert^{p-1}w - w\right) = 0$\end{document}, where \begin{document}$h$\end{document} is an strictly decreasing function having exactly one zero in this interval and \begin{document}$\ell$\end{document} is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a sequence of sign-changing solutions to the first problem which are constant on the isoparametric hypersurfaces associated to \begin{document}$f$\end{document} and blowing-up at one or two of the focal submanifolds generating the isoparametric family. Our methods apply also when \begin{document}$p>\frac{n+2}{n-2}$\end{document}, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.

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