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Discrete & Continuous Dynamical Systems

August 2021 , Volume 41 , Issue 8

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Möbius disjointness for skew products on a circle and a nilmanifold
Wen Huang, Jianya Liu and Ke Wang
2021, 41(8): 3531-3553 doi: 10.3934/dcds.2021006 +[Abstract](325) +[HTML](140) +[PDF](413.91KB)

Let \begin{document}$ \mathbb{T} $\end{document} be the unit circle and \begin{document}$ \Gamma \backslash G $\end{document} the \begin{document}$ 3 $\end{document}-dimensional Heisenberg nilmanifold. We prove that a class of skew products on \begin{document}$ \mathbb{T} \times \Gamma \backslash G $\end{document} are distal, and that the Möbius function is linearly disjoint from these skew products. This verifies the Möbius Disjointness Conjecture of Sarnak.

Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative
Olivier Ley, Erwin Topp and Miguel Yangari
2021, 41(8): 3555-3577 doi: 10.3934/dcds.2021007 +[Abstract](345) +[HTML](156) +[PDF](404.84KB)

We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order \begin{document}$ \alpha \in (0, 1) $\end{document} cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case \begin{document}$ \alpha = 1 $\end{document}, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.

Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed
Yohei Yamazaki
2021, 41(8): 3579-3614 doi: 10.3934/dcds.2021008 +[Abstract](350) +[HTML](155) +[PDF](516.69KB)

In this paper, we construct center stable manifolds around unstable line solitary waves of the Zakharov–Kuznetsov equation on two dimensional cylindrical spaces \begin{document}$ \mathbb {R} \times \mathbb {T}_L $\end{document} (\begin{document}$ {\mathbb T}_L = {\mathbb R}/2\pi L {\mathbb Z} $\end{document}). In the paper [39], center stable manifolds around unstable line solitary waves have been constructed excluding the case of critical speeds \begin{document}$ c \in \{4n^2/5L^2;n \in {\mathbb Z}, n>1\} $\end{document}. In the case of critical speeds \begin{document}$ c $\end{document}, any neighborhood of the line solitary wave with speed \begin{document}$ c $\end{document} in the energy space includes solitary waves which are depend on the direction \begin{document}$ {\mathbb T}_L $\end{document}. To construct center stable manifolds including their solitary waves and to recover the degeneracy of the linearized operator around line solitary waves with critical speed, we prove the stability condition of the center stable manifold for critical speed by applying to the estimate of the 4th order term of a Lyapunov function in [37] and [38].

On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions
Jan Březina, Eduard Feireisl and Antonín Novotný
2021, 41(8): 3615-3627 doi: 10.3934/dcds.2021009 +[Abstract](291) +[HTML](150) +[PDF](392.64KB)

We consider the barotropic Navier–Stokes system describing the motion of a compressible Newtonian fluid in a bounded domain with in and out flux boundary conditions. We show that if the boundary velocity coincides with that of a rigid motion, all solutions converge to an equilibrium state for large times.

Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions
Sishu Shankar Muni, Robert I. McLachlan and David J. W. Simpson
2021, 41(8): 3629-3650 doi: 10.3934/dcds.2021010 +[Abstract](640) +[HTML](177) +[PDF](2254.0KB)

We consider a homoclinic orbit to a saddle fixed point of an arbitrary \begin{document}$ C^\infty $\end{document} map \begin{document}$ f $\end{document} on \begin{document}$ \mathbb{R}^2 $\end{document} and study the phenomenon that \begin{document}$ f $\end{document} has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory this requires \begin{document}$ f $\end{document} to have a homoclinic tangency. We show it is also necessary for \begin{document}$ f $\end{document} to satisfy a 'global resonance' condition and for the eigenvalues associated with the fixed point, \begin{document}$ \lambda $\end{document} and \begin{document}$ \sigma $\end{document}, to satisfy \begin{document}$ |\lambda \sigma| = 1 $\end{document}. The phenomenon is codimension-three in the case \begin{document}$ \lambda \sigma = -1 $\end{document}, but codimension-four in the case \begin{document}$ \lambda \sigma = 1 $\end{document} because here the coefficients of the leading-order resonance terms associated with \begin{document}$ f $\end{document} at the fixed point must add to zero. We also identify conditions sufficient for the phenomenon to occur, illustrate the results for an abstract family of maps, and show numerically computed basins of attraction.

The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function
Juntao Sun and Tsung-fang Wu
2021, 41(8): 3651-3682 doi: 10.3934/dcds.2021011 +[Abstract](339) +[HTML](158) +[PDF](459.79KB)

In this paper, we study the multiplicity of two spikes nodal solutions for a nonautonomous Schrödinger–Poisson system with the nonlinearity \begin{document}$ f(x)\vert u\vert ^{p-2}u(2<p<6) $\end{document} in \begin{document}$ \mathbb{R}^{3} $\end{document}. By assuming that the weight function \begin{document}$ f\in C(\mathbb{R}^{3},\mathbb{R}^{+}) $\end{document} has \begin{document}$ m $\end{document} maximum points in \begin{document}$ \mathbb{R}^{3} $\end{document}, we conclude that such system admits \begin{document}$ m^{2} $\end{document} distinct nodal solutions, each of which has exactly two nodal domains. The proof is based on a natural constraint approach developed by us as well as the generalized barycenter map.

A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo"
Alessandro Fonda and Rodica Toader
2021, 41(8): 3683-3708 doi: 10.3934/dcds.2021012 +[Abstract](330) +[HTML](153) +[PDF](417.71KB)

We prove the existence of bounded and periodic solutions for planar systems by introducing a notion of lower and upper solutions which generalizes the classical one for scalar second order equations. The proof relies on phase plane analysis; after suitably modifying the nonlinearities, the Ważewski theory provides a solution which remains bounded in the future. For the periodic problem, the Massera Theorem applies, yielding the existence result. We then show how our result generalizes some well known theorems on the existence of bounded and of periodic solutions. Finally, we provide some corollaries on the existence of almost periodic solutions for scalar second order equations.

Proximality of multidimensional $ \mathscr{B} $-free systems
Aurelia Dymek
2021, 41(8): 3709-3724 doi: 10.3934/dcds.2021013 +[Abstract](281) +[HTML](144) +[PDF](431.8KB)

We characterize proximality of multidimensional \begin{document}$ \mathscr{B} $\end{document}-free systems in the case of number fields and lattices in \begin{document}$ \mathbb{Z}^m $\end{document}, \begin{document}$ m\geq2 $\end{document}.

Martin boundary of brownian motion on Gromov hyperbolic metric graphs
Soonki Hong and Seonhee Lim
2021, 41(8): 3725-3757 doi: 10.3934/dcds.2021014 +[Abstract](328) +[HTML](143) +[PDF](637.74KB)

Let \begin{document}$ \widetilde{X} $\end{document} be a locally finite Gromov hyperbolic graph whose Gromov boundary consists of infinitely many points and with a cocompact isometric action of a discrete group \begin{document}$ \Gamma $\end{document}. We show the uniform Ancona inequality for the Brownian motion which implies that the \begin{document}$ \lambda $\end{document}-Martin boundary coincides with the Gromov boundary for any \begin{document}$ \lambda \in [0, \lambda_0], $\end{document} in particular at the bottom of the spectrum \begin{document}$ \lambda_0 $\end{document}.

Stability of Broucke's isosceles orbit
Skyler Simmons
2021, 41(8): 3759-3779 doi: 10.3934/dcds.2021015 +[Abstract](285) +[HTML](129) +[PDF](474.42KB)

We extend the result of Yan to Broucke's isosceles orbit with masses \begin{document}$ m_1 $\end{document}, \begin{document}$ m_1 $\end{document}, and \begin{document}$ m_2 $\end{document} with \begin{document}$ 2m_1 + m_2 = 3 $\end{document}. Under suitable changes of variables, isolated binary collisions between the two mass \begin{document}$ m_1 $\end{document} particles are regularizable. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is reduced to computing three entries of a \begin{document}$ 4 \times 4 $\end{document} matrix related to the monodromy matrix. Additionally, it is shown that the four-degrees-of-freedom setting has a two-degrees-of-freedom invariant set, and linear stability results in the subset comes "for free" from the calculation in the full space. The final numerical analysis shows that the four-degrees-of-freedom orbit is linearly unstable except for the interval \begin{document}$ 0.595 < m_1 < 0.715 $\end{document}, whereas the two-degrees-of-freedom orbit is linearly stable for a much wider interval.

Chaotic Delone sets
Jesús A. Álvarez López, Ramón Barral Lijó, John Hunton, Hiraku Nozawa and John R. Parker
2021, 41(8): 3781-3796 doi: 10.3934/dcds.2021016 +[Abstract](336) +[HTML](135) +[PDF](916.42KB)

We present a definition of chaotic Delone set and establish the genericity of chaos in the space of \begin{document}$ (\epsilon,\delta) $\end{document}-Delone sets for \begin{document}$ \epsilon\geq \delta $\end{document}. We also present a hyperbolic analogue of the cut-and-project method that naturally produces examples of chaotic Delone sets.

On fair entropy of the tent family
Bing Gao and Rui Gao
2021, 41(8): 3797-3816 doi: 10.3934/dcds.2021017 +[Abstract](279) +[HTML](128) +[PDF](408.02KB)

The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues [13] recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy \begin{document}$ h(a) $\end{document} of the tent map \begin{document}$ f_a $\end{document}, as a function of the parameter \begin{document}$ a = \exp(h_{top}(f_a)) $\end{document}, is continuous and strictly increasing on \begin{document}$ [\sqrt{2},2] $\end{document}. In this short note, we extend the last result and characterize regularity of the function \begin{document}$ h $\end{document} precisely. We prove that \begin{document}$ h $\end{document} is \begin{document}$ \frac{1}{2} $\end{document}-Hölder continuous on \begin{document}$ [\sqrt{2},2] $\end{document} and identify its best Hölder exponent on each subinterval of \begin{document}$ [\sqrt{2},2] $\end{document}. On the other hand, parallel to a recent result on topological entropy of the quadratic family due to Dobbs and Mihalache [7], we give a formula of pointwise Hölder exponents of \begin{document}$ h $\end{document} at parameters chosen in an explicitly constructed set of full measure. This formula particularly implies that the derivative of \begin{document}$ h $\end{document} vanishes almost everywhere.

Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five
Norman Noguera and Ademir Pastor
2021, 41(8): 3817-3836 doi: 10.3934/dcds.2021018 +[Abstract](306) +[HTML](137) +[PDF](399.13KB)

In this paper we study the scattering of radial solutions to a \begin{document}$ l $\end{document}-component system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five. Our approach is based on the recent technique introduced by Dodson and Murphy, which relies on the radial Sobolev embedding and a Morawetz estimate.

On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $
Woocheol Choi and Youngwoo Koh
2021, 41(8): 3837-3867 doi: 10.3934/dcds.2021019 +[Abstract](256) +[HTML](130) +[PDF](498.15KB)

In this paper, we establish a convergence result for the operator splitting scheme \begin{document}$ Z_{\tau} $\end{document} introduced by Ignat [12], with initial data in \begin{document}$ H^1 $\end{document}, for the nonlinear Schrödinger equation:

where \begin{document}$ p >0 $\end{document}, \begin{document}$ \lambda \in \{-1,1\} $\end{document} and \begin{document}$ (x,t) \in \mathbb{R}^d \times [0,\infty) $\end{document}. We prove the \begin{document}$ L^2 $\end{document} convergence of order \begin{document}$ \mathcal{O}(\tau^{1/2}) $\end{document} for the scheme with initial data in the space \begin{document}$ H^1 (\mathbb{R}^d) $\end{document} for the energy-subcritical range of \begin{document}$ p $\end{document}.

Random substitution tilings and deviation phenomena
Scott Schmieding and Rodrigo Treviño
2021, 41(8): 3869-3902 doi: 10.3934/dcds.2021020 +[Abstract](237) +[HTML](125) +[PDF](546.36KB)

Suppose a set of prototiles allows \begin{document}$ N $\end{document} different substitution rules. In this paper we study tilings of \begin{document}$ \mathbb{R}^d $\end{document} constructed from random application of the substitution rules. The space of all possible tilings obtained from all possible combinations of these substitutions is the union of all possible tilings spaces coming from these substitutions and has the structure of a Cantor set. The renormalization cocycle on the cohomology bundle over this space determines the statistical properties of the tilings through its Lyapunov spectrum by controlling the deviation of ergodic averages of the \begin{document}$ \mathbb{R}^d $\end{document} action on the tiling spaces.

On the cardinality of collisional clusters for hard spheres at low density
Mario Pulvirenti and Sergio Simonella
2021, 41(8): 3903-3914 doi: 10.3934/dcds.2021021 +[Abstract](240) +[HTML](126) +[PDF](421.81KB)

We resume the investigation, started in [2], of the statistics of backward clusters in a gas of \begin{document}$ N $\end{document} hard spheres of small diameter \begin{document}$ \varepsilon $\end{document}. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We obtain an estimate of the average cardinality of clusters with respect to the equilibrium measure, global in time, uniform in \begin{document}$ \varepsilon, N $\end{document} for \begin{document}$ \varepsilon^2 N = 1 $\end{document} (Boltzmann-Grad regime).

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces
Andreia Chapouto
2021, 41(8): 3915-3950 doi: 10.3934/dcds.2021022 +[Abstract](268) +[HTML](129) +[PDF](536.93KB)

We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle. We first consider the real-valued setting and show global well-posedness of the (usual) renormalized mKdV equation in the Fourier-Lebesgue spaces. In the complex-valued setting, we observe that the momentum plays an important role in the well-posedness theory. In particular, we prove that the complex-valued mKdV equation is ill-posed in the sense of non-existence of solutions when the momentum is infinite, in the spirit of the work on the nonlinear Schrödinger equation by Guo-Oh (2018). This non-existence result motivates the introduction of the second renormalized mKdV equation, which we propose as the correct model in the complex-valued setting outside \begin{document}$ H^\frac12(\mathbb{T}) $\end{document}. Furthermore, imposing a new notion of finite momentum for the initial data, at low regularity, we show existence of solutions to the complex-valued mKdV equation. In particular, we require an energy estimate, from which conservation of momentum follows.

Response solutions for degenerate reversible harmonic oscillators
Wen Si
2021, 41(8): 3951-3972 doi: 10.3934/dcds.2021023 +[Abstract](268) +[HTML](136) +[PDF](397.39KB)

We consider the existence of response solutions for the quasi-periodic perturbation of degenerate reversible harmonic oscillators

where \begin{document}$ \lambda = \pm 1 $\end{document}, \begin{document}$ n>1 $\end{document} is an integer and \begin{document}$ f(-\omega t, x, -\dot x, \epsilon) = f(\omega t, x, \dot x, \epsilon) $\end{document}. With \begin{document}$ f $\end{document} satisfying certain non-degenerate conditions, we obtain the following results: (1) For \begin{document}$ \lambda = 1 $\end{document} and \begin{document}$ \epsilon $\end{document} sufficiently small, response solutions exist for each \begin{document}$ \omega $\end{document} satisfying a weak non-resonant condition; (2) For \begin{document}$ \lambda = -1 $\end{document} and \begin{document}$ \epsilon_* $\end{document} sufficiently small, there exists a Cantor set \begin{document}$ \mathcal{E}\in(0, \epsilon_*) $\end{document} with almost full Lebesgue measure such that response solutions exist for each \begin{document}$ \epsilon\in\mathcal{E} $\end{document} if \begin{document}$ \omega $\end{document} satisfies a Diophantine condition. Non-existence of response solutions is also discussed when \begin{document}$ f $\end{document} fails to satisfy the non-degenerate conditions.

Decay estimates for nonlinear Schrödinger equations
Chenjie Fan and Zehua Zhao
2021, 41(8): 3973-3984 doi: 10.3934/dcds.2021024 +[Abstract](299) +[HTML](128) +[PDF](319.07KB)

In this short note, we present some decay estimates for nonlinear solutions of 3d quintic, 3d cubic NLS, and 2d quintic NLS (nonlinear Schrödinger equations).

Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics
José A. Carrillo, Bertram Düring, Lisa Maria Kreusser and Carola-Bibiane Schönlieb
2021, 41(8): 3985-4012 doi: 10.3934/dcds.2021025 +[Abstract](319) +[HTML](131) +[PDF](1731.32KB)

In this paper, we study the equilibria of an anisotropic, nonlocal aggregation equation with nonlinear diffusion which does not possess a gradient flow structure. Here, the anisotropy is induced by an underlying tensor field. Anisotropic forces cannot be associated with a potential in general and stationary solutions of anisotropic aggregation equations generally cannot be regarded as minimizers of an energy functional. We derive equilibrium conditions for stationary line patterns in the setting of spatially homogeneous tensor fields. The stationary solutions can be regarded as the minimizers of a regularised energy functional depending on a scalar potential. A dimension reduction from the two- to the one-dimensional setting allows us to study the associated one-dimensional problem instead of the two-dimensional setting. We establish \begin{document}$ \Gamma $\end{document}-convergence of the regularised energy functionals as the diffusion coefficient vanishes, and prove the convergence of minimisers of the regularised energy functional to minimisers of the non-regularised energy functional. Further, we investigate properties of stationary solutions on the torus, based on known results in one spatial dimension. Finally, we prove weak convergence of a numerical scheme for the numerical solution of the anisotropic, nonlocal aggregation equation with nonlinear diffusion and any underlying tensor field, and show numerical results.

2019  Impact Factor: 1.338




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