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Discrete and Continuous Dynamical Systems

July 2022 , Volume 42 , Issue 7

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Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum
Claire Chavaudret and Stefano Marmi
2022, 42(7): 3077-3101 doi: 10.3934/dcds.2022009 +[Abstract](330) +[HTML](126) +[PDF](2520.4KB)
Abstract:

One considers a system on \begin{document}$ \mathbb{C}^2 $\end{document} close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded by the exponential of the negative Brjuno sum of \begin{document}$ d\alpha $\end{document}, where \begin{document}$ d\in \mathbb{N}^* $\end{document} and \begin{document}$ \alpha $\end{document} is the frequency of the linear part, and the error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.

Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation
Younghun Hong and Sangdon Jin
2022, 42(7): 3103-3118 doi: 10.3934/dcds.2022010 +[Abstract](426) +[HTML](126) +[PDF](382.44KB)
Abstract:

For the one-dimensional mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable [2]. In this study, we proved the local uniqueness and established the orbital stability of the solitary wave by improving that of the energy minimizer set. A key aspect thereof is the reformulation of the variational problem in the non-relativistic regime, which we consider to be more natural because the proof extensively relies on the subcritical nature of the limiting model. Thus, the role of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Subsequently, this limit is employed to derive the local uniqueness and orbital stability.

Singular solutions of Toda system in high dimensions
Linlin Dou
2022, 42(7): 3119-3142 doi: 10.3934/dcds.2022011 +[Abstract](332) +[HTML](153) +[PDF](374.1KB)
Abstract:

We construct some singular solutions to the four component Toda system. These solutions are almost split into two groups, each one modelled on an explicit solution to the two component Toda system (i.e. Liouivlle equation). These solutions are shown to be stable in high dimensions. This gives a sharp example on the partial regularity of stable solutions to Toda system.

Vortex collapses for the Euler and Quasi-Geostrophic models
Ludovic Godard-Cadillac
2022, 42(7): 3143-3168 doi: 10.3934/dcds.2022012 +[Abstract](381) +[HTML](123) +[PDF](427.01KB)
Abstract:

This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains two main theorems and also smaller propositions with several links between each other. The first main result focuses on the Euler point-vortex model, and under the non-neutral cluster hypothesis we prove a convergence result. The second result is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.

A KAM theorem for quasi-periodic non-twist mappings and its application
Zhichao Ma and Junxiang Xu
2022, 42(7): 3169-3185 doi: 10.3934/dcds.2022013 +[Abstract](362) +[HTML](153) +[PDF](340.92KB)
Abstract:

In this paper we consider quasi-periodic non-twist mappings with self-intersection property, which depend on a small parameter. Without assuming any twist condition, we prove that for many sufficiently small parameters the mapping admits an invariant curve. As application, we use this result to study Lagrange stability of second order systems.

Introducing sub-Riemannian and sub-Finsler billiards
Lucas Dahinden and Álvaro del Pino
2022, 42(7): 3187-3232 doi: 10.3934/dcds.2022014 +[Abstract](270) +[HTML](135) +[PDF](1375.68KB)
Abstract:

We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting, including the failure of the reflection law to be well-defined at singular points of the boundary distribution, the appearance of gliding and creeping orbits, and the behavior of reflections at wavefronts.

We then study some concrete tables in \begin{document}$ 3 $\end{document}-dimensional euclidean space endowed with the standard contact structure. These can be interpreted as planar magnetic billiards, of varying magnetic strength, for which the magnetic strength may change under reflection. For each table we provide various results regarding periodic trajectories, gliding orbits, and creeping orbits.

Regularity of almost-minimizers of Hölder-coefficient surface energies
David A. Simmons
2022, 42(7): 3233-3299 doi: 10.3934/dcds.2022015 +[Abstract](271) +[HTML](102) +[PDF](634.45KB)
Abstract:

We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.

Existence of positive solutions for a class of fractional Choquard equation in exterior domain
César E. Torres Ledesma
2022, 42(7): 3301-3328 doi: 10.3934/dcds.2022016 +[Abstract](331) +[HTML](105) +[PDF](391.9KB)
Abstract:

In this paper we show existence of positive solutions for a class of problems involving the fractional Laplacian in exterior domain and Choquard type nonlinearity. We prove the main results using variational method combined with Brouwer theory of degree and Deformation Lemma..

Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz condition
Lauren M. M. Bonaldo, Elard J. Hurtado and Olímpio H. Miyagaki
2022, 42(7): 3329-3353 doi: 10.3934/dcds.2022017 +[Abstract](303) +[HTML](98) +[PDF](442.56KB)
Abstract:

In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations \begin{document} $( \mathscr{P}_\lambda)$\end{document} in a smooth bounded domain, driven by a nonlocal integrodifferential operator \begin{document}$ \mathscr{L}_{\mathcal{A}K} $\end{document} with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem \begin{document} $( \mathscr{P}_\lambda)$\end{document} and we show that the problem treated has at least one nontrivial solution for any parameter \begin{document}$ \lambda >0 $\end{document} small enough as well as that the solution blows up, in the fractional Sobolev norm, as \begin{document}$ \lambda \to 0 $\end{document}. Moreover, for the sublinear case, by imposing some additional hypotheses on the nonlinearity \begin{document}$ f(x,\cdot) $\end{document}, and by using a new version of the symmetric Mountain Pass Theorem due to Kajikiya [18], we obtain the existence of infinitely many weak solutions which tend to zero, in the fractional Sobolev norm, for any parameter \begin{document}$ \lambda >0 $\end{document}. As far as we know, the results of this paper are new in the literature.

Linear stability of exact solutions for the generalized Kaup-Boussinesq equation and their dynamical evolutions
Ruizhi Gong, Yuren Shi and Deng-Shan Wang
2022, 42(7): 3355-3378 doi: 10.3934/dcds.2022018 +[Abstract](286) +[HTML](192) +[PDF](769.7KB)
Abstract:

The integrability, classification of traveling wave solutions and stability of exact solutions for the generalized Kaup-Boussinesq equation are studied by prolongation structure technique and linear stability analysis. Firstly, it is proved that the generalized Kaup-Boussinesq equation is completely integrable in sense of having Lax pair. Secondly, the complete classification of exact traveling wave solutions of the generalized Kaup-Boussinesq equation are given and a family of exact solutions are proposed. Finally, the stability of these exact solutions are investigated by linear stability analysis and dynamical evolutions, and some stable traveling wave solutions are found. It is shown that the results of linear stability analysis are in excellent agreement with the results from dynamical evolutions.

Polynomial ergodic averages for certain countable ring actions
Andrew Best and Andreu Ferré Moragues
2022, 42(7): 3379-3413 doi: 10.3934/dcds.2022019 +[Abstract](246) +[HTML](91) +[PDF](511.71KB)
Abstract:

A recent result of Frantzikinakis in [17] establishes sufficient conditions for joint ergodicity in the setting of \begin{document}$ \mathbb{Z} $\end{document}-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action \begin{document}$ (T_n)_{n \in F} $\end{document} of a countable field \begin{document}$ F $\end{document} with characteristic zero on a probability space \begin{document}$ (X,\mathcal{B},\mu) $\end{document} and a family \begin{document}$ \{p_1,\dots,p_k\} $\end{document} of independent polynomials, we have

where \begin{document}$ f_i \in L^{\infty}(\mu) $\end{document}, \begin{document}$ (\Phi_N) $\end{document} is a Følner sequence of \begin{document}$ (F,+) $\end{document}, and the convergence takes place in \begin{document}$ L^2(\mu) $\end{document}. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.

SRB measures of singular hyperbolic attractors
Dominic Veconi
2022, 42(7): 3415-3430 doi: 10.3934/dcds.2022020 +[Abstract](261) +[HTML](105) +[PDF](326.68KB)
Abstract:

It is known that hyperbolic maps admitting singularities have at most countably many ergodic Sinai-Ruelle-Bowen (SRB) measures. These maps include the Belykh attractor, the geometric Lorenz attractor, and more general Lorenz-type systems. In this paper, we establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary in general.

Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications
Marc Briane and Loïc Hervé
2022, 42(7): 3431-3463 doi: 10.3934/dcds.2022021 +[Abstract](239) +[HTML](102) +[PDF](466.77KB)
Abstract:

This paper deals with the long time asymptotics of the flow \begin{document}$ X $\end{document} solution to the vector-valued ODE: \begin{document}$ X'(t, x) = b(X(t, x)) $\end{document} for \begin{document}$ t\in \mathbb{R} $\end{document}, with \begin{document}$ X(0, x) = x $\end{document} a point of the torus \begin{document}$ Y_d $\end{document}. We assume that the vector field \begin{document}$ b $\end{document} reads as \begin{document}$ \rho\, \Phi $\end{document}, where \begin{document}$ \rho $\end{document} is a non negative regular function and \begin{document}$ \Phi $\end{document} is a non vanishing regular vector field in \begin{document}$ Y_d $\end{document}. In this work, the singleton condition means that the Herman rotation set \begin{document}$ {\mathsf{C}}_b $\end{document} composed of the average values of \begin{document}$ b $\end{document} with respect to the invariant probability measures for the flow \begin{document}$ X $\end{document} is a singleton \begin{document}$ \{\zeta\} $\end{document}. This first allows us to obtain the asymptotics of the flow \begin{document}$ X $\end{document} when \begin{document}$ b $\end{document} is a nonlinear current field. Then, we prove a general perturbation result assuming that \begin{document}$ \rho $\end{document} is the uniform limit in \begin{document}$ Y_d $\end{document} of a positive sequence \begin{document}$ (\rho_n)_{n\in \mathbb{N}} $\end{document} satisfying \begin{document}$ \rho\leq\rho_n $\end{document} and \begin{document}$ {\mathsf{C}}_{\rho_n\Phi} $\end{document} is a singleton \begin{document}$ \{\zeta_n\} $\end{document}. It turns out that the limit set \begin{document}$ {\mathsf{C}}_b $\end{document} either remains a singleton, or enlarges to the closed line segment \begin{document}$ [0_{ \mathbb{R}^d}, \lim_n\zeta_n] $\end{document} of \begin{document}$ \mathbb{R}^d $\end{document}. We provide various corollaries of this result according to the positivity or not of some weighted harmonic means of \begin{document}$ \rho $\end{document}. These results are illustrated by different examples which highlight the alternative satisfied by \begin{document}$ {\mathsf{C}}_b $\end{document}. Finally, the singleton condition allows us to homogenize the linear transport equation induced by the oscillating velocity \begin{document}$ b(x/{\varepsilon}) $\end{document}.

A combinatorial approach to Rauzy-type dynamics II: The labelling method and a second proof of the KZB classification theorem
Quentin De Mourgues
2022, 42(7): 3465-3538 doi: 10.3934/dcds.2022022 +[Abstract](231) +[HTML](94) +[PDF](1079.83KB)
Abstract:

Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich, and by Boissy through methods involving both combinatorics algebraic geometry, topology and dynamical systems. Our precedent paper [5] as well as the one of Fickenscher [8] proposed an ad hoc combinatorial proof of this classification.

However, unlike those two previous combinatorial proofs, we develop in this paper a general method, called the labelling method, which allows one to classify Rauzy-type dynamics in a much more systematic way. We apply the method to the Rauzy dynamics and obtain a third combinatorial proof of the classification. The method is versatile and will be used to classify three other Rauzy-type dynamics in follow-up works.

Another feature of this paper is to introduce an algorithmic method to work with the sign invariant of the Rauzy dynamics. With this method, we can prove most of the identities appearing in the literature so far ([10], [6], [2], [5]...) in an automatic way.

On a class of singularly perturbed elliptic systems with asymptotic phase segregation
Farid Bozorgnia, Martin Burger and Morteza Fotouhi
2022, 42(7): 3539-3556 doi: 10.3934/dcds.2022023 +[Abstract](263) +[HTML](87) +[PDF](1496.34KB)
Abstract:

This work is devoted to study a class of singular perturbed elliptic systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behavior of limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero, which implies that all components can not coexist simultaneously. We present a novel method, which provides an explicit solution of the limiting problem for a special choice of parameters. Moreover, we present some numerical simulations of the asymptotic problem.

On decomposition of ambient surfaces admitting $ A $-diffeomorphisms with non-trivial attractors and repellers
Vyacheslav Grines and Dmitrii Mints
2022, 42(7): 3557-3568 doi: 10.3934/dcds.2022024 +[Abstract](207) +[HTML](115) +[PDF](841.03KB)
Abstract:

It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom \begin{document}$ A $\end{document} and the topology of the ambient manifold. In the given article, this statement is considered for the class \begin{document}$ \mathbb G(M^2) $\end{document} of \begin{document}$ A $\end{document}-diffeomorphisms of closed orientable connected surfaces, the non-wandering set of each of which consists of \begin{document}$ k_f\geq 2 $\end{document} connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism \begin{document}$ f\in \mathbb G(M^2) $\end{document} is homeomorphic to the connected sum of \begin{document}$ k_f $\end{document} closed orientable connected surfaces and \begin{document}$ l_f $\end{document} two-dimensional tori such that the genus of each surface is determined by the dynamical properties of appropriating connected component of a basic set and \begin{document}$ l_f $\end{document} is determined by the number and position of bunches, belonging to all connected components of basic sets. We also prove that every diffeomorphism from the class \begin{document}$ \mathbb G(M^2) $\end{document} is \begin{document}$ \Omega $\end{document}-stable but is not structurally stable.

Particle approximation of one-dimensional Mean-Field-Games with local interactions
Marco Di Francesco, Serikbolsyn Duisembay, Diogo Aguiar Gomes and Ricardo Ribeiro
2022, 42(7): 3569-3591 doi: 10.3934/dcds.2022025 +[Abstract](242) +[HTML](81) +[PDF](609.67KB)
Abstract:

We study a particle approximation for one-dimensional first-order Mean-Field-Games (MFGs) with local interactions with planning conditions. Our problem comprises a system of a Hamilton-Jacobi equation coupled with a transport equation. As we deal with the planning problem, we prescribe initial and terminal distributions for the transport equation. The particle approximation builds on a semi-discrete variational problem. First, we address the existence and uniqueness of a solution to the semi-discrete variational problem. Next, we show that our discretization preserves some previously identified conserved quantities. Finally, we prove that the approximation by particle systems preserves displacement convexity. We use this last property to establish uniform estimates for the discrete problem. We illustrate our results for the discrete problem with numerical examples.

On information gain, Kullback-Leibler divergence, entropy production and the involution kernel
Artur O. Lopes and Jairo K. Mengue
2022, 42(7): 3593-3627 doi: 10.3934/dcds.2022026 +[Abstract](206) +[HTML](84) +[PDF](506.57KB)
Abstract:

It is well known that in Information Theory and Machine Learning the Kullback-Leibler divergence, which extends the concept of Shannon entropy, plays a fundamental role. Given an a priori probability kernel \begin{document}$ \hat{\nu} $\end{document} and a probability \begin{document}$ \pi $\end{document} on the measurable space \begin{document}$ X\times Y $\end{document} we consider an appropriate definition of entropy of \begin{document}$ \pi $\end{document} relative to \begin{document}$ \hat{\nu} $\end{document}, which is based on previous works. Using this concept of entropy we obtain a natural definition of information gain for general measurable spaces which coincides with the mutual information given from the K-L divergence in the case \begin{document}$ \hat{\nu} $\end{document} is identified with a probability \begin{document}$ \nu $\end{document} on \begin{document}$ X $\end{document}. This will be used to extend the meaning of specific information gain and dynamical entropy production to the model of thermodynamic formalism for symbolic dynamics over a compact alphabet (TFCA model). Via the concepts of involution kernel and dual potential, one can ask if a given potential is symmetric - the relevant information is available in the potential. In the affirmative case, its corresponding equilibrium state has zero entropy production.

2021 Impact Factor: 1.588
5 Year Impact Factor: 1.568
2021 CiteScore: 2.4

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