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Discrete & Continuous Dynamical Systems - A

April 2018 , Volume 38 , Issue 4

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Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior
Roland Gunesch and Philipp Kunde
2018, 38(4): 1615-1655 doi: 10.3934/dcds.2018067 +[Abstract](1274) +[HTML](443) +[PDF](683.82KB)
Abstract:

We show that on any smooth compact connected manifold of dimension \begin{document}$m≥2$\end{document} admitting a smooth non-trivial circle action \begin{document}$\mathcal{S} = \left\{S_t\right\}_{t ∈ \mathbb{R}}$\end{document}, \begin{document}$S_{t+1}=S_t$\end{document}, the set of weakly mixing \begin{document}$C^{∞}$\end{document}-diffeomorphisms which preserve both a smooth volume \begin{document}$ν$\end{document} and a measurable Riemannian metric is dense in \begin{document}${{\mathcal{A}}_{\alpha }}\left( M \right)={{\overline{\left\{ h\circ {{S}_{\alpha }}\circ {{h}^{-1}}:h\in \text{Dif}{{\text{f}}^{\infty }}\left( M,\nu \right) \right\}}}^{{{C}^{\infty }}}}$\end{document} for every Liouville number \begin{document}$α$\end{document}. The proof is based on a quantitative version of the approximation by conjugation-method with explicitly constructed conjugation maps and partitions.

Periodic measures are dense in invariant measures for residually finite amenable group actions with specification
Xiankun Ren
2018, 38(4): 1657-1667 doi: 10.3934/dcds.2018068 +[Abstract](958) +[HTML](372) +[PDF](372.71KB)
Abstract:

We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactions of a countable discrete group by automorphisms of compact abelian groups have specification property.

Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain
Fucai Li and Yanmin Mu
2018, 38(4): 1669-1705 doi: 10.3934/dcds.2018069 +[Abstract](967) +[HTML](232) +[PDF](596.28KB)
Abstract:

This paper studies the convergence of the compressible isentropic magnetohydrodynamic equations to the corresponding incompressiblemagnetohydrodynamic equations with ill-preparedinitial data in a periodic domain.We prove that the solution to the compressible isentropic magnetohydrodynamic equations with small Mach number exists uniformly in the time interval as long as that to the incompressible one does. Furthermore,we obtain the convergence result for the solutions filtered by the group of acoustics.

On the asymptotic character of a generalized rational difference equation
Esha Chatterjee and Sk. Sarif Hassan
2018, 38(4): 1707-1718 doi: 10.3934/dcds.2018070 +[Abstract](1066) +[HTML](266) +[PDF](469.53KB)
Abstract:

We investigate the global asymptotic stability of the solutions of \begin{document}$X_{n+1}=\frac{β X_{n-l} + γ X_{n-k}}{A + X_{n-k}} $\end{document} for \begin{document}$n=1,2, ...$\end{document}, where \begin{document}$l$\end{document} and \begin{document}$k$\end{document} are positive integers such that \begin{document}$l≠ k$\end{document}. The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We find necessary and sufficient conditions for the global asymptotic stability of the zero equilibrium. We also investigate the positive equilibrium and find the regions of parameters where the positive equilibrium is a global attractor of all positive solutions. Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms (\begin{document}$l$\end{document}, \begin{document}$k$\end{document}) being (odd, odd), (odd, even), (even, odd) or (even, even). In this manuscript we will investigate these aspects of the solutions for all such combinations of delay terms.

Consensus and voting on large graphs: An application of graph limit theory
Barton E. Lee
2018, 38(4): 1719-1744 doi: 10.3934/dcds.2018071 +[Abstract](753) +[HTML](228) +[PDF](546.82KB)
Abstract:

Building on recent work by Medvedev (2014) we establish new connections between a basic consensus model, called the voting model, and the theory of graph limits. We show that in the voting model if consensus is attained in the continuum limit then solutions to the finite model will eventually be close to a constant function, and a class of graph limits which guarantee consensus is identified. It is also proven that the dynamics in the continuum limit can be decomposed as a direct sum of dynamics on the connected components, using Janson's definition of connectivity for graph limits. This implies that without loss of generality it may be assumed that the continuum voting model occurs on a connected graph limit.

A formula for the boundary of chaos in the lexicographical scenario and applications to the bifurcation diagram of the standard two parameter family of quadratic increasing-increasing Lorenz maps
Rafael Labarca and Solange Aranzubia
2018, 38(4): 1745-1776 doi: 10.3934/dcds.2018072 +[Abstract](814) +[HTML](226) +[PDF](753.03KB)
Abstract:

The Geometric Lorenz Attractor has been a source of inspiration for many mathematical studies. Most of these studies deal with the two or one dimensional representation of its first return map. A one dimensional scenario (the increasing-increasing one's) can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics of any member of the standard family can be modeled by a subshift in the Lexicographical model of two symbols. These subshifts can be considered as the maximal invariant set for the shift map in some interval, in the Lexicographical model. For all of these subshifts, the lower extreme of the interval is a minimal sequence and the upper extreme is a maximal sequence. The Lexicographical world (LW) is precisely the set of sequences (lower extreme, upper extreme) of all of these subshifts. In this scenario the topological entropy is a map from LW onto the interval \begin{document}$[0, \log{2}]$\end{document}. The boundary of chaos (that is the boundary of the set of \begin{document}$ (a, b) ∈ LW$\end{document} such that \begin{document}$h_{top}(a, b)>0$\end{document}) is given by a map \begin{document}$ b = χ(a)$\end{document}, which is defined by a recurrence formula. In the present paper we obtain an explicit formula for the value \begin{document}$χ(a)$\end{document} for \begin{document}$a$\end{document} in a dense set contained in the set of minimal sequences. Moreover, we apply this computation to determine regions of positive topological entropy for the standard quadratic family of contracting increasing-increasing Lorenz maps.

Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus
Marzie Zaj, Abbas Fakhari, Fatemeh Helen Ghane and Azam Ehsani
2018, 38(4): 1777-1807 doi: 10.3934/dcds.2018073 +[Abstract](1045) +[HTML](222) +[PDF](617.77KB)
Abstract:

In this paper we address the existence and ergodicity of non-uniformly hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems have formulation as a skew product system defined by planar diffeomorphisms, with average contraction condition, forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one physical measure. In our approach, these skew product systems arising from iterated function systems which are generated by finitely many weak contractive diffeomorphisms. Under some conditions including negative fiber Lyapunov exponents, we prove the existence of unique non-uniformly hyperbolic attracting invariant graphs for these systems which attract positive orbits of almost all initial points. Also, we prove that these systems are Bernoulli and therefore they are mixing. Moreover, these properties remain true under small perturbations in the space of endomorphisms on the solid torus.

$C^1$ weak Palis conjecture for nonsingular flows
Qianying Xiao and Zuohuan Zheng
2018, 38(4): 1809-1832 doi: 10.3934/dcds.2018074 +[Abstract](807) +[HTML](242) +[PDF](475.21KB)
Abstract:

This paper focuses on generic properties of continuous dynamical systems. We prove \begin{document}$C^1$\end{document} weak Palis conjecture for nonsingular flows: Morse-Smale vector fields and vector fields admitting horseshoes are open and dense among \begin{document}$C^1$\end{document} nonsingular vector fields.

Our arguments contain three main ingredients: linear Poincaré flow, Liao's selecting lemma and the adapting of Crovisier's central model.

Firstly, by studying the linear Poincaré flow, we prove for a \begin{document}$C^1$\end{document} generic vector field away from horseshoes, any non-trivial nonsingular chain recurrent class contains a minimal set which is partially hyperbolic with 1-dimensional center with respect to the linear Poincaré flow.

Secondly, to understand the neutral behaviour of the 1-dimensional center, we adapt Crovisier's central model. The difficulties are that we can not build invariant plaque family of any time, the periodic point of a flow is not periodic for the discrete time map. Through delicate analysis of the center manifold of a periodic orbit near the partially hyperbolic set, we manage to yield nice periodic points such that their stable manifolds and unstable manifolds are well-placed for transverse intersection.

Reflected backward stochastic differential equations with perturbations
Jasmina Djordjević and Svetlana Janković
2018, 38(4): 1833-1848 doi: 10.3934/dcds.2018075 +[Abstract](1050) +[HTML](250) +[PDF](427.42KB)
Abstract:

This paper deals with a large class of reflected backward stochastic differential equations whose generators arbitrarily depend on a small parameter. The solutions of these equations, named the perturbed equations, are compared in the \begin{document} $L^p$\end{document}-sense, \begin{document} $p∈ ]1,2[$\end{document}, with the solutions of the appropriate equations of the equal type, independent of a small parameter and named the unperturbed equations. Conditions under which the solution of the unperturbed equation is \begin{document} $L^p$\end{document}-stable are given. It is shown that for an arbitrary \begin{document} $η>0$\end{document} there exists an interval \begin{document} $[t(η), T]\subset [0,T]$\end{document} on which the \begin{document} $L^p$\end{document}-difference between the solutions of both the perturbed and unperturbed equations is less than \begin{document} $η$\end{document}.

Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities
Sze-Man Ngai, Wei Tang and Yuanyuan Xie
2018, 38(4): 1849-1887 doi: 10.3934/dcds.2018076 +[Abstract](857) +[HTML](229) +[PDF](759.49KB)
Abstract:

We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps are in certain sense essentially of finite type, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by a self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu[24].

Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$
Juntao Sun, Tsung-Fang Wu and Zhaosheng Feng
2018, 38(4): 1889-1933 doi: 10.3934/dcds.2018077 +[Abstract](1282) +[HTML](418) +[PDF](644.15KB)
Abstract:

We study the existence of positive solutions for the non-autonomous Schrödinger-Poisson system:

where \begin{document} $\lambda >0$\end{document}, \begin{document} $2 < p \le 4$\end{document} and both \begin{document} $K\left( x\right) $\end{document} and \begin{document} $a\left( x\right) $\end{document} are nonnegative functions in \begin{document} $\mathbb{R}^{3}$\end{document}, which satisfy the given conditions, but not require any symmetry property. Assuming that \begin{document} $% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$\end{document} and \begin{document} $\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$\end{document}, we explore the existence of positive solutions, depending on the parameters \begin{document} $\lambda$\end{document} and \begin{document} $p$\end{document}. More importantly, we establish the existence of ground state solutions in the case of \begin{document} $3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$\end{document}.

Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity
Hailong Zhu, Jifeng Chu and Weinian Zhang
2018, 38(4): 1935-1953 doi: 10.3934/dcds.2018078 +[Abstract](1311) +[HTML](310) +[PDF](462.73KB)
Abstract:

In the setting of mean-square exponential dichotomies, we study the existence and uniqueness of mean-square almost automorphic solutions of non-autonomous linear and nonlinear stochastic differential equations.

The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos
Radosław Kurek, Paweł Lubowiecki and Henryk Żołądek
2018, 38(4): 1955-1981 doi: 10.3934/dcds.2018079 +[Abstract](866) +[HTML](236) +[PDF](542.44KB)
Abstract:

We consider a special situation of the Hess-Appelrot case of the Euler-Poisson system which describes the dynamics of a rigid body about a fixed point. One has an equilibrium point of saddle type with coinciding stable and unstable invariant 2-dimensional separatrices. We show rigorously that, after a suitable perturbation of the Hess-Appelrot case, the separatrix connection is split such that only finite number of 1-dimensional homoclinic trajectories remain and that such situation leads to a chaotic dynamics with positive entropy and to the non-existence of any additional first integral.

Large deviations for stochastic heat equations with memory driven by Lévy-type noise
Markus Riedle and Jianliang Zhai
2018, 38(4): 1983-2005 doi: 10.3934/dcds.2018080 +[Abstract](902) +[HTML](216) +[PDF](440.73KB)
Abstract:

For a heat equation with memory driven by a Lévy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with memory driven by a Lévy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.

Rarefaction waves for the Toda equation via nonlinear steepest descent
Iryna Egorova, Johanna Michor and Gerald Teschl
2018, 38(4): 2007-2028 doi: 10.3934/dcds.2018081 +[Abstract](704) +[HTML](206) +[PDF](662.62KB)
Abstract:

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice with steplike initial data corresponding to a rarefaction wave.

Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems
Yilei Tang
2018, 38(4): 2029-2046 doi: 10.3934/dcds.2018082 +[Abstract](920) +[HTML](225) +[PDF](1543.08KB)
Abstract:

In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi-homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a center in piecewise smooth quadratic quasi-homogeneous systems. Moreover, the center is global and non-isochronous, which cannot appear in smooth quadratic quasi-homogeneous systems. Then the global structures of piecewise smooth quadratic quasi-homogeneous but non-homogeneous systems are obtained. Finally we investigate limit cycle bifurcations of the piecewise quadratic quasi-homogeneous center and give the maximal number of limit cycles bifurcating from periodic orbits of the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.

Well-posedness of a model for the growth of tree stems and vines
Alberto Bressan and Michele Palladino
2018, 38(4): 2047-2064 doi: 10.3934/dcds.2018083 +[Abstract](670) +[HTML](204) +[PDF](481.01KB)
Abstract:

The paper studies a PDE model for the growth of a tree stem or a vine, having the form of a differential inclusion with state constraints. The equations describe the elongation due to cell growth, and the response to gravity and to external obstacles.

The main theorem shows that the evolution problem is well posed, until a specific "breakdown configuration" is reached. A formula is proved, characterizing the reaction produced by unilateral constraints. At a.e. time $t$, this is determined by the minimization of an elastic energy functional under suitable constraints.

The return times property for the tail on logarithm-type spaces
María Jesús Carro and Carlos Domingo-Salazar
2018, 38(4): 2065-2078 doi: 10.3934/dcds.2018084 +[Abstract](709) +[HTML](209) +[PDF](427.88KB)
Abstract:

Given a dynamical system \begin{document} $(Ω,Σ,μ, τ)$\end{document} with \begin{document} $μ$\end{document} a non-atomic probability measure and \begin{document} $τ$\end{document} an invertible measure preserving ergodic transformation, we prove that the maximal operator, considered by I. Assani, Z. Buczolich and R. D. Mauldin in 2005,

satisfies that

is bounded where the space \begin{document} $\left[ {L \log_3 L (μ)} \right]$\end{document} is defined by the condition

with \begin{document} $\log_3 x = 1+\log_+\log_+\log_+ x$\end{document} and \begin{document} $f^*_μ$\end{document} the decreasing rearrangement of \begin{document} $f$\end{document} with respect to \begin{document} $μ$\end{document}. This space is near \begin{document} $L \log_3 L (μ)$\end{document}, which is the optimal Orlicz space on which such boundedness can hold. As a consequence, the space \begin{document} $\left[ {L \log_3 L (μ)} \right]$\end{document} satisfies the Return Times Property for the Tail; that is, for every \begin{document} $f∈\left[ {L \log_3 L (μ)} \right]$\end{document}, there exists a set \begin{document} $X_0$\end{document} so that \begin{document} $μ(X_0) = 1$\end{document} and, for all \begin{document} $x_0∈ X_0$\end{document}, all dynamical systems \begin{document} $(Y,\mathcal{C},ν, S)$\end{document} and all \begin{document} $g∈ L^1(ν)$\end{document}, the sequence

Minimization of the lowest eigenvalue for a vibrating beam
Quanyi Liang, Kairong Liu, Gang Meng and Zhikun She
2018, 38(4): 2079-2092 doi: 10.3934/dcds.2018085 +[Abstract](762) +[HTML](203) +[PDF](376.98KB)
Abstract:

In this paper we solve the minimization problem of the lowest eigenvalue for a vibrating beam. Firstly, based on the variational method, we establish the basic theory of the lowest eigenvalue for the fourth order measure differential equation (MDE). Secondly, we build the relationship between the minimization problem of the lowest eigenvalue for the ODE and the one for the MDE. Finally, with the help of this built relationship, we find the explicit optimal bound of the lowest eigenvalue for a vibrating beam.

Invariance entropy, quasi-stationary measures and control sets
Fritz Colonius
2018, 38(4): 2093-2123 doi: 10.3934/dcds.2018086 +[Abstract](712) +[HTML](226) +[PDF](649.38KB)
Abstract:

For control systems in discrete time, this paper discusses measure-theoretic invariance entropy for a subset Q of the state space with respect to a quasi-stationary measure obtained by endowing the control range with a probability measure. The main results show that this entropy is invariant under measurable transformations and that it is already determined by certain subsets of Q which are characterized by controllability properties.

Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting
Xinsheng Wang, Lin Wang and Yujun Zhu
2018, 38(4): 2125-2140 doi: 10.3934/dcds.2018087 +[Abstract](763) +[HTML](221) +[PDF](402.79KB)
Abstract:

Metric entropies along a hierarchy of unstable foliations are investigated for \begin{document}$C^1 $\end{document} diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents in each hierarchy, are given.

Pullback $ \mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model
Theodore Tachim Medjo
2018, 38(4): 2141-2169 doi: 10.3934/dcds.2018088 +[Abstract](764) +[HTML](231) +[PDF](517.51KB)
Abstract:

The existence and final fractal dimension of a pullback attractor in the space \begin{document}$ \mathbb{V}$\end{document} for a three dimensional system of a non-autonomous globally modified two phase flow on a bounded domain is established under appropriate properties on the time depending forcing term. The model consists of the globally modified Navier-Stokes equations proposed in [6] for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. The existence of the pullback attractors is obtained using the flattening property. Furthermore, we prove that the fractal dimension in \begin{document}$ \mathbb{V}$\end{document} of the pullback attractor is finite.

Theory of rotated equations and applications to a population model
Maoan Han, Xiaoyan Hou, Lijuan Sheng and Chaoyang Wang
2018, 38(4): 2171-2185 doi: 10.3934/dcds.2018089 +[Abstract](934) +[HTML](335) +[PDF](440.42KB)
Abstract:

We consider a family of scalar periodic equations with a parameter and establish theory of rotated equations, studying the behavior of periodic solutions with the change of the parameter. It is shown that a stable (completely unstable) periodic solution of a rotated equation varies monotonically with respect to the parameter and a semi-stable periodic solution splits into two periodic solutions or disappears as the parameter changes in one direction or another. As an application of the obtained results, we give a further study of a piecewise smooth population model verifying the existence of saddle-node bifurcation.

New periodic orbits in the planar equal-mass three-body problem
Rongchang Liu, Jiangyuan Li and Duokui Yan
2018, 38(4): 2187-2206 doi: 10.3934/dcds.2018090 +[Abstract](936) +[HTML](299) +[PDF](506.51KB)
Abstract:

It is known that there exist two sets of nontrivial periodic orbits in the planar equal-mass three-body problem: retrograde orbit and prograde orbit. By introducing topological constraints to a two-point free boundary value problem, we show that there exists a new set of periodic orbits for a small interval of rotation angle \begin{document}$ \mathit{\theta }$\end{document}.

Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data
Chenmin Sun, Hua Wang, Xiaohua Yao and Jiqiang Zheng
2018, 38(4): 2207-2228 doi: 10.3934/dcds.2018091 +[Abstract](971) +[HTML](254) +[PDF](451.96KB)
Abstract:

The aim of this paper is to adapt the strategy in [8] [ See, B. Dodson, J. Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for certain focusing fractional NLS. The main ingredient is to apply the fractional virial identity proved in [3] [ See, T. Boulenger, D. Himmelsbach, E. Lenzmann, Blow up for fractional NLS, J. Func. Anal, 271(2016), 2569-2603 ] to exclude the concentration of mass near the origin.

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