# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

September 2018 , Volume 38 , Issue 9

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2018, 38(9): 4243-4257 doi: 10.3934/dcds.2018185 +[Abstract](2132) +[HTML](109) +[PDF](347.98KB)
Abstract:

We study the displacement function of homeomorphisms isotopic to the identity of the universal one-dimensional solenoid and we get a characterization of the lifting property for an open and dense subgroup of the isotopy component of the identity. The dynamics of an element in this subgroup is also described using rotation theory.

2018, 38(9): 4259-4278 doi: 10.3934/dcds.2018186 +[Abstract](2052) +[HTML](92) +[PDF](621.38KB)
Abstract:

We determine all triples \begin{document}$(α, β, λ)$\end{document} such that \begin{document}$C_α- λ C_β$\end{document} forms a closed interval, where \begin{document}$C_α$\end{document} and \begin{document}$C_β$\end{document} are middle Cantor sets. This follows from a new recurrence type result for certain renormalization operators. We also consider the affine Cantor sets \begin{document}$K$\end{document} and \begin{document}$K'$\end{document} defined by two increasing maps which the product of their thicknesses is bigger than one. Then we construct a recurrent set for their renormalization operators. This leads us to characterize all \begin{document}$λ$\end{document} that \begin{document}$K- λ K'$\end{document} is a closed interval.

2018, 38(9): 4279-4304 doi: 10.3934/dcds.2018187 +[Abstract](1765) +[HTML](83) +[PDF](515.88KB)
Abstract:

We study the zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic (MHD) equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the resistivity coefficient, for the solutions to the viscous incompressible MHD equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible MHD equations converge to that of the ideal incompressible MHD equations as the viscosity and resistivity coefficients go to zero. Moreover, the convergence rate is also given.

2018, 38(9): 4305-4327 doi: 10.3934/dcds.2018188 +[Abstract](1688) +[HTML](87) +[PDF](863.58KB)
Abstract:

Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads to \begin{document}$Ω$\end{document}-stable flows on surfaces, which are not structurally stable. However, in the present paper we prove that a topological classification of such flows is also reduced to a combinatorial problem. Our complete topological invariant is a multigraph, and we present a polynomial-time algorithm for the distinction of such graphs up to an isomorphism. We also present a graph criterion for orientability of the ambient manifold and a graph-associated formula for its Euler characteristic. Additionally, we give polynomial-time algorithms for checking the orientability and calculating the characteristic.

2018, 38(9): 4329-4351 doi: 10.3934/dcds.2018189 +[Abstract](2348) +[HTML](106) +[PDF](469.17KB)
Abstract:

This paper deals with traveling wave solutions for time periodic reaction-diffusion systems. The existence of traveling wave solutions is established by combining the fixed point theorem with super- and sub-solutions, which reduces the existence of traveling wave solutions to the existence of super- and sub-solutions. The asymptotic behavior is determined by the stability of periodic solutions of the corresponding initial value problems. To illustrate the abstract results, we investigate a time periodic Lotka-Volterra system with two species by presenting the existence and nonexistence of traveling wave solutions, which connect the trivial steady state to the unique positive periodic solution of the corresponding kinetic system.

2018, 38(9): 4353-4390 doi: 10.3934/dcds.2018190 +[Abstract](1966) +[HTML](103) +[PDF](602.33KB)
Abstract:

For $n≥ 3$ and $p = (n+2)/(n-2),$ we consider the Hénon equation with the homogeneous Neumann boundary condition

where \begin{document}$Ω \subset B(0,1) \subset \mathbb{R}^n, n ≥ 3$, $α≥ 0$ and $\partial^*Ω \equiv \partialΩ \cap \partial B(0,1) \ne \emptyset.$\end{document} It is well known that for \begin{document}$α = 0,$\end{document} there exists a least energy solution of the problem. We are concerned on the existence of a least energy solution for \begin{document}$α > 0$\end{document} and its asymptotic behavior as the parameter \begin{document}$α$\end{document} approaches from below to a threshold \begin{document}$α_0 ∈ (0,∞]$\end{document} for existence of a least energy solution.

2018, 38(9): 4391-4419 doi: 10.3934/dcds.2018191 +[Abstract](2347) +[HTML](75) +[PDF](571.67KB)
Abstract:

Global asymptotic and exponential stability of equilibria for the following class of functional differential equations with distributed delay is investigated

We make our analysis by introducing a new approach, combining a Lyapunov functional and monotone semiflow theory. The relevance of our results is illustrated by studying the well-known integro-differential Nicholson's blowflies and Mackey-Glass equations, where some delay independent stability conditions are provided. Furthermore, new results related to exponential stability region of the positive equilibrium for these both models are established.

2018, 38(9): 4421-4431 doi: 10.3934/dcds.2018192 +[Abstract](1527) +[HTML](70) +[PDF](335.96KB)
Abstract:

We look at the maximal entropy measure (MME) of the boundaries of connected components of the Fatou set of a rational map of degree $≥ 2$. We show that if there are infinitely many Fatou components, and if either the Julia set is disconnected or the map is hyperbolic, then there can be at most one Fatou component whose boundary has positive MME measure. We also replace hyperbolicity by the more general hypothesis of geometric finiteness.

2018, 38(9): 4433-4447 doi: 10.3934/dcds.2018193 +[Abstract](1370) +[HTML](56) +[PDF](397.14KB)
Abstract:

In this article we give sufficient conditions for Komuro expansivity to imply the rescaled expansivity recently introduced by Wen and Wen. Also, we show that a flow on a compact metric space is expansive in the sense of Katok-Hasselblatt if and only if it is separating in the sense of Gura and the set of fixed points is open.

2018, 38(9): 4449-4465 doi: 10.3934/dcds.2018194 +[Abstract](1561) +[HTML](78) +[PDF](464.46KB)
Abstract:

The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system possessing \begin{document}$N$\end{document}-peakon weak solutions, for all \begin{document}$N≥ 1$\end{document}, in the setting of an integral formulation which is used in analysis for studying local well-posedness, global existence, and wave breaking for non-peakon solutions. Unlike the original Camassa-Holm equation, the two Hamiltonians of the mCH equation do not reduce to conserved integrals (constants of motion) for \begin{document}$2$\end{document}-peakon weak solutions. This perplexing situation is addressed here by finding an explicit conserved integral for \begin{document}$N$\end{document}-peakon weak solutions for all \begin{document}$N≥ 2$\end{document}. When \begin{document}$N$\end{document} is even, the conserved integral is shown to provide a Hamiltonian structure with the use of a natural Poisson bracket that arises from reduction of one of the Hamiltonian structures of the mCH equation. But when \begin{document}$N$\end{document} is odd, the Hamiltonian equations of motion arising from the conserved integral using this Poisson bracket are found to differ from the dynamical equations for the mCH \begin{document}$N$\end{document}-peakon weak solutions. Moreover, the lack of conservation of the two Hamiltonians of the mCH equation when they are reduced to \begin{document}$2$\end{document}-peakon weak solutions is shown to extend to \begin{document}$N$\end{document}-peakon weak solutions for all \begin{document}$N≥ 2$\end{document}. The connection between this loss of integrability structure and related work by Chang and Szmigielski on the Lax pair for the mCH equation is discussed.

2018, 38(9): 4467-4482 doi: 10.3934/dcds.2018195 +[Abstract](1859) +[HTML](75) +[PDF](389.59KB)
Abstract:

In this paper we generalize Katok's entropy formula to a large class of infinite countably amenable group actions.

2018, 38(9): 4483-4507 doi: 10.3934/dcds.2018196 +[Abstract](1677) +[HTML](65) +[PDF](604.48KB)
Abstract:

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.

2018, 38(9): 4509-4536 doi: 10.3934/dcds.2018197 +[Abstract](1379) +[HTML](63) +[PDF](495.33KB)
Abstract:

Let \begin{document} $\Omega$\end{document} be a smooth bounded axisymmetric set in \begin{document} $\mathbb{R}^3$\end{document}. In this paper we investigate the existence of minimizers of the so-called neo-Hookean energy among a class of axisymmetric maps. Due to the appearance of a critical exponent in the energy we must face a problem of lack of compactness. Indeed as shown by an example of Conti-De Lellis in [12,Section 6], a phenomenon of concentration of energy can occur preventing the strong convergence in \begin{document} $W^{1,2}(\Omega ,\mathbb{R}^3)$\end{document} of a minimizing sequence along with the equi-integrability of the cofactors of that sequence. We prove that this phenomenon can only take place on the axis of symmetry of the domain. Thus if we consider domains that do not contain the axis of symmetry then minimizers do exist. We also provide a partial description of the lack of compactness in terms of Cartesian currents. Then we study the case where \begin{document} $\Omega$\end{document} is not necessarily axisymmetric but the boundary data is affine. In that case if we do not allow cavitation (nor in the interior neither at the boundary) then the affine extension is the unique minimizer, that is, quadratic polyconvex energies are \begin{document} $W^{1,2}$\end{document}-quasiconvex in our admissible space. At last, in the case of an axisymmetric domain not containing its symmetry axis, we obtain for the first time the existence of weak solutions of the energy-momentum equations for 3D neo-Hookean materials.

2018, 38(9): 4537-4554 doi: 10.3934/dcds.2018198 +[Abstract](1453) +[HTML](68) +[PDF](435.54KB)
Abstract:

We consider the semilinear elliptic equation \begin{document} $Δ u + K(|x|)e^u = 0$\end{document} in \begin{document} $\mathbf{R}^N$\end{document} for \begin{document} $N > 2$\end{document}, and investigate separation phenomena of radial solutions. In terms of intersection and separation, we classify the solution structures and establish characterizations of the structures. These observations lead to sufficient conditions for partial separation. For \begin{document} $N = 10+4\ell$\end{document} with \begin{document} $\ell>-2$\end{document}, the equation changes its nature drastically according to the sign of the derivative of \begin{document} $r^{-\ell}K(r)$\end{document} when \begin{document} $r^{-\ell}K(r)$\end{document} is monotonic in \begin{document} $r$\end{document} and \begin{document} $r^{-\ell} K(r)\to1$\end{document} as \begin{document} $r\to∞$\end{document}.

2018, 38(9): 4555-4570 doi: 10.3934/dcds.2018199 +[Abstract](1401) +[HTML](76) +[PDF](397.82KB)
Abstract:

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. Let \begin{document} $P$\end{document} be a Borel probability measure on \begin{document} $\mathbb R$\end{document} such that \begin{document} $P = \frac 12 P\circ S_1^{-1}+\frac 12 P\circ S_2^{-1},$\end{document} where \begin{document} $S_1$\end{document} and \begin{document} $S_2$\end{document} are two contractive similarity mappings given by \begin{document} $S_1(x) = rx$\end{document} and \begin{document} $S_2(x) = rx+1-r$\end{document} for \begin{document} $0<r<\frac 12$\end{document} and \begin{document} $x∈ \mathbb R$\end{document}. Then, \begin{document} $P$\end{document} is supported on the Cantor set generated by \begin{document} $S_1$\end{document} and \begin{document} $S_2$\end{document}. The case \begin{document} $r = \frac 13$\end{document} was treated by Graf and Luschgy who gave an exact formula for the unique optimal quantization of the Cantor distribution \begin{document} $P$\end{document} (Math. Nachr., 183 (1997), 113-133). In this paper, we compute the precise range of \begin{document} $r$\end{document}-values to which Graf-Luschgy formula extends.

2018, 38(9): 4571-4602 doi: 10.3934/dcds.2018200 +[Abstract](1467) +[HTML](73) +[PDF](745.86KB)
Abstract:

The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal isotopy such that we can give a fine description of the dynamics of its transverse foliation. We will define a sort of identity isotopies: torsion-low isotopies. In particular, when \begin{document} $f$ \end{document} is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy \begin{document} $I$ \end{document} of \begin{document} $f$ \end{document} is torsion-low if and only if for every point \begin{document} $z$ \end{document} fixed along the isotopy, the (real) rotation number \begin{document} $ρ(I, z)$ \end{document} (which is well defined when one blows up \begin{document} $f$ \end{document} at \begin{document} $z$ \end{document}) is contained in \begin{document} $(-1, 1)$ \end{document}. We will prove the existence of torsion-low maximal isotopies, and will deduce the local dynamics of the transverse foliations of any torsion-low maximal isotopy near any isolated singularity.

2018, 38(9): 4603-4615 doi: 10.3934/dcds.2018201 +[Abstract](2226) +[HTML](77) +[PDF](393.64KB)
Abstract:

In this paper, we study symmetry properties of positive solutions to the fractional Laplace equation with negative powers on the whole space. We can use the direct method of moving planes introduced by Jarohs-Weth-Chen-Li-Li to prove one particular result below. If \begin{document} $u∈ C^{1, 1}_{loc}(\mathbb{R}^{n})\cap L_{α}$ \end{document} satisfies

with the growth/decay property

where \begin{document} $\frac{α}{β+1}<m<1$ \end{document}, \begin{document} $a>0$ \end{document} is a constant, then the positive solution \begin{document} $u(x)$ \end{document} must be radially symmetric about some point in \begin{document} $\mathbb{R}^{n}$ \end{document}. Similar result is also true for Hénon type nonlinear fractional Laplace equation with negative powers.

2018, 38(9): 4617-4635 doi: 10.3934/dcds.2018202 +[Abstract](1579) +[HTML](593) +[PDF](430.79KB)
Abstract:

We study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty, not only in the context of the Ericksen-Leslie model, is that the relative energy inequality is proved for a system with a nonconvex energy.

2018, 38(9): 4637-4655 doi: 10.3934/dcds.2018203 +[Abstract](1888) +[HTML](84) +[PDF](420.57KB)
Abstract:

In this paper, we investigate a hydrodynamic system that models the dynamics of incompressible magneto-viscoelastic flows. First, we prove the local well-posedness of the initial boundary value problem in the periodic domain. Then we establish a blow-up criterion in terms of the temporal integral of the maximum norm of the velocity gradient. Finally, an analog of the Beale-Kato-Majda criterion is derived.

2018, 38(9): 4657-4674 doi: 10.3934/dcds.2018204 +[Abstract](1492) +[HTML](83) +[PDF](473.36KB)
Abstract:

We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order. As an application, we present some results on the existence and stability of \begin{document}$T$\end{document}-periodic solutions of a Liebau-type equation.

2018, 38(9): 4675-4692 doi: 10.3934/dcds.2018205 +[Abstract](1372) +[HTML](130) +[PDF](1225.15KB)
Abstract:

In this paper we study a degenerate parabolic system of reaction-diffusion equations arising in cellular biology models. Its specificity lies in the fact that one of the concentrations does not diffuse. Under realistic conditions on the reaction term, we prove existence and uniqueness of a nonnegative solution to the considered system, and we study its regularity. Moreover, we discuss the existence and linear stability of the steady solutions (equilibria), and give a sufficient condition on the reaction term for Turing-like instabilities to be triggered. These results are finally illustrated by some numerical simulations.

2018, 38(9): 4693-4713 doi: 10.3934/dcds.2018206 +[Abstract](1548) +[HTML](57) +[PDF](421.1KB)
Abstract:

We study the convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential. Even without uniformly bounded distortion in this case, which makes the study much harder, we are still able to obtain a very nice estimation of the convergence speed under a certain quasi-gap condition.

2018, 38(9): 4715-4726 doi: 10.3934/dcds.2018207 +[Abstract](1484) +[HTML](106) +[PDF](360.31KB)
Abstract:

This paper deals with the periodic solutions of second order equations with asymptotical non-resonance. Using the point of view that the force is a perturbation, we can think that, asymptotically, the solutions of forced non-autonomous equation behave as those of the autonomous equation. Then, under a sharp integral condition, we prove that the periodic solution of non-autonomous equation can be estimated by using time map of autonomous equation. The existence of periodic solutions is thus proved via qualitative analysis and topological degree theory. The main result in this paper generalize a existence result obtained by Capietto, Mawhin and Zanolin.

2018, 38(9): 4727-4744 doi: 10.3934/dcds.2018208 +[Abstract](1694) +[HTML](73) +[PDF](417.37KB)
Abstract:

The question how chaotic is an almost mean equicontinuous system is addressed. It is shown that every topological dynamical system can be embedded into an almost mean equicontinuous system with the same entropy which is an almost one-to-one extension of some mean equicontinuous system. Besides, there is an almost mean equicontinous system that is topologically K and Devaney chaotic, and as this consequence we know that every ergodic measure of such a topologically K system does not have full support.

2018, 38(9): 4745-4765 doi: 10.3934/dcds.2018209 +[Abstract](1898) +[HTML](98) +[PDF](484.66KB)
Abstract:

Considering the stochastic 3-D incompressible anisotropic Navier-Stokes equations, we prove the local existence of strong solution in \begin{document}$H^2(\mathbb{T}^3)$\end{document}. Moreover, we express the probabilistic estimate of the random time interval for the existence of a local solution in terms of expected values of the initial data and the random noise, and establish the global existence of strong solution in probability if the initial data and the random noise are sufficiently small.

2018, 38(9): 4767-4817 doi: 10.3934/dcds.2018210 +[Abstract](2074) +[HTML](114) +[PDF](731.75KB)
Abstract:

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

2018  Impact Factor: 1.143