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Discrete and Continuous Dynamical Systems

June 2013 , Volume 33 , Issue 6

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Global dynamics for symmetric planar maps
Begoña Alarcón, Sofia B. S. D. Castro and Isabel S. Labouriau
2013, 33(6): 2241-2251 doi: 10.3934/dcds.2013.33.2241 +[Abstract](2353) +[PDF](332.6KB)
We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a compact Lie group, it is possible to describe the local dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This allows us to use results based on the theory of free homeomorphisms to describe the global dynamical behaviour. We briefly discuss the case when reflections are absent, for which global dynamics may not follow from local dynamics near the unique fixed point.
Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle
David Burguet and Todd Fisher
2013, 33(6): 2253-2270 doi: 10.3934/dcds.2013.33.2253 +[Abstract](2814) +[PDF](449.8KB)
We relate the symbolic extension entropy of a partially hyperbolic dynamical system to the entropy appearing at small scales in local center manifolds. In particular, we prove the existence of symbolic extensions for $\mathcal{C}^2$ partially hyperbolic diffeomorphisms with a $2$-dimensional center bundle. 200 words.
Existence of smooth solutions to coupled chemotaxis-fluid equations
Myeongju Chae, Kyungkeun Kang and Jihoon Lee
2013, 33(6): 2271-2297 doi: 10.3934/dcds.2013.33.2271 +[Abstract](4179) +[PDF](473.5KB)
We consider a system coupling the parabolic-parabolic chemotaxis equations to the incompressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criterions. For two dimensional chemotaxis-Navier-Stokes equations, regular solutions constructed locally in time are, in reality, extended globally under some assumptions pertinent to experimental observations in [21] on the consumption rate and chemotactic sensitivity. We also show the existence of global weak solutions in spatially three dimensions with stronger restriction on the consumption rate and chemotactic sensitivity.
Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas
Jann-Long Chern, Zhi-You Chen and Yong-Li Tang
2013, 33(6): 2299-2318 doi: 10.3934/dcds.2013.33.2299 +[Abstract](2433) +[PDF](421.7KB)
Arising from one-particle distribution functions of stationary dissipative plasmas, we consider a coupled elliptic system with singular data in the plane. The existence and uniqueness of solutions to the Dirichlet boundary value problem are proved. In addition, the structure of other solutions, including blow-up solutions, is also clarified.
Schauder estimates for a class of non-local elliptic equations
Hongjie Dong and Doyoon Kim
2013, 33(6): 2319-2347 doi: 10.3934/dcds.2013.33.2319 +[Abstract](3763) +[PDF](508.8KB)
We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or Hölder continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.
Thermal runaway for a nonlinear diffusion model in thermal electricity
Lili Du and Mingshu Fan
2013, 33(6): 2349-2368 doi: 10.3934/dcds.2013.33.2349 +[Abstract](3787) +[PDF](503.5KB)
In this paper, we consider the phenomena of the thermal runaway and the asymptotic runaway in a nonlocal nonlinear model, which is raised from the thermal-electricity and it is so-called an Ohmic heating model. The model prescribes the dimensionless temperature when the electric current flows through two conductors, subject to a fixed potential difference. The electrical resistivity of the one of the conductors depends on the temperature and the other one remains constant. The problem will be mathematically formulated to a quasilinear nonlocal parabolic equation with Dirichlet boundary condition. An analysis of the problem shows that the solution of the problem exists globally, provided that the conductor with constant resistivity is connected. Furthermore, for some special temperature-resistivity relations, the unique stationary solution is shown to be global asymptotically stable. The results assert a physical fact that the thermal produced by the Ohmic heating process will runaway from the surfaces of the conductor, the temperature of the conductor remains bounded and solution of the system converges asymptotically to the unique equilibrium.
Unique periodic orbits of a delay differential equation with piecewise linear feedback function
Ábel Garab
2013, 33(6): 2369-2387 doi: 10.3934/dcds.2013.33.2369 +[Abstract](3766) +[PDF](428.2KB)
In this paper we study the scalar delay differential equation \linebreak $\dot{x}(t)=-ax(t) + bf(x(t-\tau))$ with feedback function $f(\xi)=\frac{1}{2}(|\xi+1|-|\xi-1|)$ and with real parameters $a>0,\ \tau>0$ and $b\neq 0$, which can model a single neuron or a group of synchronized neurons. We give necessary and sufficient conditions for existence and uniqueness of periodic orbits with prescribed oscillation frequencies. We also investigate the period of the slowly oscillating periodic solution as a function of the delay. Based on the obtained results we state an analogous theorem concerning existence and uniqueness of periodic orbits of a certain type of system of delay differential equations. The proofs are based among others on theory of monotone systems and discrete Lyapunov functionals.
On the stability of standing waves of Klein-Gordon equations in a semiclassical regime
Marco Ghimenti, Stefan Le Coz and Marco Squassina
2013, 33(6): 2389-2401 doi: 10.3934/dcds.2013.33.2389 +[Abstract](3281) +[PDF](396.4KB)
We investigate the orbital stability and instability of standing waves for two classes of Klein-Gordon equations in the semi-classical regime.
Rényi entropy and recurrence
Milton Ko
2013, 33(6): 2403-2421 doi: 10.3934/dcds.2013.33.2403 +[Abstract](2741) +[PDF](428.1KB)
This paper studies the relationship between the return time $\tau_n$ and the Rényi Entropy Function of order $s$, $R(s)$. For a dynamical system with an invariant $\alpha$-mixing measure $\mu$ and a measurable partition, we consider the sum $W$ of measures of cylinders along orbit segments of length $\tau_n$ and relate that growth/decay rate to the R$\acute{\textrm{e}}$nyi Entropy. The key strategy is to introduce the hitting number $\nu_x(A) = | \{1 \leq i \leq \tau_n(x) : T^i(x) \in A\}|$, the number of times that $x$ hits the set $A$ when $x$ travels along its orbit of length $\tau_n(x)$, and write $W=\sum \nu_x(A) \mu(A)^s$, where the sum is taken over the $n$-cylinders. Then we show that $\nu_x(A) \approx \exp(n h_{\mu}) \mu(A)$ for most $n$-cylinders $A$. Hence $W \approx \exp(nh_{\mu}) \sum \mu(A)^{1+s}$, which relates $\tau_n(x)$ to $R(s)$, as the sum $\sum \mu(A)^{1+s} \approx \exp(-nsR(s))$.
Global dynamics of the nonradial energy-critical wave equation above the ground state energy
Joachim Krieger, Kenji Nakanishi and Wilhelm Schlag
2013, 33(6): 2423-2450 doi: 10.3934/dcds.2013.33.2423 +[Abstract](3622) +[PDF](562.4KB)
In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions $3$ and $5$ assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in $\dot H^{1}\times L^{2}$ with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as $t → ±∞$.
Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts
Dominik Kwietniak
2013, 33(6): 2451-2467 doi: 10.3934/dcds.2013.33.2451 +[Abstract](3787) +[PDF](430.6KB)
Positive topological entropy and distributional chaos are characterized for hereditary shifts. A hereditary shift has positive topological entropy if and only if it is DC$2$-chaotic (or equivalently, DC$3$-chaotic) if and only if it is not uniquely ergodic. A hereditary shift is DC$1$-chaotic if and only if it is not proximal (has more than one minimal set). As every spacing shift and every beta shift is hereditary the results apply to those classes of shifts. Two open problems on topological entropy and distributional chaos of spacing shifts from an article of Banks et al. are solved thanks to this characterization. Moreover, it is shown that a spacing shift $\Omega_P$ has positive topological entropy if and only if $\mathbb{N}\setminus P$ is a set of Poincaré recurrence. Using a result of Kříž an example of a proximal spacing shift with positive entropy is constructed. Connections between spacing shifts and difference sets are revealed and the methods of this paper are used to obtain new proofs of some results on difference sets.
Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term
Sophia Th. Kyritsi and Nikolaos S. Papageorgiou
2013, 33(6): 2469-2494 doi: 10.3934/dcds.2013.33.2469 +[Abstract](2795) +[PDF](503.6KB)
We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator. We assume that the Carathéodory reaction term $f(z,x)$ exhibits an asymmetric behavior on the two semiaxes of $\mathbb{R}$. Namely, $f(z,\cdot)$ is $(p-1)$-linear near $-\infty$ and $(p-1)$-superlinear near $+\infty$, but without satisfying the well-known Ambrosetti--Rabinowitz condition (AR-condition). Combining variational methods based on critical point theory, with suitable truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative).
Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations
Meng Liu and Ke Wang
2013, 33(6): 2495-2522 doi: 10.3934/dcds.2013.33.2495 +[Abstract](4754) +[PDF](690.3KB)
This paper is concerned with two $n$-species stochastic cooperative systems. One is autonomous, the other is non-autonomous. For the first system, we prove that for each species, there is a constant which can be represented by the coefficients of the system. If the constant is negative, then the corresponding species will go to extinction with probability 1; If the constant is positive, then the corresponding species will be persistent with probability 1. For the second system, sufficient conditions for stochastic permanence and global attractivity are established. In addition, the upper- and lower-growth rates of the positive solution are investigated. Our results reveal that, firstly, the stochastic noise of one population is unfavorable for the persistence of all species; secondly, a population could be persistent even the growth rate of this population is less than the half of the intensity of the white noise.
Ergodicity of certain cocycles over certain interval exchanges
David Ralston and Serge Troubetzkoy
2013, 33(6): 2523-2529 doi: 10.3934/dcds.2013.33.2523 +[Abstract](2781) +[PDF](284.9KB)
We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.
Global well-posedness of the Chern-Simons-Higgs equations with finite energy
Sigmund Selberg and Achenef Tesfahun
2013, 33(6): 2531-2546 doi: 10.3934/dcds.2013.33.2531 +[Abstract](3302) +[PDF](420.7KB)
We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global well-posedness for more regular data. Moreover, we prove local well-posedness even below the energy regularity, using the the null structure of the system in Lorenz gauge and bilinear space-time estimates for wave-Sobolev norms.
Localized Birkhoff average in beta dynamical systems
Bo Tan, Bao-Wei Wang, Jun Wu and Jian Xu
2013, 33(6): 2547-2564 doi: 10.3934/dcds.2013.33.2547 +[Abstract](3221) +[PDF](478.1KB)
In this note, we investigate the localized multifractal spectrum of Birkhoff average in the beta-dynamical system $([0,1], T_{\beta})$ for general $\beta>1$, namely the dimension of the following level sets $$ \Big\{x\in [0,1]: \lim_{n\to \infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(T^jx)=f(x)\Big\}, $$ where $f$ and $\psi$ are two continuous functions defined on the unit interval $[0,1]$. Instead of a constant function in the classical multifractal cases, the function $f$ here varies with $x$. The method adopted in the proof indicates that the multifractal analysis of Birkhoff average in a general $\beta$-dynamical system can be achieved by approximating the system by its subsystems.
Unsteady flows of non-Newtonian fluids in generalized Orlicz spaces
Aneta Wróblewska-Kamińska
2013, 33(6): 2565-2592 doi: 10.3934/dcds.2013.33.2565 +[Abstract](3503) +[PDF](553.0KB)
Our purpose is to show existence of weak solutions to unsteady flow of non-Newtonian incompressible nonhomogeneous fluids with nonstandard growth conditions of the stress tensor. We are motivated by the fluids of anisotropic behaviour and characterised by rapid shear thickening. Since we are interested in flows with the rheology more general than power-law-type, we describe the growth conditions with the help of an $x$--dependent convex function and formulate our problem in generalized Orlicz spaces.
On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems
Junxiang Xu
2013, 33(6): 2593-2619 doi: 10.3934/dcds.2013.33.2593 +[Abstract](3142) +[PDF](443.8KB)
In this paper we consider two-dimensional nonlinear quasi-periodic system with small perturbations. Assume that the unperturbed system has a hyperbolic-type degenerate equilibrium point and the frequency satisfies the Diophantine conditions. Using the KAM iteration we prove that for sufficiently small perturbations, the system can be reduced by a nonlinear quasi-periodic transformation to a suitable normal form with an equilibrium point at the origin. Hence, for the system we can obtain a small quasi-periodic solution.

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




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