Discrete and Continuous Dynamical Systems
June 2018 , Volume 38 , Issue 6
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The paper is primarily concerned with the asymptotic behavior as
We study partially hyperbolic sets of
We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to
Starting from the stochastic Cucker-Smale model introduced in [
We present a non-trivial lower bound for the critical coupling strength to the Cucker-Smale model with unit speed constraint and short-range communication weight from the viewpoint of a mono-cluster(global) flocking. For a long-range communication weight, the critical coupling strength is zero in the sense that the mono-cluster flocking emerges from any initial configurations for any positive coupling strengths, whereas for a short-range communication weight, a mono-cluster flocking can emerge from an initial configuration only for a sufficiently large coupling strength. Our main interest lies on the condition of non-flocking. We provide a positive lower bound for the critical coupling strength. We also present numerical simulations for the upper and lower bounds for the critical coupling strength depending on initial configurations and compare them with analytical results.
In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation
In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.
The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.
In this paper we consider the system
The present paper is devoted to the compressible nematic liquid crystal flow in the whole space
We study supersonic flow past a convex corner which is surrounded by quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a strong rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to separate the supersonic flow behind the rarefaction wave from the static gas. In this paper, we employ a wave front tracking method to establish structural stability of such a flow pattern under non-smooth perturbations of the upcoming supersonic flow. It is an initial-value/free-boundary problem for the two-dimensional steady non-isentropic compressible Euler system. The main ingredients are careful analysis of wave interactions and construction of suitable Glimm functional, to overcome the difficulty that the strong rarefaction wave has a large total variation.
In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving Hardy-Leray potential
We classify the isolated singularities and obtain the existence and stability of positive solutions of (1). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.
We define a broad class of piecewise smooth plane homeomorphisms which have properties similar to the properties of Lozi maps, including the existence of a hyperbolic attractor. We call those maps Lozi-like. For those maps one can apply our previous results on kneading theory for Lozi maps. We show a strong numerical evidence that there exist Lozi-like maps that have kneading sequences different than those of Lozi maps.
This paper is devoted to study the front propagation for a class of discrete periodic monostable equations with delay and nonlocal interaction. We first establish the existence of rightward and leftward spreading speeds and prove their coincidence with the minimal wave speeds of the pulsating traveling fronts in the right and left directions, respectively. The dependency of the speeds of propagation on the heterogeneity of the medium and the delay term is also investigated. We find that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium; while the delay decreases the invasion speed. Further, we prove the uniqueness of all noncritical pulsating traveling fronts. Finally, we show that all noncritical pulsating traveling fronts are globally exponentially stable, as long as the initial perturbations around them are uniformly bounded in a weight space.
In this paper we are concerned with the existence of positive high energy solutions of the Choquard equation. Under certain assumptions, the ground state of Choquard equation does not exist. However, by global compactness analysis, we prove that there exists a positive high energy solution.
We consider a phase-field system modeling phase transition phenomena, where the Cahn-Hilliard-Oono equation for the order parameter is coupled with the Coleman-Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.
A coupled parabolic-hyperbolic system of partial differential equations modeling the interaction of a structure submerged in a fluid is studied. The system being considered incorporates delays in the interaction on the interface between the fluid and the solid. We study the stability properties of the interaction model under suitable assumptions between the competing strengths of the delays and the feedback controls.
We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point
Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE). We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for other infinite-dimensional small noise stochastic equations.
We are interested in the existence of sign-changing multi-bump solutions for the following Kirchhoff equation
In this paper we study, for certain problems in the calculus of variations and optimal control, two different questions related to uniqueness of multipliers appearing in first order necessary conditions. One deals with conditions under which a given multiplier associated with an extremal of a fixed function is unique, a property which, in nonlinear programming, is known to be equivalent to the strict Mangasarian-Fromovitz constraint qualification. We show that, for isoperimetric problems in the calculus of variations, a similar characterization holds, but not in optimal control where the corresponding condition is only sufficient for the uniqueness of the multiplier. The other question is related to the set of multipliers associated with all functions for which a solution to the constrained problem is given. We prove that, for both types of problems, this set is a singleton if and only if a strong normality assumption holds.
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