Stationary critical points of the heat flow in the plane

In previous papers [MS 1, 2], we considered stationary critical points of solutions of the initial-boundary value problems for the heat equation on bounded domains in ℝ^N,N ≧ 2. In [MS 1], we showed that a solutionu has a stationary critical pointO if and only ifu satisfies a certain balance law with respect toO for any time. Furthermore, we proved necessary and sufficient conditions relating the symmetry of the domain to the initial datau ₀; in this way, we gave a characterization of the ball in ℝ^N([MS 1]) and of centrosymmetric domains ([MS 2]). In the present paper, we consider a rotationA _dby an angle 2π/d,d ≧ 2 for planar domains and give some necessary and some sufficient conditions onu ₀ which relate to domains invariant underA _d. We also establish some conjectures. This research was partially supported by a Grant-in-Aid for Scientific Research (C) (# 10640175) and (B) (# 12440042) of the Japan Society for the Promotion of Science. The first author was supported also by the Italian MURST.     

Homogenization of Nonlinear Degenerate Non-monotone Elliptic Operators in Domains Perforated with Tiny Holes

The paper deals with the homogenization problem beyond the periodic setting, for a degenerated nonlinear non-monotone elliptic type operator on a perforated domain Ω ^ε in ℝ ^N with isolated holes. While the space variable in the coefficients a ₀ and a is scaled with size ε (ε>0 a small parameter), the system of holes is scaled with ε ² size, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The homogenization problem is formulated in terms of the general, so-called deterministic homogenization theory combining real homogenization algebras with the Σ-convergence method. We present a new approach based on the Besicovitch type spaces to solve deterministic homogenization problems, and we obtain a very general abstract homogenization results. We then illustrate this abstract setting by providing some concrete applications of these results to, e.g., the periodic homogenization, the almost periodic homogenization, and others.     

Hodge–Helmholtz decompositions of weighted Sobolev spaces in irregular exterior domains with inhomogeneous and anisotropic media

We study in detail Hodge–Helmholtz decompositions in nonsmooth exterior domains Ω??^N filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms of rank q belonging to the weighted L²‐space L_s^{2, q}(Ω), s∈?, into irrotational and solenoidal q‐forms. These decompositions are essential tools, for example, in electro‐magnetic theory for exterior domains. To the best of our knowledge, these decompositions in exterior domains with nonsmooth boundaries and inhomogeneous and anisotropic media are fully new results. In the Appendix, we translate our results to the classical framework of vector analysis N=3 and q=1, 2. Copyright © 2008 John Wiley & Sons, Ltd.     

Tautness and Fatou Components in ?2

Hyperbolicity played an important role in the classification of Fatou components for rational functions in the Riemann sphere. In higher dimensions Fatou components are not nearly as well understood. We investigate the Kobayashi completeness and tautness of invariant Fatou components for holomorphic endomorphisms of ?² and for Hénon mappings. We show that basins of attraction and domains with an attracting Riemann surface, previously known to be taut, are also complete, which is strictly stronger. We also prove tautness for a class of Siegel domains.     

Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems

Using Grozman’s formalism of invariant differential operators we demonstrate the derivation of N=2 Camassa-Holm equation from the action of Vect(S ^1|2) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N=2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H ¹-metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N=1 super peakon type equations, known as N=1 super b-field equations, are derived from the action of Vect(S ^1|1) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S ^1|1) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N=1 super b-field equations.     

Lipschitz Estimates in Almost‐Periodic Homogenization

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C^1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W^1,p estimates.© 2016 Wiley Periodicals, Inc.     

Polygonal invariant curves for a planar piecewise isometry

We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.

     

Minimal dynamical system with givenD-function and topological entropy

TheD-function is a new topological invariant introduced by the author in [3] to classify the minimal dynamical system and to generalize Sharkovskii's theorem on the coexistence of periodic orbits. We show that theD-function and the topological entropy are independent.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 287–292, February, 1993.     

A case of the dynamical Mordell–Lang conjecture

We prove a special case of a dynamical analogue of the classical Mordell–Lang conjecture. Specifically, let φ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of φ on (\mathbb P¹)^g{(\mathbb P^1)^g}. If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (\mathbb P¹)^g{(\mathbb P^1)^g} has only finite intersection with any curve contained in (\mathbb P¹)^g{(\mathbb P^1)^g}. We also show that our result holds for indecomposable polynomials φ with coefficients in \mathbb C{\mathbb C}. Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over \mathbb C{\mathbb C} uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (φ, φ) on \mathbb A²{\mathbb A^2}.     

Aubry‐Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ¹. It is well‐known that its smooth invariant curves correspond to smooth Z²‐periodic solutions of the PDE u_t + H(x, t, u)_x = 0. In this paper, we establish a connection between the Aubry‐Mather theory of invariant sets of the Hamiltonian system and Z²‐periodic weak solutions of this PDE by realizing the Aubry‐Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry‐Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry‐Mather set, defined in (2.24). The graph itself is a backward‐invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry‐Mather theory into the characteristic fields of the above PDE. This is done by making use of one‐ and two‐sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z²‐periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two‐sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.     

Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four

A class of degree four differential systems that have an invariant conic x ² + Cy ² = 1, C ∈ ℝ, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.     

Invariant curves of one-step methods

This paper considers the invariant sets of numerical one-step integration methods in a neighbourhood of a hyperbolic periodic solution of a nonlinear ODE. Using results from the dynamical systems theory it was possible to show that for the usual one-step methods the invariant sets areC ^k-circles (closed curves) for small enough stepsizeh. Here we give a direct proof for that and also show that they areO(h ^p)C ^k-close to the true periodic trajectory, wherep is the order of the method.Most of this work was done during the author's visit at the Mittag-Leffler Institute, Djursholm, Sweden, academic year 1985–86.     

Global asymptotic stability of a class of nonautonomous integro-differential systems and applications

In this paper, we study the global asymptotic stability of a class of nonautonomous integro-differential systems. By constructing suitable Lyapunov functionals, we establish new and explicit criteria for the global asymptotic stability in the sense of Definition 2.1. In the autonomous case, we discuss the global asymptotic stability of a unique equilibrium of the system, and in the case of periodic system, we establish sufficient criteria for existence, uniqueness and global asymptotic stability of a periodic solution. Also explored are applications of our main results to some biological and neural network models. The examples show that our criteria are more general and easily applicable, and improve and generalize some existing results.     

Invariant measure for a class of parabolic degenerate equations

In this paper we consider a stochastic flow in Rⁿ which leaves a closed convex set K invariant. By using a recent characterization of the invariance, involving the distance function, we study the corresponding transition semigroup P_t and its infinitesimal generator N. Due to the invariance property, N is a degenerate elliptic operator. We study existence of an invariant measure ν of P_t and the realization of N in L² (H, ν).     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.     

Low-order resonances in weakly dissipative spin-orbit models

Second-order differential equations with small nonlinearity and weak dissipation, such as the spin-orbit model of celestial mechanics, are considered. Explicit conditions for the coexistence of periodic orbits and estimates on the measure of the basins of attraction of stable periodic orbits are discussed.     

Multistability of neural networks with discontinuous activation function

In this paper, the multistability is studied for two-dimensional neural networks with multilevel activation functions. And it is showed that the system has n² isolated equilibrium points which are locally exponentially stable, where the activation function has n segments. Furthermore, evoked by periodic external input, n² periodic orbits which are locally exponentially attractive, can be found. And these results are extended to k-neuron networks, which is really enlarge the capacity of the associative memories. Examples and simulation results are used to illustrate the theory.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.