Mathematics

This note is an attempt with the open conjecture 8 proposed by the authors of\cite{G. LADAS} which states: Assume α,β,λ?�[0,?? . Then every positive solution of the difference equation : z n+1 = α+ z n β+ z n?? λ z n?? ,n=0,1,??is bounded if and only if β=λ . We will use a construction of sub-energy function and properties of Todd's difference equation to disprove that conjecture in general.

We use techniques of proof mining to obtain a computable and uniform rate of metastability (in the sense of Tao) for the mean ergodic theorem for a finite number of commuting linear contractive operators on a uniformly convex Banach space.

In this paper, we consider the asymptotic behavior of traveling wave solutions of the degenerate nonlinear parabolic equation: u t = u p ( u xx +u)−δu ( δ=0 or 1 ) for ξ≡x−ct→−∞ with c>0 . We give a refined one of them, which was not obtain in the preceding work [Ichida-Sakamoto, 2020], by an appropriate asymptotic study and properties of the Lambert W function.

We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of Hörmander's hypoelliptic regularity theory in an L 1 framework which estimates this (degenerate) Fisher information from below by an $W^{1,s}_{\loc}$ Sobolev norm. This method is applicable to a wide range of systems beyond the reach of currently existing mathematically rigorous methods. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE; in this paper we prove that this class includes the Lorenz 96 model in any dimension, provided the additive stochastic driving is applied to any consecutive pair of modes.

Marcus stochastic delay differential equations (SDDEs) are often used to model stochastic dynamical systems with memory in science and engineering. Since no infinitesimal generators exist for Marcus SDDEs due to the non-Markovian property, conventional Fokker-Planck equations, which govern the evolution behavior of density, are not available for Marcus SDDEs. In this paper, we identify the Marcus SDDE with some Marcus stochastic differential equation (SDE) without delays but subject to extra constraints. This provides an efficient way to establish existence and uniqueness for the solution, and obtain a representation formula for probability density of the Marcus SDDE. In the formula, the probability density for Marcus SDDE is expressed in terms of that for Marcus SDE without delay.

In this paper, we consider a mass-spring-friction oscillator with the friction modelled by a regularized stiction model in the limit where the ratio of the natural spring frequency and the forcing frequency is on the same order of magnitude as the scale associated with the regularized stiction model. The motivation for studying this situation comes from \cite{bossolini2017b} which demonstrated new friction phenomena in this regime. The results of Bossolini et al 2017 led to some open problems, that we resolve in this paper. In particular, using GSPT and blowup we provide a simple geometric description of the bifurcation of stick-slip limit cycles through a combination of a canard and a global return mechanism. We also show that this combination leads to a canard-based horseshoe and are therefore able to prove existence of chaos in this fundamental oscillator system.

We explicitly construct a strongly aperiodic subshift of finite type for the discrete Heisenberg group. Our example builds on the classical aperiodic tilings of the plane due to Raphael Robinson. Extending those tilings to the Heisenberg group by exploiting the group's structure and posing additional local rules to prune out remaining periodic behavior we maintain a rich projective subdynamics on Z 2 cosets. In addition the obtained subshift is an almost 1-to-1 extension of a strongly aperiodic, minimal sofic shift. As a consequence of our construction we establish the undecidability of the emptiness as well as the extension problem for shifts of finite type on the Heisenberg group.

We consider piecewise monotone maps, we show that an ergodic measure for which the map is invertible almost everywhere can not be mixing. It follows that every ergodic measure for an interval translation mapping is not mixing. We also show that double rotations without periodic points have an ergodic but not weakly mixing invariant measure. This article is dedicated to the memory of Anatoly Mikhailovich Stepin.

In this paper, we study the SRB measures of generalized horseshoe map. We prove that under the conditions of transversality and fatness, the SRB measure is actually absolutely continuous with respect to the Lebesgue measure.

Stable accessibility for partially hyperbolic diffeomorphisms is central to their ergodic theory, and we establish its \(C^1\)-density among 1. all, 2. volume-preserving, 3. symplectic, and 4. contact partially hyperbolic flows. As applications, we obtain in each of these 4 categories \(C^1\)-density of \(C^1\)-stable topological transitivity, ergodicity, and triviality of the centralizer.