Nonautonomous attractors and Young measures

Motivated by the problem of identifying a mathematical framework for the formal definition of concepts such as weather, climate and connections between them, we discuss a question of convergence of short-time time averages for random nonautonomous dynamical systems depending on a parameter. The problem is formulated by means of Young measures. Using the notion of pull-back attractor, we prove a general theorem giving a sufficient condition for the tightness of the law of the approximating problems. In a specific example, we show that the theorem applies and we characterize the unique limit point.

Singularly Perturbed Oscillators with Exponential Nonlinearities

Singular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ ϵ > 0 small occur in many different applications. These terms have essential singularities for $$\epsilon =0$$ ϵ = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $$\epsilon \rightarrow 0$$ ϵ → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for both systems—existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

Global Dynamics of a Planar Filippov System with Symmetry

Chen [2016a, 2016b] studied global dynamics of the Filippov systems [Formula: see text], respectively. To study the global dynamics of [Formula: see text] completely, since the dynamics of [Formula: see text] is very simple, we are only interested in the global dynamics of [Formula: see text] in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system [Formula: see text] is totally different from systems [Formula: see text].

A Discontinuous ODE Model of the Glacial Cycles with Diffusive Heat Transport

We present a new discontinuous ordinary differential equation (ODE) model of the glacial cycles. Model trajectories flip from a glacial to an interglacial state, and vice versa, via a switching mechanism motivated by ice sheet mass balance principles. Filippov’s theory of differential inclusions is used to analyze the system, which can be viewed as a nonsmooth geometric singular perturbation problem. We prove the existence of a unique limit cycle, corresponding to the Earth’s glacial cycles. The diffusive heat transport component of the model is ideally suited for investigating the competing temperature gradient and transport efficiency feedbacks, each associated with ice-albedo feedback. It is the interplay of these feedbacks that determines the maximal extent of the ice sheet. In the nonautonomous setting, model glacial cycles persist when subjected to external forcing brought on by changes in Earth’s orbital parameters over geologic time. The system also exhibits various bifurcation scenarios as key parameters vary.

Admissibility of a solution to generalized Chaplygin gas

It is known that there is a solution to the Riemann problem for generalized Chaplygin gas model and that it contains the Dirac delta function in some cases. In some cases, usual admissible criteria can not extract a unique weak solution as it was shown in [4]. The aim of this paper is to use a solution to perturbed generalized Chaplygin model by a small constant ?? > 0 and obtain a its unique limit. A weak solution to the unperturbed system that equals that limit is called admissible. The perturbation is made by using the modified model of Chaplygin gas defined in [5].

EQUIVALENT DEFINITIONS OF SUPERSTABILITY IN TAME ABSTRACT ELEMENTARY CLASSES

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:(1)For all high-enough λ,Khas no long splitting chains.(2)For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.(3)For all high-enough λ,Khas a unique limit model of cardinality λ.(4)For all high-enough λ,Khas a superlimit model of cardinality λ.(5)For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.(6)There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

On the Limit Cycles for a Class of Continuous Piecewise Linear Differential Systems with Three Zones

Lima and Llibre [2012] have studied a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, they proved that this class admits always a unique limit cycle, which is hyperbolic. The class studied in [Lima & Llibre, 2012] belongs to a larger set of planar continuous piecewise linear vector fields with three zones that can be separated into four other classes. Here, we consider some of these classes and we prove that some of them always admit a unique limit cycle, which is hyperbolic. However we find a class that does not have limit cycles.

FRAÏSSÉ LIMITS OF METRIC STRUCTURES

We develop Fraïssé theory, namely the theory of Fraïssé classes and Fraïssé limits, in the context of metric structures. We show that a class of finitely generated structures is Fraïssé if and only if it is the age of a separable approximately homogeneous structure, and conversely, that this structure is necessarily the unique limit of the class, and is universal for it.We do this in a somewhat new approach, in which “finite maps up to errors” are coded by approximate isometries.