We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. By replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both for random forcing by measure-preserving dynamical systems and for deterministic forcing by homeomorphisms of compact metric spaces. Additional assumptions like ergodicity or minimality of the forcing process then yield further information about the dynamics.The main difference to the unforced situation is that at the critical bifurcation parameter, two alternatives exist. In addition to the possibility of a unique neutral invariant graph, corresponding to a neutral fixed point, a pair of so-called pinched invariant graphs may occur. In quasiperiodically forced systems, these are often referred to as 'strange non-chaotic attractors'. The results on deterministic forcing can be considered as an extension of the work of Novo, Núñez, Obaya and Sanz on nonautonomous convex scalar differential equations. As a by-product, we also give a generalisation of a result by Sturman and Stark on the structure of minimal sets in forced systems.
Abstract. Let T be the angle-doubling map on the circle T, and consider the 1-parameter family of piecewise-linear cosine functions f θ : T → R, defined by f θ (x) = 1 − 4d T (x, θ). We identify the maximizing T -invariant measures for this family: for each θ the f θ -maximizing measure is unique and Sturmian (i.e. with support contained in some closed semi-circle). For rational p/q, we give an explicit formula for the set of functions in the family whose maximizing measure is the Sturmian measure of rotation number p/q. This allows us to analyse the variation with θ of the maximum ergodic average for f θ .
Abstract. For a given beta-shift, the lexicographic order induces a partial order (known as first-order stochastic dominance) on the collection of its shift-invariant probability measures. We characterise those beta-shifts for which this partial order has a largest element. These beta-shifts are all of finite type, and their lexicographically largest point is a periodic sequence of a particular kind: it is Sturmian (i.e. its shift-orbit is combinatorially equivalent to a rotation) with weight-per-symbol either an integer, or equal to p/(ap + 1) for some a, p ≥ 1, or equal to A + p/(p + 1) for some p ≥ 1, A ≥ 2. In these cases, the largest invariant measure is precisely the unique one supported by the shift-orbit of the lexicographically largest point in the beta-shift.
Inspired by an example of Grebogi et al [1], we study a class of model systems which exhibit the full two-step scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold [2]. The specific structure of these models allows a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant 'generalised torus' splitting off a previously stable central manifold after the second bifurcation point.The scenario is described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew product systems on a compact product space. Secondly, we treat randomly forced systems, which lead to skew products over a measure-preserving base transformation. In the random case, a semiuniform ergodic theorem for random dynamical systems is required, to make up for the lack of compactness.(1.7)is the global attractor outside Θ × {0}, in the sense thatfor all sufficiently small δ > 0.2 That is, homeomorphic to the closed unit disk D = {z ∈ C | |z| ≤ 1}.
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