Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure
Todd Young
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 359-378 doi: 10.3934/dcds.2003.9.359
We consider Birkhoff averages of an observable $\phi$ along orbits of a continuous map $f:X \rightarrow X$ with respect to a non-invariant measure $m$. In the simple case where the averages converge $m$-almost everywhere, one may discuss the distribution of values of the average in a natural way. We extend this analysis to the case where convergence does not hold $m$-almost everywhere. The case that the averages converge $m$-almost everywhere is shown to be related to the recently defined notion of "predictable" behavior, which is a condition on the existence of pointwise asymptotic measures (SRB measures). A heteroclinic attractor is an example of a system which is not predictable. We define a more general notion called "statistically predictable" behavior which is weaker than predictability, but is strong enough to allow meaningful statistical properties, i.e. distribution of Birkhoff averages, to be analyzed. Statistical predictability is shown to imply the existence of an asymptotic measure, but not vice versa. We investigate the relationship between the various notions of asymptotic measures and distributions of Birkhoff average. Analysis of the heteroclinic attractor is used to illustrate the applicability of the concepts.
keywords: historical behavior statistically predictable. Predictable dynamics
Intermittency and Jakobson's theorem near saddle-node bifurcations
Ale Jan Homburg Todd Young
Discrete & Continuous Dynamical Systems - A 2007, 17(1): 21-58 doi: 10.3934/dcds.2007.17.21
We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to an atomic measure supported on the orbit of the saddle-node point. Using these measures we analyze the intermittent time series that result from the destruction of the periodic attractor in the saddle-node bifurcation and prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the periodic attractor.
keywords: absolutely continuous invariant measure saddle node bifurcation nonuniform hyperbolicity.
A result in global bifurcation theory using the Conley index
Todd Young
Discrete & Continuous Dynamical Systems - A 1996, 2(3): 387-396 doi: 10.3934/dcds.1996.2.387
We study a saddle-node bifurcation in a Lipschitz family of diffeomorphisms on a manifold, in the case that the stable set and unstable set of the fixed point intersect transversally in a countable collection of one-dimensional manifolds diffeomorphic to circles. We formulate generic conditions on the circles stated in terms of standard coordinates, a recently defined tool for the study of saddle-node bifurcations. Under the conditions, it is shown that there is a decreasing sequence of intervals $[\underline{\mu_j},\overline{\mu_j}]$ of parameter values for which the diffeomorphism is semi-conjugated to shift dynamics on the space of binary sequences. The semi-conjugacy is implied by a recent result in the Conley index theory.
keywords: global bifurcations. global dynamics
Partially hyperbolic sets from a co-dimension one bifurcation
Todd Young
Discrete & Continuous Dynamical Systems - A 1995, 1(2): 253-275 doi: 10.3934/dcds.1995.1.253
We study the saddle-node bifurcation of a partially hyperbolic fixed point in a Lipschitz family of $C^{k}$ diffeomorphisms on a Banach manifold (possibly infinite dimensional) in the case that the fixed point is a saddle along hyperbolic directions and has multiple curves of homoclinic orbits. We show that this bifurcation results in an invariant set which consists of a countable collection of closed invariant curves and an uncountable collection of nonclosed invariant curves which are the topological limits of the closed curves. In addition, it is shown that these curves are $C^k$-smooth and that this invariant set is uniformly partially hyperbolic.
keywords: saddle-node bifurcation homoclinic orbits. diffeomorphisms partially hyperbolic fixed point

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