1,036 research outputs found
Coupled skinny baker's maps and the Kaplan-Yorke conjecture
The Kaplan-Yorke conjecture states that for "typical" dynamical systems with
a physical measure, the information dimension and the Lyapunov dimension
coincide. We explore this conjecture in a neighborhood of a system for which
the two dimensions do not coincide because the system consists of two uncoupled
subsystems. We are interested in whether coupling "typically" restores the
equality of the dimensions. The particular subsystems we consider are skinny
baker's maps, and we consider uni-directional coupling. For coupling in one of
the possible directions, we prove that the dimensions coincide for a prevalent
set of coupling functions, but for coupling in the other direction we show that
the dimensions remain unequal for all coupling functions. We conjecture that
the dimensions prevalently coincide for bi-directional coupling. On the other
hand, we conjecture that the phenomenon we observe for a particular class of
systems with uni-directional coupling, where the information and Lyapunov
dimensions differ robustly, occurs more generally for many classes of
uni-directionally coupled systems (also called skew-product systems) in higher
dimensions.Comment: 33 pages, 3 figure
Stretched exponential estimates on growth of the number of periodic points for prevalent diffeomorphisms, I.
For diffeomorphisms of smooth compact finite-dimensional manifolds, we consider the problem of how fast the number of periodic points with period grows as a function of . In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for or smoother diffeomorphisms. In the present work we show that, by contrast, for a measure-theoretic notion of genericity we call "prevalence", the growth is not much faster than exponential. Specifically, we show that for each , there is a prevalent set of (or smoother) diffeomorphisms for which the number of periodic points is bounded above by for some independent of . We also obtain a related bound on the decay of hyperbolicity of the periodic points as a function of , and obtain the same results for -dimensional endomorphisms. The contrast between topologically generic and measure-theoretically generic behavior for the growth of the number of periodic points and the decay of their hyperbolicity show this to be a subtle and complex phenomenon, reminiscent of KAM theory. Here in Part I we state our results and describe the methods we use. We complete most of the proof in the -dimensional -smooth case and outline the remaining steps, deferred to Part II, that are needed to establish the general case.
The novel feature of the approach we develop in this paper is the introduction of Newton Interpolation Polynomials as a tool for perturbing trajectories of iterated maps
Comment on "Long Time Evolution of Phase Oscillator Systems" [Chaos 19,023117 (2009), arXiv:0902.2773]
A previous paper (arXiv:0902.2773, henceforth referred to as I) considered a
general class of problems involving the evolution of large systems of globally
coupled phase oscillators. It was shown there that, in an appropriate sense,
the solutions to these problems are time asymptotically attracted toward a
reduced manifold of system states (denoted M). This result has considerable
utility in the analysis of these systems, as has been amply demonstrated in
recent papers. In this note, we show that the analysis of I can be modified in
a simple way that establishes significant extensions of the range of validity
of our previous result. In particular, we generalize I in the following ways:
(1) attraction to M is now shown for a very general class of oscillator
frequency distribution functions g(\omega), and (2) a previous restriction on
the allowed class of initial conditions is now substantially relaxed
Synchronization in large directed networks of coupled phase oscillators
We extend recent theoretical approximations describing the transition to
synchronization in large undirected networks of coupled phase oscillators to
the case of directed networks. We also consider extensions to networks with
mixed positive/negative coupling strengths. We compare our theory with
numerical simulations and find good agreement
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