1,601 research outputs found
Reversors and Symmetries for Polynomial Automorphisms of the Plane
We obtain normal forms for symmetric and for reversible polynomial
automorphisms (polynomial maps that have polynomial inverses) of the plane. Our
normal forms are based on the generalized \Henon normal form of Friedland and
Milnor. We restrict to the case that the symmetries and reversors are also
polynomial automorphisms. We show that each such reversor has finite-order, and
that for nontrivial, real maps, the reversor has order 2 or 4. The normal forms
are shown to be unique up to finitely many choices. We investigate some of the
dynamical consequences of reversibility, especially for the case that the
reversor is not an involution.Comment: laTeX with 5 figures. Added new sections dealing with symmetries and
an extensive discussion of the reversing symmetry group
Efficient Computation of Invariant Tori in Volume-Preserving Maps
In this paper we implement a numerical algorithm to compute codimension-one
tori in three-dimensional, volume-preserving maps. A torus is defined by its
conjugacy to rigid rotation, which is in turn given by its Fourier series. The
algorithm employs a quasi-Newton scheme to find the Fourier coefficients of a
truncation of the series. This technique is based upon the theory developed in
the accompanying article by Blass and de la Llave. It is guaranteed to converge
assuming the torus exists, the initial estimate is suitably close, and the map
satisfies certain nondegeneracy conditions. We demonstrate that the growth of
the largest singular value of the derivative of the conjugacy predicts the
threshold for the destruction of the torus. We use these singular values to
examine the mechanics of the breakup of the tori, making comparisons to
Aubry-Mather and anti-integrability theory when possible
Heteroclinic orbits and transport in a perturbed integrable Suris map
Explicit formulae are given for the saddle connection of an integrable family
of standard maps studied by Y. Suris. When the map is perturbed this connection
is destroyed, and we use a discrete version of Melnikov's method to give an
explicit formula for the first order approximation of the area of the lobes of
the resultant turnstile. These results are compared with computations of the
lobe area.Comment: laTex file with 6 eps figure
Characterizing and modeling the dynamics of online popularity
Online popularity has enormous impact on opinions, culture, policy, and
profits. We provide a quantitative, large scale, temporal analysis of the
dynamics of online content popularity in two massive model systems, the
Wikipedia and an entire country's Web space. We find that the dynamics of
popularity are characterized by bursts, displaying characteristic features of
critical systems such as fat-tailed distributions of magnitude and inter-event
time. We propose a minimal model combining the classic preferential popularity
increase mechanism with the occurrence of random popularity shifts due to
exogenous factors. The model recovers the critical features observed in the
empirical analysis of the systems analyzed here, highlighting the key factors
needed in the description of popularity dynamics.Comment: 5 pages, 4 figures. Modeling part detailed. Final version published
in Physical Review Letter
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