528 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
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
Transport in Transitory Dynamical Systems
We introduce the concept of a "transitory" dynamical system---one whose
time-dependence is confined to a compact interval---and show how to quantify
transport between two-dimensional Lagrangian coherent structures for the
Hamiltonian case. This requires knowing only the "action" of relevant
heteroclinic orbits at the intersection of invariant manifolds of "forward" and
"backward" hyperbolic orbits. These manifolds can be easily computed by
leveraging the autonomous nature of the vector fields on either side of the
time-dependent transition. As illustrative examples we consider a
two-dimensional fluid flow in a rotating double-gyre configuration and a simple
one-and-a-half degree of freedom model of a resonant particle accelerator. We
compare our results to those obtained using finite-time Lyapunov exponents and
to adiabatic theory, discussing the benefits and limitations of each method.Comment: Updated and corrected version. LaTeX, 29 pages, 21 figure
Transport in Transitory, Three-Dimensional, Liouville Flows
We derive an action-flux formula to compute the volumes of lobes quantifying
transport between past- and future-invariant Lagrangian coherent structures of
n-dimensional, transitory, globally Liouville flows. A transitory system is one
that is nonautonomous only on a compact time interval. This method requires
relatively little Lagrangian information about the codimension-one surfaces
bounding the lobes, relying only on the generalized actions of loops on the
lobe boundaries. These are easily computed since the vector fields are
autonomous before and after the time-dependent transition. Two examples in
three-dimensions are studied: a transitory ABC flow and a model of a
microdroplet moving through a microfluidic channel mixer. In both cases the
action-flux computations of transport are compared to those obtained using
Monte Carlo methods.Comment: 30 pages, 16 figures, 1 table, submitted to SIAM J. Appl. Dyn. Sy
Chaos in a well : Effects of competing length scales
A discontinuous generalization of the standard map, which arises naturally as
the dynamics of a periodically kicked particle in a one dimensional infinite
square well potential, is examined. Existence of competing length scales,
namely the width of the well and the wavelength of the external field,
introduce novel dynamical behaviour. Deterministic chaos induced diffusion is
observed for weak field strengths as the length scales do not match. This is
related to an abrupt breakdown of rotationally invariant curves and in
particular KAM tori. An approximate stability theory is derived wherein the
usual standard map is a point of ``bifurcation''.Comment: 15 pages, 5 figure
Quantum Breaking Time Scaling in the Superdiffusive Dynamics
We show that the breaking time of quantum-classical correspondence depends on
the type of kinetics and the dominant origin of stickiness. For sticky dynamics
of quantum kicked rotor, when the hierarchical set of islands corresponds to
the accelerator mode, we demonstrate by simulation that the breaking time
scales as with the transport exponent
that corresponds to superdiffusive dynamics. We discuss also other
possibilities for the breaking time scaling and transition to the logarithmic
one with respect to
New Class of Eigenstates in Generic Hamiltonian Systems
In mixed systems, besides regular and chaotic states, there are states
supported by the chaotic region mainly living in the vicinity of the hierarchy
of regular islands. We show that the fraction of these hierarchical states
scales as and relate the exponent to the
decay of the classical staying probability . This is
numerically confirmed for the kicked rotor by studying the influence of
hierarchical states on eigenfunction and level statistics.Comment: 4 pages, 3 figures, Phys. Rev. Lett., to appea
Quantum Poincar\'e Recurrences
We show that quantum effects modify the decay rate of Poincar\'e recurrences
P(t) in classical chaotic systems with hierarchical structure of phase space.
The exponent p of the algebraic decay P(t) ~ 1/t^p is shown to have the
universal value p=1 due to tunneling and localization effects. Experimental
evidence of such decay should be observable in mesoscopic systems and cold
atoms.Comment: revtex, 4 pages, 4 figure
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