38,763 research outputs found
Stability of real parametric polynomial discrete dynamical systems
We extend and improve the existing characterization of the dynamics of
general quadratic real polynomial maps with coefficients that depend on a
single parameter , and generalize this characterization to cubic real
polynomial maps, in a consistent theory that is further generalized to real
-th degree real polynomial maps. In essence, we give conditions for the
stability of the fixed points of any real polynomial map with real fixed
points. In order to do this, we have introduced the concept of Canonical
Polynomial Maps which are topologically conjugate to any polynomial map of the
same degree with real fixed points. The stability of the fixed points of
canonical polynomial maps has been found to depend solely on a special function
termed Product Position Function for a given fixed point. The values of this
product position determine the stability of the fixed point in question, when
it bifurcates, and even when chaos arises, as it passes through what we have
termed stability bands. The exact boundary values of these stability bands are
yet to be calculated for regions of type greater than one for polynomials of
degree higher than three.Comment: 23 pages, 4 figures, now published in Discrete Dynamics in Nature and
Societ
Metastable States in Spin Glasses and Disordered Ferromagnets
We study analytically M-spin-flip stable states in disordered short-ranged
Ising models (spin glasses and ferromagnets) in all dimensions and for all M.
Our approach is primarily dynamical and is based on the convergence of a
zero-temperature dynamical process with flips of lattice animals up to size M
and starting from a deep quench, to a metastable limit. The results (rigorous
and nonrigorous, in infinite and finite volumes) concern many aspects of
metastable states: their numbers, basins of attraction, energy densities,
overlaps, remanent magnetizations and relations to thermodynamic states. For
example, we show that their overlap distribution is a delta-function at zero.
We also define a dynamics for M=infinity, which provides a potential tool for
investigating ground state structure.Comment: 34 pages (LaTeX); to appear in Physical Review
Discrete Symmetry and Stability in Hamiltonian Dynamics
In this tutorial we address the existence and stability of periodic and
quasiperiodic orbits in N degree of freedom Hamiltonian systems and their
connection with discrete symmetries. Of primary importance in our study are the
nonlinear normal modes (NNMs), i.e periodic solutions which represent
continuations of the system's linear normal modes in the nonlinear regime. We
examine the existence of such solutions and discuss different methods for
constructing them and studying their stability under fixed and periodic
boundary conditions. In the periodic case, we employ group theoretical concepts
to identify a special type of NNMs called one-dimensional "bushes". We describe
how to use linear combinations such NNMs to construct s(>1)-dimensional bushes
of quasiperiodic orbits, for a wide variety of Hamiltonian systems and exploit
the symmetries of the linearized equations to simplify the study of their
destabilization. Applying this theory to the Fermi Pasta Ulam (FPU) chain, we
review a number of interesting results, which have appeared in the recent
literature. We then turn to an analytical and numerical construction of
quasiperiodic orbits, which does not depend on the symmetries or boundary
conditions. We demonstrate that the well-known "paradox" of FPU recurrences may
be explained in terms of the exponential localization of the energies Eq of
NNM's excited at the low part of the frequency spectrum, i.e. q=1,2,3,....
Thus, we show that the stability of these low-dimensional manifolds called
q-tori is related to the persistence or FPU recurrences at low energies.
Finally, we discuss a novel approach to the stability of orbits of conservative
systems, the GALIk, k=2,...,2N, by means of which one can determine accurately
and efficiently the destabilization of q-tori, leading to the breakdown of
recurrences and the equipartition of energy, at high values of the total energy
E.Comment: 50 pages, 13 figure
On resilient control of dynamical flow networks
Resilience has become a key aspect in the design of contemporary
infrastructure networks. This comes as a result of ever-increasing loads,
limited physical capacity, and fast-growing levels of interconnectedness and
complexity due to the recent technological advancements. The problem has
motivated a considerable amount of research within the last few years,
particularly focused on the dynamical aspects of network flows, complementing
more classical static network flow optimization approaches. In this tutorial
paper, a class of single-commodity first-order models of dynamical flow
networks is considered. A few results recently appeared in the literature and
dealing with stability and robustness of dynamical flow networks are gathered
and originally presented in a unified framework. In particular, (differential)
stability properties of monotone dynamical flow networks are treated in some
detail, and the notion of margin of resilience is introduced as a quantitative
measure of their robustness. While emphasizing methodological aspects --
including structural properties, such as monotonicity, that enable tractability
and scalability -- over the specific applications, connections to
well-established road traffic flow models are made.Comment: accepted for publication in Annual Reviews in Control, 201
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
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