120,083 research outputs found
Unifying discrete and continuous Weyl-Titchmarsh theory via a class of linear Hamiltonian systems on Sturmian time scales
In this study, we are concerned with introducing Weyl-Titchmarsh theory for a
class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian
time scales. After developing fundamental properties of solutions and regular
spectral problems, we introduce the corresponding maximal and minimal operators
for the system. Matrix disks are constructed and proved to be nested and
converge to a limiting set. Some precise relationships among the rank of the
matrix radius of the limiting set, the number of linearly independent square
summable solutions, and the defect indices of the minimal operator are
established. Using the above results, a classification of singular dynamic
linear Hamiltonian nabla systems is given in terms of the defect indices of the
minimal operator, and several equivalent conditions on the cases of limit point
and limit circle are obtained, respectively. These results unify and extend
certain classic and recent results on the subject in the continuous and
discrete cases, respectively, to Sturmian time scales.Comment: 34 page
Nonlinear hybrid-mode resonant forced oscillations of sagged inclined cables at avoidances
We investigate non-linear forced oscillations of sagged inclined cables under planar 1:1 internal resonance at avoidance. To account for frequency avoidance phenomena and associated hybrid modes actually distinguishing inclined cables from horizontal cables, asymmetric inclined static configurations are considered. Emphasis is placed on highlighting nearly tuned 1:1 resonant interactions involving coupled hybrid modes. The inclined cable is subjected to a uniformly distributed vertical harmonic excitation at primary resonance of a high-frequency mode. Approximate non-linear partial-differential equations of motion, capturing overall displacement coupling and dynamic extensibility effect, are analytically solved based on a multi-mode discretization and a second-order multiple scales approach. Bifurcation analyses of both equilibrium and dynamic solutions are carried out via a continuation technique, highlighting the influence of system parameters on internally resonant forced dynamics of avoidance cables. Direct numerical integrations of modulation equations are also performed to validate the continuation prediction and characterize non-linear coupled dynamics in post-bifurcation states. Depending on the elasto-geometric (cable sag and inclination) and control parameters, and on assigned initial conditions, the hybrid modal interactions undergo several kinds of bifurcations and non-linear phenomena, along with meaningful transition from periodic to quasi-periodic and chaotic responses. Moreover, corresponding spatio-temporal distributions of cable non-linear dynamic displacement and tension are manifested
Sturm-Liouville operators on time scales
We establish the connection between Sturm-Liouville equations on time scales
and Sturm--Liouville equations with measure-valued coefficients. Based on this
connection we generalize several results for Sturm-Liouville equations on time
scales which have been obtained by various authors in the past.Comment: 12 page
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables
This study aims at comparing non-linear modal interactions in shallow horizontal cables with kinematically non-condensed vs. condensed modeling, under simultaneous primary external and internal resonances. Planar 1:1 or 2:1 internal resonance is considered. The governing partial-differential equations of motion of non-condensed model account for spatio-temporal modification of dynamic tension, and explicitly capture non-linear coupling of longitudinal/ vertical displacements. On the contrary, in the condensed model, a single integro-differential equation is obtained by eliminating the longitudinal inertia according to a quasi-static cable stretching assumption, which entails spatially uniform dynamic tension. This model is largely considered in the literature. Based on a multi-modal discretization and a second-order multiple scales solution accounting for higher-order quadratic effects of a infinite number of modes, coupled/uncoupled dynamic responses and the associated stability are evaluated by means of frequency- and force-response diagrams. Direct numerical integrations confirm the occurrence of amplitude-steady or -modulated responses. Non-linear dynamic configurations and tensions are also examined. Depending on internal resonance condition, system elasto-geometric and control parameters, the condensed model may lead to significant quantitative and/or qualitative discrepancies, against the non-condensed model, in the evaluation of resonant dynamic responses, bifurcations and maximal/minimal stresses. Results of even shallow cables reveal meaningful drawbacks of the kinematic condensation and allow us to detect cases where the more accurate non-condensed model has to be used
Dissipative Time Evolution of Observables in Non-equilibrium Statistical Quantum Systems
We discuss differential-- versus integral--equation based methods describing
out--of thermal equilibrium systems and emphasize the importance of a well
defined reduction to statistical observables. Applying the projection operator
approach, we investigate on the time evolution of expectation values of linear
and quadratic polynomials in position and momentum for a statistical anharmonic
oscillator with quartic potential. Based on the exact integro-differential
equations of motion, we study the first and naive second order approximation
which breaks down at secular time-scales. A method is proposed to improve the
expansion by a non--perturbative resummation of all quadratic operator
correlators consistent with energy conservation for all times. Motion cannot be
described by an effective Hamiltonian local in time reflecting non-unitarity of
the dissipative entropy generating evolution. We numerically integrate the
consistently improved equations of motion for large times. We relate entropy to
the uncertainty product, both being expressible in terms of the observables
under consideration.Comment: 20 pages, 6 Figure
A geometric method for model reduction of biochemical networks with polynomial rate functions
Model reduction of biochemical networks relies on the knowledge of slow and
fast variables. We provide a geometric method, based on the Newton polytope, to
identify slow variables of a biochemical network with polynomial rate
functions. The gist of the method is the notion of tropical equilibration that
provides approximate descriptions of slow invariant manifolds. Compared to
extant numerical algorithms such as the intrinsic low dimensional manifold
method, our approach is symbolic and utilizes orders of magnitude instead of
precise values of the model parameters. Application of this method to a large
collection of biochemical network models supports the idea that the number of
dynamical variables in minimal models of cell physiology can be small, in spite
of the large number of molecular regulatory actors
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