24,552 research outputs found
Toda Lattice Solutions of Differential-Difference Equations for Dissipative Systems
In a certain class of differential-difference equations for dissipative
systems, we show that hyperbolic tangent model is the only the nonlinear system
of equations which can admit some particular solutions of the Toda lattice. We
give one parameter family of exact solutions, which include as special cases
the Toda lattice solutions as well as the Whitham's solutions in the Newell's
model. Our solutions can be used to describe temporal-spatial density patterns
observed in the optimal velocity model for traffic flow.Comment: Latex, 13 pages, 1 figur
Periodic Travelling Waves in Dimer Granular Chains
We study bifurcations of periodic travelling waves in granular dimer chains
from the anti-continuum limit, when the mass ratio between the light and heavy
beads is zero. We show that every limiting periodic wave is uniquely continued
with respect to the mass ratio parameter and the periodic waves with the
wavelength larger than a certain critical value are spectrally stable.
Numerical computations are developed to study how this solution family is
continued to the limit of equal mass ratio between the beads, where periodic
travelling waves of granular monomer chains exist
Collective phase description of oscillatory convection
We formulate a theory for the collective phase description of oscillatory
convection in Hele-Shaw cells. It enables us to describe the dynamics of the
oscillatory convection by a single degree of freedom which we call the
collective phase. The theory can be considered as a phase reduction method for
limit-cycle solutions in infinite-dimensional dynamical systems, namely, stable
time-periodic solutions to partial differential equations, representing the
oscillatory convection. We derive the phase sensitivity function, which
quantifies the phase response of the oscillatory convection to weak
perturbations applied at each spatial point, and analyze the phase
synchronization between two weakly coupled Hele-Shaw cells exhibiting
oscillatory convection on the basis of the derived phase equations.Comment: 16 pages, 4 figures, to appear in Chao
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
convergence result for nonlocal elliptic-type problems via Tartar's method
In this work we obtain a compactness result for the convergence of a
family of nonlocal and nonlinear monotone elliptic-type problems by means of
Tartar's method of oscillating test functions.Comment: In this revision we added a new section that shows the
Gamma-convergence of the associated energy functional
- âŚ