Asymptotic behavior and periodic forcing in time for the solutions of a nonlinear system representing a phase change with dissipation

1996 ◽  
pp. 2281-2288
Keyword(s):  
Asymptotic Behavior ◽  
Nonlinear System ◽  
Phase Change ◽  
Periodic Forcing

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO CONSERVATION LAWS WITH PERIODIC FORCING

2006 ◽  
Vol 03 (02) ◽  
pp. 387-401 ◽  
Author(s):  
DEBORA AMADORI ◽  
DENIS SERRE
Keyword(s):  
Asymptotic Behavior ◽  
Steady State ◽  
Initial Data ◽  
Conservation Law ◽  
Inflection Point ◽  
Global Minimum ◽  
Asymptotic Behavior Of Solutions ◽  
Periodic Forcing ◽  
Scalar Conservation Law ◽  
Scalar Conservation

We consider the asymptotic behavior of the solution of a forced scalar conservation law, where both the forcing and the initial data are periodic. We prove that there exists a steady state toward which the solution converges in the L1 norm, as t → ∞. We do not assume any smallness or smoothness of the initial data. The limit steady state can be discontinuous, as effect of resonance, and it can be identified when the potential of the forcing term has a unique global minimum, thanks to the conservation of mass. The flux is assumed to be strictly convex; the relevance of this assumption is justified by the construction of solutions with a lattice of periods for a flux with an inflection point.

ON SOLVABILITY OF A NONLINEAR DISCRETE SYSTEM IN THE SPREAD THEORY OF INFECTION

2020 ◽  
Vol 54 (2 (252)) ◽  
pp. 87-95
Author(s):  
M.H. Avetisyan
Keyword(s):  
Asymptotic Behavior ◽  
Nonlinear System ◽  
Special Class ◽  
Discrete System ◽  
Mathematical Theory ◽  
Algebraic Equations ◽  
Uniqueness Of The Solution ◽  
Nonlinear Discrete System ◽  
Bounded Sequences ◽  
Temporal Spread

In this paper a special class of infinite nonlinear system of algebraic equations with Teoplitz matrix is studied. The mentioned system arises in the mathematical theory of the spatial temporal spread of the epidemic. The existence and the uniqueness of the solution in the space of bounded sequences are proved. It is studied also the asymptotic behavior of the constructed solution at infinity. At the end of the work specific examples are given.

scholarly journals Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Nasser-Eddine Tatar
Keyword(s):  
Asymptotic Behavior ◽  
Wave Equation ◽  
Nonlinear System ◽  
Initial Energy ◽  
Viscoelastic Damping ◽  
Unbounded Coefficients ◽  
Kirchhoff Type ◽  
Usual Manner

A wave equation of the Kirchhoff type with several nonlinearities is stabilized by a viscoelastic damping. We consider the case of nonconstant (and unbounded) coefficients. This is a nondissipative case, and as a consequence the nonlinear terms cannot be estimated in the usual manner by the initial energy. We suggest a way to get around this difficulty. It is proved that if the solution enters a certain region, which we determine, then it will be attracted exponentially by the equilibrium.

STRONG SENSITIVITY OF THE VIBRATIONAL RESONANCE INDUCED BY FRACTAL STRUCTURES

2013 ◽  
Vol 23 (07) ◽  
pp. 1350129 ◽  
Author(s):  
ALVAR DAZA ◽  
ALEXANDRE WAGEMAKERS ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Phase Space ◽  
Nonlinear System ◽  
Oscillatory Regime ◽  
Fractal Structures ◽  
Periodic Forcing ◽  
Vibrational Resonance ◽  
Extreme Sensitivity

We consider a nonlinear system perturbed by two harmonic forcings of different frequencies. The slow forcing drives the system into an oscillatory regime while the fast perturbation enhances the effect of the slow periodic drive. The vibrational resonance occurs when this enhancement is optimal, usually when the fast perturbation has an amplitude much higher than the slow periodic forcing. We show that this resonance can also happen when the amplitude of the fast perturbation is far below the amplitude of the slow periodic forcing due to a peculiar condition of the phase space. Moreover, this resonance presents an extreme sensitivity to small variations of the fast perturbation. We explore here this phenomenon that we call ultrasensitive vibrational resonance.

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