scholarly journals A new characterization of nonisotropic chaotic vibrations of the one-dimensional linear wave equation with a van der Pol boundary condition

2003 ◽  
Vol 288 (1) ◽  
pp. 78-96 ◽  
Author(s):  
Yu Huang
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Linear Wave ◽  
Van Der Pol ◽  
Linear Wave Equation ◽  
One Dimensional ◽  
Chaotic Vibrations ◽  
The One

RAPID FLUCTUATIONS OF SNAPSHOTS OF ONE-DIMENSIONAL LINEAR WAVE EQUATION WITH A VAN DER POL NONLINEAR BOUNDARY CONDITION

2005 ◽  
Vol 15 (02) ◽  
pp. 567-580 ◽  
Author(s):  
YU HUANG ◽  
JUN LUO ◽  
ZUOLING ZHOU
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Dynamical Behavior ◽  
Linear Wave ◽  
Van Der Pol ◽  
Linear Wave Equation ◽  
Energy Pumping ◽  
Long Run

In this paper, we consider a linear wave equation on an interval with a van der Pol nonlinear boundary condition at one end and an energy-pumping condition at the other end. We study the dynamical behavior of the Riemann invariants (u,v) of the wave equation in terms of the growth rates of the total variations of the snapshots on the spatial interval. Our main contributions here are the detection of rapid fluctuations of the snapshots of u and v in the long run. The results here sharpen those in the earlier works of [Chen et al., 2001] and [Huang, 2003].

Estimates for the controls of the wave equation with a potential

2018 ◽  
Vol 24 (1) ◽  
pp. 289-309 ◽  
Author(s):  
Sorin Micu ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Space Variable ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
One Dimensional ◽  
Boundary Controls ◽  
The One ◽  
Variable Potential ◽  
Uniformly Bounded

This article studies the L2-norm of the boundary controls for the one dimensional linear wave equation with a space variable potential a = a(x). It is known these controls depend on a and their norms may increase exponentially with ||a||L∞. Our aim is to make a deeper study of this dependence in correlation with the properties of the initial data. The main result of the paper shows that the minimal L2−norm controls are uniformly bounded with respect to the potential a, if the initial data have only sufficiently high eigenmodes.

scholarly journals Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

2020 ◽  
Vol 64 (6) ◽  
pp. 657-662
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Initial Condition ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

Chaotic Vibrations of the One-Dimensional Wave Equation Due to a Self-Excitation Boundary Condition.

1998 ◽  
Vol 08 (03) ◽  
pp. 423-445 ◽  
Author(s):  
Goong Chen ◽  
Sze-Bi Hsu ◽  
Jianxin Zhou
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Period Doubling ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Boundary Conditions ◽  
Homoclinic Orbits ◽  
Left Endpoint ◽  
Linear Wave ◽  
Van Der Pol

Consider the initial-boundary value problem of the linear wave equation wtt-wxx=0 on an interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system, while the boundary condition at the right endpoint has cubic nonlinearity of a van der Pol type. We show that the interactions of these linear and nonlinear boundary conditions can cause chaos to the Riemann invariants (u,v) of the wave equation when the parameters enter a certain regime. Period-doubling routes to chaos and homoclinic orbits are established. We further show that when the initial data are smooth satisfying certain compatibility conditions at the boundary points, the space-time trajectory or the state of the wave equation, which satisfies another type of the van der Pol boundary condition, can be chaotic. Numerical simulations are also illustrated.

A linear wave equation with a nonlinear boundary condition of viscoelastic type

2010 ◽  
Vol 72 (3-4) ◽  
pp. 1865-1885
Author(s):  
Le Thi Phuong Ngoc ◽  
Nguyen Anh Triet ◽  
Nguyen Thanh Long
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Linear Wave ◽  
Linear Wave Equation

PERTURBATION OF THE CHAOTIC 1D MIXED WAVE SYSTEM CAUSES THE UNBOUNDED GROWTH OF THE SNAPSHOTS OF THE GRADIENT

2011 ◽  
Vol 21 (03) ◽  
pp. 685-701
Author(s):  
CHUNG-CHE HU
Keyword(s):  
Boundary Condition ◽  
Linear Displacement ◽  
Wave System ◽  
Cubic Nonlinearity ◽  
Van Der Pol ◽  
One Dimensional ◽  
Chaotic Vibration ◽  
Mixed Wave ◽  
The One ◽  
The Right

Consider the one-dimensional mixed wave equation on a unit interval, where the left-end boundary condition is linear, pumping energy into the system, while the right-end boundary condition is self-regulating of the van der Pol type with a cubic nonlinearity. First, we show a certain parameter range of the chaotic vibration of the system. Furthermore, if the right-end van der Pol boundary contains an extra linear displacement feedback term, then we show that under some suitable conditions, the system is still chaotic in the sense of unbounded growth of the snapshots of the gradient.

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