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.

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].

scholarly journals ANALYZING DISPLACEMENT TERM'S MEMORY EFFECT IN A VAN DER POL TYPE BOUNDARY CONDITION TO PROVE CHAOTIC VIBRATION OF THE WAVE EQUATION

2002 ◽  
Vol 12 (05) ◽  
pp. 965-981 ◽  
Author(s):  
GOONG CHEN ◽  
SZE-BI HSU ◽  
TINGWEN HUANG
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Memory Effect ◽  
Homoclinic Orbits ◽  
Linear Displacement ◽  
Parameter Range ◽  
Perturbation Approach ◽  
Van Der Pol ◽  
Chaotic Vibration ◽  
The Right

Consider the one-dimensional 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. Then for a certain parameter range it is now known that chaotic vibration occurs. However, if the right-end van der Pol boundary condition contains an extra linear displacement feedback term, then it induces a memory effect and considerable technical difficulty arises as to how to define and determine chaotic vibration of the system. In this paper, we take advantage of the extra margin property of the reflection map and utilize properties of homoclinic orbits coupled with a perturbation approach to show that for a small parameter range, chaotic vibrations occur in the sense of unbounded growth of snapshots of the gradient. The work also has significant implications to the occurrence of chaotic vibration for the wave equation on a 3D annular domain.

CHAOTIC VIBRATIONS OF THE ONE-DIMENSIONAL MIXED WAVE SYSTEM

2009 ◽  
Vol 19 (02) ◽  
pp. 579-590 ◽  
Author(s):  
CHUNG-CHE HU
Keyword(s):  
Boundary Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Left Endpoint ◽  
Wave System ◽  
One Dimensional ◽  
Interval Map ◽  
Odd Degree ◽  
Mixed Wave ◽  
The One

In this paper, we consider the initial-boundary value problem of the one-dimensional linear mixed wave equation ωtt - dωtx - c2ωxx = 0 (d ∈ ℝ, c > 0) on an interval, where the boundary condition at the left endpoint is linear, pumping energy into the system, while the boundary condition at the right endpoint has odd-degree nonlinearity. This problem is said to be the one-dimensional mixed wave system. The solution of the one-dimensional mixed wave system corresponds to the iteration of an interval map h. Thus, the mixed wave system is said to be chaotic if the interval map h is chaotic in the sense of Li–Yorke. In this paper, we show that the mixed wave system is chaotic under some conditions.

A shock problem involving a nonlinear viscoelastic bar associated with a nonlinear boundary condition

2008 ◽  
Vol 41 (1) ◽  
Author(s):  
Nguyen Thanh Long ◽  
Vo Giang Giai ◽  
Le Xuan Truong
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Boundary Value Problem ◽  
Nonlinear Wave Equation ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Viscoelastic ◽  
Initial Boundary Value

AbstractWe study the initial-boundary value problem for a nonlinear wave equation given by

scholarly journals CLASSICAL SOLUTION FOR INITIAL–BOUNDARY VALUE PROBLEM FOR WAVE EQUATION WITH INTEGRAL BOUNDARY CONDITION

2012 ◽  
Vol 17 (3) ◽  
pp. 309-329 ◽  
Author(s):  
Victor Korzyuk ◽  
Victor Erofeenko ◽  
Julia Sheika
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Boundary Value Problem ◽  
Classical Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
Initial Boundary Value

The unique existence of classical solution of initial–boundary value problem for wave equation with a special integral boundary condition is proved in the work. Classical solution of the problem in analytical form is also found in the article. This problem arises at the modeling of electromagnetic fields with arbitrary time dependence when interaction between the field and solids is simulated with impedance boundary conditions.

scholarly journals A Wave Equation Associated with Mixed Nonhomogeneous Conditions: The Compactness and Connectivity of Weak Solution Set

2007 ◽  
Vol 2007 ◽  
pp. 1-17
Author(s):  
Nguyen Thanh Long ◽  
Le Thi Phuong Ngoc
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Boundary Value Problem ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Initial Boundary Value

The purpose of this paper is to show that the set of weak solutions of the initial-boundary value problem for the linear wave equation is nonempty, connected, and compact.

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