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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Chaotic Vibrations
of the One-Dimensional Wave Equation
Due to a Self-Excitation Boundary Condition
Part I: Controlled hysteresis


Authors: Goong Chen, Sze-Bi Hsu, Jianxin Zhou, Guanrong Chen and Giovanni Crosta
Journal: Trans. Amer. Math. Soc. 350 (1998), 4265-4311
MSC (1991): Primary 35L05, 35L70, 58F39, 70L05.
MathSciNet review: 1443867
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Abstract: The study of nonlinear vibrations/oscillations in mechanical and electronic systems has always been an important research area. While important progress in the development of mathematical chaos theory has been made for finite dimensional second order nonlinear ODEs arising from nonlinear springs and electronic circuits, the state of understanding of chaotic vibrations for analogous infinite dimensional systems is still very incomplete.

The 1-dimensional vibrating string satisfying $w_{tt}- w_{xx}=0$ on the unit interval $x \in (0,1)$ is an infinite dimensional harmonic oscillator. Consider the boundary conditions: at the left end $x=0$, the string is fixed, while at the right end $x=1$, a nonlinear boundary condition $w_{x}= \alpha w_t - \beta w_{t}^{3}, \ \alpha, \beta>0$, takes effect. This nonlinear boundary condition behaves like a van der Pol oscillator, causing the total energy to rise and fall within certain bounds regularly or irregularly. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary condition. Since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates, the problem is reduced to a discrete iteration problem of the type $u_{n+1}=F(u_n)$, where $F$ is the nonlinear reflection relation. We say that the PDE system is chaotic if the mapping $F$ is chaotic as an interval map. Algebraic, asymptotic and numerical techniques are developed to tackle the cubic nonlinearities. We then define a rotation number, following J.P. Keener , and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of M.I. Malkin to determine the chaotic regime of $\alpha$ for the nonlinear reflection relation $F$, thereby rigorously proving chaos. Nonchaotic cases for other values of $\alpha$ are also classified. Such cases correspond to limit cycles in nonlinear second order ODEs. Numerical simulations of chaotic and nonchaotic vibrations are illustrated by computer graphics.


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Additional Information

Goong Chen
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: gchen@math.tamu.edu

Sze-Bi Hsu
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C.
Email: sbhsu@am.nthu.edu.tw

Jianxin Zhou
Affiliation: Department of Mathematics, Texas A& M University, College Station, Texas 77843
Email: jzhou@math.tamu.edu

Guanrong Chen
Affiliation: Department of Electrical Engineering University of Houston, Houston, Texas 77204-4793
Email: chengr@tree.egr.uh.edu

Giovanni Crosta
Affiliation: Department of Environmental Science, University of Milan, Milan I-20126, Italy
Email: giovanni@alpha.disat.unimi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02022-4
PII: S 0002-9947(98)02022-4
Received by editor(s): July 20, 1995
Received by editor(s) in revised form: October 16, 1996
Additional Notes: The first and third authors’ work was supported in part by NSF Grant DMS 9404380, Texas ARP Grant 010366-046, and Texas A&M University Interdisciplinary Research Initiative IRI 96-39. Work completed while the first author was on sabbatical leave at the Institute of Applied Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C. The second author’s work was supported in part by Grant NSC 83-0208-M-007-003 from the National Council of Science of the Republic of China.
Article copyright: © Copyright 1998 American Mathematical Society