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 is an element of (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(wt) - 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 a for the nonlinear reflection relation F, thereby rigorously proving chaos. Nonchaotic cases for other values of a 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.

Chen, G., Hsu, S., Zhou, J., Chen, G., Crosta, G. (1998). Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition - Part I: Controlled hysteresis. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 350(11), 4265-4311 [10.1090/S0002-9947-98-02022-4].

### Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition - Part I: Controlled hysteresis

#### 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 is an element of (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(wt) - 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 a for the nonlinear reflection relation F, thereby rigorously proving chaos. Nonchaotic cases for other values of a 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.
##### Scheda breve Scheda completa Scheda completa (DC) Articolo in rivista - Articolo scientifico
nonlinear oscillations; infinite dimensional systems; vibrating string; nonlinear boundary condition; method of characteristics; asymptotic techniques; rotation number.
English
1998
350
11
4265
4311
none
Chen, G., Hsu, S., Zhou, J., Chen, G., Crosta, G. (1998). Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition - Part I: Controlled hysteresis. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 350(11), 4265-4311 [10.1090/S0002-9947-98-02022-4].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/93714`
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