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

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 [J.P. Keener, Chaotic behavior in piecewise continuous difference equations, Transactions Amer. Math. Soc., 261 (1980), 589–604], and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of M.I. Malkin [M.I. Malkin, Rotation intervals and the dynamics of Lorenz type mappings, Selecta Math. Sovietica 10(3) (1991), 265–275] 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.