The authors consider the one-dimensional vibrating string satisfying \(w_{tt}- w_{xx}= 0\) on the unit interval \(x\in (0,1)\) with following boundary conditions: at the left end \(x= 0\) the string is fixed while at the right end \(x= 1\), a nonlinear boundary condition \(w_x= \alpha w_t- \beta w^3_t\), \(\alpha,\beta> 0\), takes effect. This nonlinear boundary condition behaves like a van der Pol oscillator.
The problem is formulated as an equivalent first-order hyperbolic system, and the method of characteristics is used to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. 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. 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, and the chaos is rigorously proved for some \(\alpha\) values. Nonchaotic case 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.