A novel approach to induce chaos for vibratory PDE systems

In this paper, we propose a novel method for inducing chaos in infinite-dimensional systems. The new approach, referred to as the kernel function method, regulates a carefully chosen performance output to exhibit chaotic dynamics governed by a pre-specified ODE system. Unlike the method of characteristics, the proposed technique is applicable not only to wave equations but also to the Euler-Bernoulli beam equation. Moreover, by appropriately selecting the pre-specified chaotic ODE system, a wide variety of chaotic oscillations can be generated. Specifically, by choosing the well-known Van der Pol equation, we derive a new nonlinear boundary condition for an Euler-Bernoulli beam that demonstrates chaotic oscillations. Numerical simulations are provided to help visualize the theoretical results.