Uniqueness and stability of the simultaneous detection of the nonlinear restoring and excitation of a forced nonlinear oscillation

• A new inverse problem for the simultaneous identification is presented.
• It identifies the restoring and excitations of a nonlinear physical system.
• We proved that the inverse problem has a unique solution and stability properties.

This paper presents a new inverse problem for the simultaneous identification of the restoring characteristic and the excitations of a nonlinear physical system. An original mathematical formalism for the identification is developed based on the notion of a zero-crossing time and a new function, called the J-function. We analyze the developed formalism to prove that the inverse problem presented has a unique solution and the stability properties. That is, it is mathematically shown that the nonlinear restoring and excitation of a forced nonlinear oscillation can be detected in a unique and stable manner.