EIGENVALUES OF THE JACOBIAN OF A GALERKIN-PROJECTED UNCERTAIN ODE SYSTEM

Projection onto polynomial chaos (PC) basis functions is often used to reformulate a system of ordinary differential equations (ODEs) with uncertain parameters and initial conditions as a deterministic ODE system that describes the evolution of the PC modes. The deterministic Jacobian of this projected system is different and typically much larger than the random Jacobian of the original ODE system. This paper shows that the location of the eigenvalues of the projected Jacobian is largely determined by the eigenvalues of the original Jacobian, regardless of PC order or choice of orthogonal polynomials. Specifically, the eigenvalues of the projected Jacobian always lie in the convex hull of the numerical range of the Jacobian of the original system.