Faà Di Bruno's formula and volterra series

Volterra series are used for modelling nonlinear systems with memory effects. The n th -order impulse response and the kernels in the series can be determined with Fréchet derivatives of Volterra series operators. Consequently, we can determine the kernels of composite systems by taking higher-order Fréchet derivatives of composite series. The generalisation of the higher-order chain rule, Faà di Bruno's formula for variational calculus, was recently determined and this note demonstrates how it can be used to determine kernels for composite Volterra series operators