Determining the periodic response of nonlinear systems by a gradient method

Abstract

Abstract For systems of differential equations of the form ẋ = f(x) or x = f(x, t ) , a periodic response may be identified by the requirement that x( kT ) = x(0) , where k = 1, 2, … and T is the period, x(0) = x 0 being the initial‐condition vector. We describe a gradient method for finding this x 0 vector by minimizing the square magnitude of the ‘discrepancy vector’ δ(x 0 ) = x( T )–x 0 . The gradient of the scalar function P (x 0 ) = δ t (x 0 )δ(x 0 ) with respect to x 0 is calculated by one full‐period forward integration of the original differential equation to obtain δ(x 0 ), and then one full‐period backward integration of the adjoint variational equations, using δ(x 0 ) as the initial‐condition vector. The gradient of P (x 0 ) is then twice the adjoint discrepancy vector. We use Fletcher's method of optimization to minimize P (x 0 ) .