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) = x0 being the initial‐condition vector. We describe a gradient method for finding this x0 vector by minimizing the square magnitude of the ‘discrepancy vector’ δ(x0) = x(T)–x0. The gradient of the scalar function P(x0) = δt(x0)δ(x0) with respect to x0 is calculated by one full‐period forward integration of the original differential equation to obtain δ(x0), and then one full‐period backward integration of the adjoint variational equations, using δ(x0) as the initial‐condition vector. The gradient of P(x0) is then twice the adjoint discrepancy vector. We use Fletcher's method of optimization to minimize P(x0). Copyright

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Journal International Journal of Circuit Theory and Applications
Nakhla, M.S, & Branin, F.H. (1977). Determining the periodic response of nonlinear systems by a gradient method. International Journal of Circuit Theory and Applications, 5(3), 255–273. doi:10.1002/cta.4490050307