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Order Reduction of Nonlinear Delay-Differential Equations with Periodic Coefficients

Eric Butcher
A technique for order reduction of nonlinear delay differential equations with timeperiodic
coefficients is presented. The DDEs considered here have at most cubic nonlinearities
multiplied by a perturbation parameter. The periodic terms and matrices are not assumed to have
predetermined norm bounds, thus making the method applicable to systems with strong
parametric excitation. Perturbation expansion converts the nonlinear response problem into
solutions of a series of non-homogenous linear ordinary differential equations with time periodic
coefficients. One set of linear non-homogenous ODEs is solved for each power of the
perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus
we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The
accuracy of the method is demonstrated with a nonlinear delayed Mathieu equation, a milling
model, and a single inverted pendulum with a periodic retarded follower force and nonlinear
restoring force in which the amplitude of the limit cycle associated with a flip bifurcation is
found analytically and compared to that obtained from direct numerical simulation.
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