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How does the nonlinear dynamics of networked systems respond to external time-dependent perturbations? As direct numerical simulations alone are unsuitable to comprehensively explore high-dimensional multi-parameter systems such as networks, theory combined with computer algebra and numerical analysis needs to help out. We start by showing how first and second order perturbation theory helps predicting local responses to fluctuations. Tipping points, indicating the transition to non-local responses, however, are not predictable by standard perturbation theory of any finite order. We thus propose a self-consistency condition in terms of harmonic balance equations, yielding algebraic expressions for the tipping point. Yet, the number of coupled nonlinear constraint equations rapidly grows with system dimension, the order of the harmonic approximation and the disparity of the frequency content of the driving signal. We illustrate how a combination of theory, computer algebra and numerical approaches may help making progress in predicting tipping points.