Normal form analysis of nonlinear oscillator equations with automated arbitrary order expansions

Arbitrary order expansions for the automatic reduction and solutions of nonlinear vibratory systems have been developed successfully within the realm of the direct parametrisation of invariant manifolds. Whereas the method has been used with high-order expansions and large dimensional systems, this article proposes to look at the same problem from the opposite view angle. By using low-dimensional systems, symbolic computations, analytical developments and numerical verifications, this contribution analyzes the reduced dynamics appearing in cases where a single master mode is involved, reviewing typical scenarios in nonlinear vibrations: primary resonance, sub- and superharmonic resonances and parametric excitation. To achieve this task, the normal form style is preferentially used. A symbolic open-source package is also provided to generalize the presented results to other styles, higher orders, and different scenarios. It is shown how the low-order terms allow recovering the classical solutions given by perturbation methods, and how the automated expansions allow one to generalize the analysis to arbitrary orders. When analytical solutions are not tractable anymore, numerical solutions are employed to underline how converged solutions are at hand when the validity limit of the expansions is not reached. All the results presented in this paper can thus be used to better understand the nonlinear dynamical solutions occurring in nonlinear vibrations, as well as from a system identification perspective, since the normal form is the simplest dynamical system displaying a given resonance scenario.