Koopman-based stability analysis of differential-algebraic equations with applications to frictional multibody systems

What the math gives to ML

The paper provides a constructive Floquet-stability method for periodic solutions of linear time-periodic differential-algebraic equations (DAEs): Fourier-expand the periodic Jacobian, form a generalized Hill matrix, and infer the monodromy spectrum while correctly handling singular descriptor matrices through a Drazin-inverse evolution. This transfers to recurrent and state-space neural networks operated under periodic inputs, periodic forcing, or cyclic learning-rate schedules, especially when hidden states obey hard algebraic constraints. The most useful implementation is a Fourier-domain stability monitor and regularizer that penalizes unstable Floquet exponents of the learned periodic orbit, with a directly falsifiable prediction that long-horizon growth changes sign when the dominant Hill exponent crosses zero.