Control-aided construction of basins of attraction of nonlinear dynamical systems Université de Liège

Abstract

Nonlinear mechanical systems subject to harmonic excitation can exhibit multistability, where qualitatively different steady-state responses coexist under identical operating conditions. The set of initial conditions converging to a given attractor, known as its basin of attraction (BoA), determines which response is observed in practice and how robust it is to perturbations. While numerical methods for basin reconstruction are well established, their experimental counterpart remains comparatively undevelopped: mapping a basin requires placing a physical system at many precisely specified states in the phase plane, simultaneously in displacement and velocity, which no existing experimental technique achieves in a systematic and automated way. This thesis proposes and demonstrates a control-aided method for the experimental reconstruction of basins of attraction of a Duffing oscillator. A proportional-derivative controller tracks a reference trajectory to drive the system from rest to the neighbourhood of any prescribed target state. A systematic reference design strategy is introduced to ensure energetic consistency between the reference and the true system dynamics across the full phase plane. Control gains are scheduled automatically from the local backbone frequency, and a stroboscopic Poincaré criterion monitors convergence after the controller is released. A trajectory-based backfilling strategy further reduces the number of required experimental runs by propagating attractor labels along confirmed free-evolution paths. The method is validated numerically on four Duffing configurations of increasing complexity and implemented in real time on a dSPACE MicroLabBox controlling an electronic analogue Duffing oscillator. Experimental basin maps agree closely with numerical references across all configurations, including cases with coexisting subharmonic attractors.