Dynamical Integrity of Periodic Orbits: A Computationally Efficient Approach

The dynamical integrity of a solution in a dynamical system reflects its robustness against external perturbations, with a larger basin of attraction indicating greater robustness. However, computing the entire basin of attraction can be computationally expensive. The local integrity measure offers an efficient alternative by quantifying the compact region of the basin of attraction around the steady state of interest. Recently, a rapid algorithm was introduced to estimate the dynamical integrity of equilibrium points using the local integrity measure. This paper extends this method to estimate the dynamical integrity of stable periodic orbits in both autonomous and periodically excited systems. The enhanced algorithm, implemented in the DynIn MATLAB Toolbox, is evaluated on four engineering systems; specifically, a Duffing--van der Pol oscillator with an attached tuned mass damper, a pitch-and-plunge wing profile, a harmonically excited Duffing oscillator, and a passive vertical hopping model. The results demonstrate that the algorithm is both rapid and effective, yielding meaningful engineering insights and paving the way for incorporating dynamical integrity as a design parameter.