Improved Stability Estimates and Flight Time Predictions Using Higher-Order Transverse Discontinuity Mapping in Hybrid Dynamical Systems

TL;DR

This paper introduces a higher-order transverse discontinuity mapping (TDM) method to improve stability estimates and impact time predictions in hybrid dynamical systems, addressing limitations of first-order linearization.

Contribution

The paper proposes a higher-order TDM approach that enhances impact timing accuracy and stability analysis in hybrid systems, overcoming linearization pitfalls.

Findings

Higher-order TDM improves impact time estimates.

The method avoids divergence near grazing impacts.

Stability analysis results align with bifurcation diagrams.

Abstract

This article emphasizes on inconsistencies in the dynamical estimates obtained by first-order transverse discontinuity mapping (TDM) and direct numerical observations for hybrid dynamical systems. Pitfalls of locally linearizing hybrid nonlinear dynamical systems near discontinuity boundaries are demonstrated along with examples of how such linearization could lead to incorrect estimates of impact occurrences for transverse interactions with a rigid barrier. A higher-order TDM is proposed to overcome this shortcoming, allowing for better analytical estimation of impact occurrence times, state transitions, and, consequently, the evolution of trajectories. The difference in flight times of two closely initiated trajectories in the local neighbourhood of a discontinuity boundary is estimated up to $\mathcal{O}(2)$. The resulting quadratic equation implies that the orbits local to the impacting state, corresponding to a negative discriminant, won't reach the discontinuity boundary. Further, the $\mathcal{O}(2)$ correction terms to the analytical expression of the TDM ensure that the flight time estimates do not diverge for low-velocity impacts near grazing, thereby avoiding overestimation of the mapped state. A numerical method is subsequently developed to estimate a saltation matrix incorporating the proposed higher-order TDM to avoid incorrect impact occurrences. Modifications to the existing algorithms used to numerically quantify local stability, namely the Lyapunov spectra and Floquet multipliers, are proposed. Stability analyses using the proposed higher-order approach are carried out for representative cases of a hard impact oscillator and a pair impact oscillator, with results consistent with numerically obtained bifurcation diagrams.