The VIVID function for numerically continuing periodic orbits arising from grazing bifurcations of hybrid dynamical systems

TL;DR

The paper tackles the numerical difficulty of continuing periodic orbits near grazing bifurcations in hybrid systems, where square-root singularities impede standard fixed-point methods. It introduces the VIVID function, a smooth surrogate $V$ that maps impact velocity to post-loop displacement variation, enabling Newton-based zeros-finding to locate maximal periodic orbits and their bifurcations. By applying VIVID to a linear one-degree-of-freedom oscillator with hard impacts, the authors demonstrate robust one- and two-parameter continuations that reveal grazing, saddle-node, and period-doubling phenomena and expose codimension-two resonant grazing. The approach provides a simple, effective alternative to collocation for near-grazing dynamics, with practical implications for predicting and tracking complex behaviors in mechanical hybrid systems, and it offers a foundation for further theoretical development via the implicit function theorem.

Abstract

Periodic orbits of systems of ordinary differential equations can be found and continued numerically by following fixed points of Poincaré maps. However, this often fails near grazing bifurcations where a periodic orbit collides tangentially with a boundary of phase space. Failure occurs when the map contains a square-root singularity and the root-finding algorithm searches beyond the domain of viable values. We show that by instead following the zeros of a function that maps Velocity Into Variation In Displacement (VIVID) this issue is circumvented and there is no such failure. We illustrate this with a prototypical one-degree-of-freedom impact oscillator model by applying Newton's method to the VIVID function to follow periodic orbits collapsing into grazing bifurcations. We also follow curves of saddle-node and period-doubling bifurcations of periodic orbits that issue from a codimension-two resonant grazing bifurcation. The VIVID function provides a simple alternative to the more sophisticated collocation method and enables periodic orbits and their bifurcations to be resolved easily and accurately near grazing bifurcations.

Figures (4)

• Figure 1: A typical trajectory of the hybrid system \ref{['eq:hybridSystem']} including its extensions (dashed) following $f$ into $x > 0$. The trajectory intersects the impacting surface $\Sigma$ at $(y_1,{\bf z}_1)$, then its extension intersects the Poincaré section $\Pi$ at $(x_0,{\bf z}_0) = P_{\rm virt}(y_1,{\bf z}_1)$. The impact point $(y_1,{\bf z}_1)$ maps under the reset law $\Phi$ to $(y_2,{\bf z}_2)$, then the (backwards) extension of the trajectory from $(y_2,{\bf z}_2)$ intersects $\Pi$ at $(x_3,{\bf z}_3) = P_{\rm virt}(y_2,{\bf z}_2)$. The map from $(x_0,{\bf z}_0)$ to $(x_3,{\bf z}_3)$ is the right piece $P_{{\rm disc},R}$ of the discontinuity map $P_{\rm disc}$. The map from $(x_3,{\bf z}_3)$ to the next intersection $(x_4,{\bf z}_4)$ of the trajectory (including its extensions) with $\Pi$ after one loop following close to the grazing trajectory is the global map $P_{\rm global}$.
• Figure 2: A schematic diagram of the linear impact oscillator modelled by \ref{['eq:impactOsc']}.
• Figure 3: One-parameter bifurcation diagrams showing the impact velocity $y_{\rm imp}$ and stability multipliers $\lambda_{1,2}$ of a two-loop maximal periodic orbit of \ref{['eq:impactOsc']} with $\zeta = 0.02$ and $\epsilon = 0.9$ (PD: period-doubling bifurcation; SN: saddle-node bifurcation). The curves are blue where the orbit is stable, red where the orbit is unstable, and dashed where the orbit is virtual. The green line indicates the grazing bifurcation of the non-impacting periodic orbit \ref{['eq:phip']}.
• Figure 4: Panel (a) is a two-parameter bifurcation diagram of \ref{['eq:impactOsc']} with $\zeta = 0.02$ and $\epsilon = 0.9$; panel (b) is a magnification of the area within the boundary portrayed by the broken lines in panel (a) (PD: period-doubling bifurcation; SN: saddle-node bifurcation).