Resonant grazing bifurcations revisited

Abstract

In vibro-impact mechanics, the division between an impact and a near miss is a zero-velocity grazing event. Grazing bifurcations of stable periodic motions often produce complicated attractors when grazing generates a square-root term in the Poincaré map. This paper concerns codimension-two scenarios for which the square-root term vanishes in some iterate of the Poincaré map. For forced one-degree-of-freedom oscillators, this occurs when the forcing frequency is a certain rational multiple of the damped natural frequency, i.e., the system is in resonance. In two-parameter bifurcation diagrams, curves of saddle-node and period-doubling bifurcations of single-impact periodic motions emanate from the codimension-two points. In this paper we prove these curves are quadratically tangent to the curve of grazing bifurcations, and derive explicit formulas for their quadratic coefficients. This is achieved by modifying the Poincaré map in a way that circumvents the square-root singularity, enabling us to use the implicit function theorem to demonstrate smoothness and perform asymptotic calculations of the saddle-node and period-doubling bifurcation curves. In doing so we resolve a long-standing conjecture on the admissibility of single-impact periodic motions by supplementing raw asymptotic computations with geometric and topological arguments. We illustrate the results with a linear impact oscillator model, matching the theoretical unfolding to numerically computed bifurcation curves. The results explain why previously reported physical experiments reveal an absence of chaos shortly past the grazing bifurcation.

Figures (9)

• Figure 1: A sketch of phase space for the class of impacting hybrid systems introduced in §\ref{['sec:results']}. Orbits (solid curves) obey the ODEs until reaching the impacting surface $\Sigma$ where they are mapped under the reset law. The impacting surface is partitioned into an incoming set, an outgoing set, and a tangency set, as determined by the direction of the ODEs relative to the surface. The reset law maps the incoming set to the outgoing set, and is the identity map on the tangency set.
• Figure 2: A bifurcation diagram of the linear impact oscillator model \ref{['eq:oscf']}--\ref{['eq:oscResetLaw']} with $\zeta = 0.02$, $\epsilon = 0.9$, and $\omega = 0.854$. The horizontal axis uses the forcing amplitude $\mathcal{A}$, while the vertical axis uses the $x$-value of $P_{\rm global}$ (so points with $x > 0$ indicate the occurrence of an impact). The right plot is a magnification; the upper plots are phase portraits. Branches of $p$-loop MPSs were computed by numerical continuation, and are indicated with thick curves where the solutions are stable, and thin curves where they are unstable. Four bifurcations are labelled: $\mathcal{A}_{\rm graz}$: grazing bifurcation of the non-impacting periodic solution; PD: period-doubling bifurcation of the two-loop MPS; SN: saddle-node bifurcation of the three-loop MPS; GZ: grazing bifurcation of the three-loop MPS. Between GZ and PD we overlay a numerical bifurcation diagram showing the long-term behaviour of forward orbits of random initial points.
• Figure 3: A sketch illustrating the $P_{\rm global}$: the return map to $\Pi$ induced by the flow of \ref{['eq:f']}. We also sketch the grazing periodic solution $\Gamma$ of Assumption \ref{['ass:grazing']}.
• Figure 4: The curves $\tau = g_p(\delta)$ and $\tau = h_p(\delta)$ for $p = 2,3,\ldots,6$ for values within the trapesium $\mathcal{T}$\ref{['eq:cT']}.
• Figure 5: A sketch of the bifurcation curves described by Theorems \ref{['th:codim21']} and \ref{['th:codim2p']} in the case $c_{{\rm SN},p} < 0$, $c_{{\rm PD},p} > 0$, and $\beta > 0$. Here a stable non-impacting periodic solution exists for $\mu < 0$.

Theorems & Definitions (19)

• Lemma 2.1
• Theorem 2.2: generic grazing, $p=1$
• Theorem 2.3: generic grazing, $p \ge 2$
• Theorem 2.4: resonant grazing, $p=1$
• Theorem 2.5: resonant grazing, $p \ge 2$
• Remark 2.1
• Proposition 3.1
• Proposition 3.2: resonant grazing, $p=1$
• Proposition 3.3: resonant grazing, $p \ge 2$
• Lemma 4.1