Near-grazing dynamics in tapping-mode atomic-force microscopy

In tapping-mode atomic-force microscopy, non-linear effects due to large variations in the force field on the probe tip over very small length scales and the intermittency of contact may induce strong dynamical instabilities. In this paper, a discontinuity-mapping-based analysis is employed to investigate the destabilizing effects of low-velocity contact on a lumped-mass model of an oscillating atomic-force-microscope cantilever tip interacting with a typical sample surface. As illustrated using two tip-sample force models, the analysis qualitatively captures the potential loss of stability and disappearance of a low-contact-velocity steady-state response. The quantitative agreement of the predictions of the discontinuity-mapping-based analysis with direct numerical simulations, at least for sufficiently low contact velocity, supports its use in the passive redesign or active control of the tip-sample mechanism for purposes of preventing such a loss of stability.     

Degenerate discontinuity-induced bifurcations in tapping-mode atomic-force microscopy

This paper documents the existence of degenerate bifurcation scenarios in the low-contact-velocity dynamics during tapping-mode atomic-force microscopy. Specifically, numerical analysis of a model of the microscope dynamics shows branch point and isola bifurcations associated with the emergence of two families of saddle–node bifurcation points along a branch of low-amplitude oscillations. The paper argues for the origin of the degenerate bifurcations in the existence of a periodic steady-state trajectory that (i) achieves tangential contact with a discontinuity surface in a piecewise smooth model of the cantilever response and (ii) retracts from the surface under variations in either direction along a line segment in parameter space. Specifically, the discontinuity-mapping technique is here rigorously applied to a general situation of such degenerate contact showing the codimension-two nature of these bifurcations for appropriately chosen parameter values. The discontinuity-mapping-based normal form derived here is a novel extension of that derived in Dankowicz and Nordmark (2000) [28] in the case that (ii) does not hold. In addition, the paper includes a quantitative reflection on the relative importance of discontinuities in the attractive and repulsive force components in producing the predicted bifurcations.     

Local models of spatio-temporally complex fields

We investigate the ability of local models of the one space dimensional Kuramoto-Sivashinsky partial differential equation with periodic boundary conditions, obtained by projection on a small set of Fourier modes on a short subinterval, to reproduce coherent events typical of solutions of the same equation on a much longer interval. We find that systems containing as few as two linearly unstable modes can produce realistic local events in the short term, but that for more reliable short time tracking and long term statistics, three or four interacting modes are required, and that the length of the short interval plays a subtle rôle, certain “resonant” lengths giving superior results.     

On various representations of higher order approximations of the free oscillatory response of nonlinear dynamical systems

This paper investigates the flexibility afforded by the application of regular perturbation methods (in particular, the method of multiple scales) for the purpose of obtaining higher order approximations of the oscillatory response of a nonlinear dynamical system. It is shown that the non-uniqueness of these higher order approximations can be removed by enforcing additional conditions while the relationship between the frequency of oscillation and measurable quantities (the Hamiltonian, the time-averaged kinetic or stored energy) is unique and is thus not affected by these additional conditions.     

Discontinuity-induced bifurcations in systems with impacts and friction: Discontinuities in the impact law

This paper concerns the non-smooth dynamics of planar mechanical systems with isolated contact in the presence of Coulomb friction. Following Stronge [Impact Mechanics, Cambridge University Press, Cambridge, 2000], a set of closed-form analytic formulae is derived for a rigid-body impact law based on an energetic coefficient of restitution and a resolution of the impact phase into distinct segments of relative slip and stick. Thus, the impact behavior is consistent both with the assumption of Coulomb friction and with the dissipative nature of impacts. The analysis highlights the presence of boundaries between open regions of initial conditions and parameter values corresponding to distinct forms of the impact law and investigates the smoothness properties of the impact law across these boundaries. It is shown how discontinuities in the impact law are associated with discontinuity-induced bifurcations of periodic trajectories, including non-smooth folds and persistence scenarios. Numerical analysis of an example mechanical model is used to illustrate and validate the conclusions.     

Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay

Nonlinear Dynamics - This paper generalizes a previously conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and...     

On the purposeful coarsening of smooth vector fields

This paper argues for the possibility of purposely approximating smooth vector fields with highly localized variability in terms of piecewise smooth vector fields for the purpose of analyzing the bifurcation characteristics of the corresponding dynamical systems. Here, emphasis is placed on the changes in system response that result as a periodic trajectory begins to incorporate a brief flow segment in the region of high variability under variations in some system parameter. In particular, it is shown that tools from the theory of grazing bifurcations in piecewise-smooth systems may be employed to qualitatively predict the bifurcation scenario associated with such a transition both in terms of the shape of the branch of periodic trajectories and in terms of the persistence of a local attractor in the vicinity of the original periodic trajectory.     

Adaptive control designs for control-based continuation of periodic orbits in a class of uncertain linear systems

Nonlinear Dynamics - This paper proposes two novel adaptive control designs for the feedback signals used in the control-based continuation paradigm to track families of periodic orbits of...