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This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package COCO demonstrate successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.

Through a behavioral coordination game played by groups of humans and simulated with agent-based models, we investigated a social network dilemma that we call {\em fraughtness}. Seven players, remotely connected to one another in various topologies via a computer network, each had to move a slider to the left or right along a horizontal bar on their screen. The goal was for all the players to move their slider to the same end of the bar. Players received feedback indicating the degree to which they and their neighbors in the topology agreed about the choice of side. When the topology had a hierarchical branching structure, the groups often got stuck in fraughtness: players on one branch favored one end of the bar, while players on the other branch favored the other; because all were receiving supportive local feedback, nobody wanted to change. Nevertheless, after being stuck in fraughtness for some time, most groups managed to escape. Fraughtness is arguably an analog of generally negatively viewed social phenomena like polarization and echo chambers. Our analyses suggest that while fraughtness is problematic, it is closely linked to successful structure formation---it thus may be most effective to focus not on how to banish it, but on how to resolve it.

This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of the COCO software package and its explicit support for community-driven development. To this end, the paper first formalizes the COCO construction paradigm for augmented continuation problems compatible with simultaneous analysis of implicitly defined manifolds of solutions to nonlinear equations and the corresponding adjoint variables associated with optimization of scalar objective functions along such manifolds. The paper uses applications to data assimilation from finite time histories and phase response analysis of periodic orbits to identify a universal paradigm of construction that permits abstraction and generalization. It then details the theoretical framework for a COCO-compatible toolbox able to support the analysis of a large family of delay-coupled multi-segment boundary-value problems, including periodic orbits, quasiperiodic orbits, connecting orbits, initial-value problems, and optimal control problems, as illustrated in a suite of numerical examples. The paper aims to present a pedagogical treatment that is accessible to the novice and inspiring to the expert by appealing to the many senses of the applied nonlinear dynamicist. Sprinkled among a systematic discussion of problem construction, graph representations of delay-coupled problems, and vectorized formulas for problem discretization, the paper includes an original derivation using Lagrangian sensitivity analysis of phase-response functionals for periodic-orbit problems in abstract Banach spaces, as well as a demonstration of the regularizing benefits of multi-dimensional manifold continuation for near-singular problems.

In this paper, we discuss an original covariance boundary-value problem that captures the effects of white noise on the local behavior near stable periodic orbits of the corresponding deterministic dynamical systems. The methodology relies on suitably constructed adjoint variables to project the dynamics onto locally transversal hyperplanes and ensure the existence of a unique solution. For each such hyperplane, the computed co-variance matrix describes an approximately Gaussian, stationary distribution that highlights directions of particular sensitivity to noise. In contrast to previous formulations, the boundary-value problem analyzed in this paper makes predictions in the original state-space variables rather than in terms of a reduced set of local coordinates for hyperplanes perpendicular to the local vector field. The formulation is compatible with the general form of an augmented continuation problem for the software package COCO; a non-adaptive version has been implemented as a general-purpose constructor for the periodic-orbit toolbox included in the COCO release. We illustrate the efficacy of the proposed formulation and toolbox by analyzing noise-induced behavior near limit cycles in two examples of nonlinear dynamical systems with autonomous drift terms. The analysis presented in this study can be used to design safe operating regimes for systems working in stochastic environments, as well as to design optimum working conditions for systems utilizing noise, such as energy-harvesting applications.

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 periodically excited dynamical systems, including black box simulation models and physical experiments. The proposed control designs rely on modifications to the classical model reference adaptive control framework and the more recent $${\mathscr {L}}_1$$ adaptive control architecture, in which an additional low-pass filter is used to ensure guaranteed transient performance and robustness to time delays in the control input even in the limit of arbitrarily large adaptive gains. In contrast to the proportional control formulations that have been used in the literature on control-based continuation, the proposed control designs achieve stable performance with a minimum of parameter tuning. In the context of a class of linear systems with matched uncertainties, the paper demonstrates the successful integration of adaptive control feedback in control-based continuation. Specifically, the control designs are shown to ensure that the control input stabilizes the sought periodic orbits of the uncontrolled system and vanishes along these orbits, provided that an a priori unknown reference input is chosen appropriately. Numerical results obtained using the coco software package demonstrate how the combination of a nonlinear solver (Newton’s method) with the pseudo-arclength parameter continuation scheme can be used to trace the correct choice for the reference input under variations in an excitation parameter.

SUMMARYThis paper reports on laboratory and field experimental results for controlled robotic manipulators operating on moving platforms with unmodeled dynamics. The aim is to validate theoretical predictions for the dependence on control parameters of an adaptive control strategy. In addition, the results provide insight into different discretizations of the continuous-time formulation, suggesting the most suitable discretization scheme for hardware implementation. The second set of experimental results, obtained from an implementation of the control framework for synchronization and consensus in networks of robotic manipulators, similarly validate theoretical predictions on the sensitivity to network communication delays.

This article proposes a methodology for integrating adaptive control with the control-based continuation paradigm for a class of uncertain, linear, discrete-time systems. The proposed adaptive control strategies aim to stabilize the closed-loop dynamics with convergence toward a known reference input, such that the dynamics approach the open-loop fixed point if the reference input is chosen to make the steady-state control input equal 0. This enables the tracking of a parameterized branch of open-loop fixed points using methods of numerical continuation without specific knowledge about the system. We implement two different adaptive control strategies: model-reference adaptive control and pole-placement adaptive control. Both implementations achieve the desired objectives for the closed-loop dynamics and support parameter continuation. These properties, as well as the boundedness of system states and control inputs, are guaranteed provided that certain stability conditions are satisfied. Besides, the tuning effort is significantly reduced in the adaptive control schemes compared with traditional proportional–derivative controllers and linear state-space feedback controllers.

Opinion dynamics on networks has wide applications to empirical and engineered systems and profound prospects in the general study of complex systems. Many efforts have been devoted to understanding how opinion dynamics is affected by network topology. However, human social interactions are best characterized as temporal networks in which ordering of interactions cannot be ignored. Temporal activity patterns including heterogeneous contact strength and interevent times, turnover edge/node dynamics and daily patterns could have significant effects that would not be captured by static aggregate network representations. In this paper, we study the effects of such temporal patterns on the speed of consensus formation in various models of continuous opinion dynamics using three empirical human face-to-face networks from different real-world settings. We find that static, aggregated networks consistently overestimate the speed of simulated consensus formation while weight heterogeneity associated with frequency of interactions has an inhibitory effect on consensus formation relative to the behavior on unweighted networks. Moreover, the speed of consensus formation is found to be highly sensitive to nodal lifetimes, suggesting that randomization protocols that dramatically alter the distribution of lifetimes cannot be relied upon as reference models. On the other hand, temporal patterns including burstiness of interevent times and the lifetime of edges are found to have insignificant effects on consensus formation., Comment: 15 pages, 8 figures

This paper investigates the near-resonance response to exogenous excitation of a class of networks of coupled linear and nonlinear oscillators with emphasis on the dependence on network topology, distribution of nonlinearities, and damping ratios. The analysis shows a qualitative transition between the behaviors associated with the extreme cases of all linear and all nonlinear oscillators, respectively, even allowing for such a transition under continuous variations in the damping ratios but for fixed topology. Theoretical predictions for arbitrary members of the network class using the multiple-scales perturbation method are validated against numerical results obtained using parameter continuation techniques. The latter include the tracking of families of quasi-periodic invariant tori emanating from saddle-node and Hopf bifurcations of periodic orbits. In networks in the class of interest with special topology, 1:1 and 1:3 internal resonances couple modes of oscillation, and the conditions to suppress the influence of these resonances are explored.

Parameter continuation of finitely parameterized, approximate solutions to integro-differential boundary-value problems typically involves regular adaptive updates to the number and meaning of the unknowns and/or the associated constraints. Different continuation steps produce solutions with different discretizations or to formally different sets of equations. Existing general-purpose, multidimensional continuation algorithms fail to account for such differences without significant additional coding and are therefore prone to redundant coverage of the set of solutions. We describe a new algorithm, implemented in the software package coco, which overcomes this problem by characterizing the solution set in an invariant, finite dimensional, projected geometry rather than in the space of unknowns corresponding to any particular discretization. It is in this geometry that distances between solutions and angles between tangent spaces are quantified and used to construct possible directions of outward expansion. A pointwise lift identifies such directions in the projected geometry with directions of continuation in the full set of unknowns, used by a nonlinear predictor-corrector algorithm to expand into uncharted parts of the solution set. Several benchmark problems from the analysis of periodic orbits in autonomous dynamical systems are used to illustrate the theory.

To study how a certain network feature affects processes occurring on a temporal network, one often compares properties of the original network against those of a randomized reference model that lacks the feature in question. The randomly permuted times (PT) reference model is widely used to probe how temporal features affect spreading dynamics on temporal networks. However, PT implicitly assumes that edges and nodes are continuously active during the network sampling period – an assumption that does not always hold in real networks. We systematically analyze a recently-proposed restriction of PT that preserves node lifetimes (PTN), and a similar restriction (PTE) that also preserves edge lifetimes. We use PT, PTN, and PTE to characterize spreading dynamics on (i) synthetic networks with heterogeneous edge lifespans and tunable burstiness, and (ii) four real-world networks, including two in which nodes enter and leave the network dynamically. We find that predictions of spreading speed can change considerably with the choice of reference model. Moreover, the degree of disparity in the predictions reflects the extent of node/edge turnover, highlighting the importance of using lifetime-preserving reference models when nodes or edges are not continuously present in the network.

Significance Interaction patterns in human communication networks are characterized by intermittency and unpredictable timing (burstiness). Simulated spreading dynamics through such networks are slower than expected. A technology for automated recording of social interactions of individual honeybees, developed by the authors, enables one to study these two phenomena in a nonhuman society. Specifically, by analyzing more than 1.2 million bee social interactions, we demonstrate that burstiness is not a human-specific interaction pattern. We furthermore show that spreading dynamics on bee social networks are faster than expected, confirming earlier theoretical predictions that burstiness and fast spreading can co-occur. We expect that these findings will inform future models of large-scale social organization, spread of disease, and information transmission., Social networks mediate the spread of information and disease. The dynamics of spreading depends, among other factors, on the distribution of times between successive contacts in the network. Heavy-tailed (bursty) time distributions are characteristic of human communication networks, including face-to-face contacts and electronic communication via mobile phone calls, email, and internet communities. Burstiness has been cited as a possible cause for slow spreading in these networks relative to a randomized reference network. However, it is not known whether burstiness is an epiphenomenon of human-specific patterns of communication. Moreover, theory predicts that fast, bursty communication networks should also exist. Here, we present a high-throughput technology for automated monitoring of social interactions of individual honeybees and the analysis of a rich and detailed dataset consisting of more than 1.2 million interactions in five honeybee colonies. We find that bees, like humans, also interact in bursts but that spreading is significantly faster than in a randomized reference network and remains so even after an experimental demographic perturbation. Thus, while burstiness may be an intrinsic property of social interactions, it does not always inhibit spreading in real-world communication networks. We anticipate that these results will inform future models of large-scale social organization and information and disease transmission, and may impact health management of threatened honeybee populations.

The objectives of this paper are threefold. First, the paper defines a staged construction paradigm for the necessary conditions for locally optimal solutions to integro-differential boundary-value problems that mirrors the natural decomposition of the original constraints. Second, it details a library of realizations of the zero functions, monitor functions, and contributions to the adjoint equations associated with each stage of problem construction. Third, it illustrates the application of this framework to multisegment trajectory problems, for example, in the analysis of spacecraft optimal control problems. The proposed construction paradigm generalizes a formulation for the construction of integro-differential-algebraic continuation problems in the software package COCO to also include automatically generated adjoint terms. The implementation enables arbitrarily many levels of nested construction with larger problems assembled from smaller ones, and multiple instances of a problem class combined insi...

This paper studies the delay robustness of an $\mathcal {L}_1$ adaptive controller designed for systems with unknown matched nonlinearities and unknown input-gain matrices. The analysis establishes rigorously the existence of a positive lower bound for the closed-loop system's time-delay margin (TDM), provided that a filter bandwidth and an adaptive gain are chosen sufficiently large. In this case, if the input delay is below a critical value, then the state and control input of the control system follow those of a nonadaptive, robust reference system closely. The analysis also suggests a way to estimate this lower bound for the delay robustness using Pade approximants. Results from forward simulation are consistent with the Pade estimate and with an explicit upper bound on the TDM which decays to 0 as the filter bandwidth grows without bound.

In rigid-body mechanics, models that capture collisional contact as an instantaneous exchange of momentum may exhibit dynamics that include infinite sequences of impacts accumulating in finite time to a state of persistent contact, often referred to as chatter. In this paper, we review theoretical tools for the analysis of transient and steady-state behavior in the vicinity of critical periodic orbits for which chatter terminates at a point corresponding to the imminent release from persistent contact, and illustrate the application of this theory to a simplified model of a mechanical pressure relief valve. A general theory for single-degree-of-freedom impact oscillators, previously described in an unpublished manuscript by Nordmark and Kisitu1, is shown to yield both qualitative and quantitative agreement with model simulation results. The predicted bifurcation structure shows that the border orbit unfolds supercritically into a universal cascade of local attractors with nontrivial scaling relationships.

This paper aims to analyze the emergence of task differentiation in a model complex system, characterized by an absence of hierarchical control, yet able to exhibit coordinated behavior and collective function. The analysis focuses on linear network filters, i.e., networks of coupled linear oscillators with a differentiated steady-state response to exogenous harmonic excitation. It demonstrates how an optimal allocation of excitation sensitivities across the network nodes in a condition of resonance may be constructed either using global information about the network topology and spectral properties or through the iterated dynamics of a nonlinear, nonsmooth learning paradigm that only relies on local information within the network. Explicit conditions on the topology and desired resonant mode shape are derived to guarantee local asymptotic stability of fixed points of the learning dynamics. The analysis demonstrates the possibly semistable nature of the fixed point with all zero excitation sensitivities, ...

Stability analysis of an L1 adaptive control design decomposes into i) the derivation of an inequality condition imposed on a filter in the control channel that guarantees transient performance bounds on the state and control input for a theoretical nonadaptive reference system, and ii) a Lyapunov-type proof of the decay of a predictor error as the inverse square root of the adaptive gain, which in turn can be made arbitrarily large without sacrificing robustness to time delay in the control input. This paper reviews recent results on the robustness to input and communication time delays for L1 control systems with uncertain nonlinearities and in nontrivial network configurations, as well as a possible scheme for compensating for large, constant input delays.

We generalize the successive continuation paradigm introduced by Kernevez and Doedel [1] for locating locally optimal solutions of constrained optimization problems to the case of simultaneous equality and inequality constraints. The analysis shows that potential optima may be found at the end of a sequence of easily-initialized separate stages of continuation, without the need to seed the first stage of continuation with nonzero values for the corresponding Lagrange multipliers. A key enabler of the proposed generalization is the use of complementarity functions to define relaxed complementary conditions, followed by the use of continuation to arrive at the limit required by the Karush-Kuhn-Tucker theory. As a result, a successful search for optima is found to be possible also from an infeasible initial solution guess. The discussion shows that the proposed paradigm is compatible with the staged construction approach of the coco software package. This is evidenced by a modified form of the coco core used to produce the numerical results reported here. These illustrate the efficacy of the continuation approach in locating optimal solutions of an objective function along families of two-point boundary value problems and in optimal control problems.

This paper generalizes a previously-conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipliers. As a consequence, it is shown how critical points on the constraint manifold can be found through several stages of continuation along a sequence of connected one-dimensional manifolds of solutions to increasing subsets of the necessary optimality conditions. Due to the presence of delayed and advanced arguments in the original and adjoint differential equations, care must be taken to determine the degree of smoothness of the Lagrange multipliers with respect to time. Such considerations naturally lead to a formulation in terms of multi-segment boundary-value problems (BVPs), including the possibility that the number of segments may change, or that their order may permute, during continuation. The methodology is illustrated using the software package coco on periodic orbits of both linear and nonlinear delay-differential equations, keeping in mind that closed-form solutions are not typically available even in the linear case. Finally, we demonstrate optimization on a family of quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with delay and parametric forcing. The quasiperiodic case is a further original contribution to the literature on optimization constrained by partial differential BVPs., Comment: preprint, 17 pages, 9 figures

This paper considers an opportunity for robotic automation in the seeding of field crops. It reports on progress in designing a partially automated seed refilling system that transfers seeds to a centralized tank on a tractor-pulled planter during travel, while the tractor continues to follow a trajectory determined by a desired seeding strategy The paper describes challenges associated with accommodating variability among commercial planters, as well as with formulating a control strategy for a robotic manipulator that compensates for time-dependent uncertainty and unknown disturbance loads due to travel of a mounting platform across uneven terrain. To address such challenges, the paper reviews theoretical predictions for a recently proposed adaptive control framework regarding the existence of transient performance bounds in the presence of fast uncertainty, as well as robustness to unmodeled dynamics in the form of time delays in the control signals. In particular, the theoretical treatment guarantees the existence of a positive time-delay margin, below which stable operation can be expected. Several experimental designs involving one or multiple robotic manipulators are considered in order to validate the theoretical results and to explore possible seed transfer strategies.

The article Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay, written by Zaid Ahsan, Harry Dankowicz, and Jan Sieber, was originally published electronically on 5 November 2019 without open access.

This paper develops an adaptive controller for underactuated robotic systems with unmodeled dynamics. The control scheme is motivated by the applications of manipulators operating on dynamic platforms. The design decouples the system's adaptation and control loops to allow for fast estimation rates, while guaranteeing bounded deviation from a nonadaptive reference system. The proposed formulation is independent of detailed information about the system model. The control scheme is tested in different trajectory-tracking scenarios: (i) a manipulator installed on a ship operating in a high-sea state with uncertain environmental disturbances and (ii) a mobile manipulator moving across a rough terrain of unknown geometry. The simulation results illustrate the tracking performance of the proposed control algorithm, its ability to deal with unmodeled dynamics, and its robustness to measurement noise and time delay, while maintaining smooth control signals. An adaptive control formulation for underactuated robotic systems with unmodeled dynamics is proposed.The design allows for fast estimation while guaranteeing bounded deviation from a nonadaptive reference system.The proposed formulation is independent of detailed information about the system model.The proof of stability is established by the analysis based on input-output maps.The system's robustness to measurement noise and time delay is demonstrated.

This paper investigates the transient growth of aerosol particles in a humid environment. It seeks to explore the dependence of the fraction of droplet-forming particles on statistical properties of the distribution of dry particle diameters, as well as on the rate of temperature decay associated with vertical motion through the atmosphere. Low-dimensional, autonomous models are investigated using basic tools of dynamical systems analysis that establish the parameter-dependent existence and stability of families of equilibrium distributions of wet particle diameters. In the fully nonautonomous case, an original heuristic parameterization of the fraction of droplet-forming particles is derived in terms of a scalar, nonlinear boundary-value problem. To address the failure of the heuristic parameterization to account for the potential of a dynamic reversal of growth following an initial increase in particle diameter, a finely resolved, high-dimensional boundary-value formulation for the aerosol dynamics is i...

We consider a scheduling and sequencing optimization problem with application to an agricultural seeding operation. The problem is motivated by potential deployment of a recently proposed mobile seed refilling system in large-scale field operations. In this setting, mobile refiller vehicles are used to minimize any interruptions to seeding that result from the need to refill the seed tanks on the planters. A feature of this refilling system is that the two types of vehicles can move together during the seed transfer operation. The optimization task seeks to determine locations where seed transfer should be initiated, as well as an ordering of visits of refiller vehicles to such locations, such that the total downtime is minimized. In this paper, we consider such an optimization task for the case of multiple planters and a single refiller that, in turn, can refill itself at a single, fixed, off-field location. We rely on a stochastic solution strategy for solving this problem by applying the genetic algorithm to a binary integer programming model that represents the decision of whether to initiate refilling or not at a discrete set of possible locations along each planter path. The results are compared with a conventional static refilling scheme (with no mobile refiller), as well as a greedy (but generally suboptimal) algorithm that produces a unique solution to the selection and ordering of rendezvous locations. The computational analysis provides deep and useful insights to this optimization problem.

Human participants and recurrent ("connectionist") neural networks were both trained on a categorization system abstractly similar to natural language systems involving irregular ("strong") classes and a default class. Both the humans and the networks exhibited staged learning and a generalization pattern reminiscent of the Elsewhere Condition (Kiparsky, 1973). Previous connectionist accounts of related phenomena have often been vague about the nature of the networks' encoding systems. We analyzed our network using dynamical systems theory, revealing topological and geometric properties that can be directly compared with the mechanisms of non-connectionist, rule-based accounts. The results reveal that the networks "contain" structures related to mechanisms posited by rule-based models, partly vindicating the insights of these models. On the other hand, they support the one mechanism (OM), as opposed to the more than one mechanism (MOM), view of symbolic abstraction by showing how the appearance of MOM behavior can arise emergently from one underlying set of principles. The key new contribution of this study is to show that dynamical systems theory can allow us to explicitly characterize the relationship between the two perspectives in implemented models.

A modification to the ${\cal L}_1$ control framework for uncertain systems with actuator delay is presented. Specifically, a time delay is introduced in the control input of the state predictor to compensate for the destabilizing effect of input delay in the plant. For this modified framework, the analysis shows that the output of the adaptive system closely follows the behavior of a suitably defined, nonadaptive, stable reference system provided that a delay-dependent stability condition is satisfied and the adaptive gain is chosen sufficiently large. The set of combinations of input delay and compensation delay for which the stability condition is satisfied contains an open set of pairs of positive values provided that a filter bandwidth, characteristic of ${\cal L}_1$ adaptive control is chosen sufficiently large. The efficacy of the delay compensation is illustrated by a simple example. A numerical continuation is also performed to explore the stability region for a case where this can be approximated a priori., 18 pages, 6 figures

Summary This paper presents an analysis of the synchronization and consensus problems of networked manipulators operating on an underactuated dynamic platform in the presence of communication delays. The proposed formulation does not require detailed information about the system model. A theoretical formulation based on input–output maps of functional differential equations shows that the control system's behavior matches closely that of a non-adaptive reference system. The tracking synchronization objective is achieved despite the effects of the communication delay and the unknown dynamics of the platform. When there is no common desired trajectory, the modified controller drives all robots to average consensus for an unsigned graph and to bipartite consensus for a structurally balanced signed digraph. In addition, a leader–follower scheme is proposed that allows for the control of the constant and time-varying consensus values. Simulation results illustrate the performance of the proposed control algorithms. Copyright © 2016 John Wiley & Sons, Ltd.

This paper collects four distinct instances of grazing contact of a periodic trajectory in a hybrid dynamical system under a common abstract framework and establishes selected general properties of the associated near-grazing dynamics. In particular, it is shown that for critical choices of parameter values, commonly used physical models of rigid or compliant mechanical contact, capillary adhesion, and cell division satisfy the conditions required by the general framework. The paper relies on the well-known discontinuity-mapping formalism. In contrast to previous treatments, the proposed abstract framework more clearly establishes the origin of the large state-space stretching in the initial (and possibly only) step of the construction of a discontinuity mapping. It further highlights the nonuniqueness in the formulation of the discontinuity mapping and its connection to the choice of a locally smooth map with which the discontinuity mapping is composed to describe the near-grazing dynamics. The analysis is illustrated with examples from tapping-mode atomic force microscopy in the presence of thin fluid layers on the sample and the probe tip and from the study of protein activity during a eukaryotic cell cycle.

We hypothesize that spatiotemporal joint coupling patterns during gait are closely associated with musculoskeletal injury mechanics. Previous studies examining joint coupling, have primarily focused on coupling between single pairs of neighboring body segments or joints; thus falling short of characterizing the full spatiotemporal complexity across the entire gait apparatus. This study proposes the reliance on properties of the temporal cross-correlation of distinct joint variables as a means to characterize and detect differences in multiple segmental coupling pairs and to quantify how these couplings change between different gait conditions or test groups. In particular, for each subject, a characteristic diagram array is obtained whose entries include the maximum values of the crosscorrelation between all pairs of joint variables as well as the associated phase shifts at which these maxima are recorded. Paired t-tests are then used to highlight significant differences in the corresponding entries between two gait conditions. In the present study, this technique was applied to angular displacement and velocity histories across 12 lower extremity joint variables, for healthy subjects with and without a brace on the right knee. As expected, the statistical analysis indicated that the temporal cross-correlations associated with the right knee-angle variables differed the most between the two gait conditions. In addition, significant differences (p < 0.01) were found in the coupling between other pairs of joint variables, establishing a characteristic spatiotemporal signature for the changes from normative gait that result from reduced mobility at the knee.

In this paper we detail the design, construction, experimental investigation, theoretical modeling, and numerical exploration of a laboratory-scale electromechanical device, which includes the possibility of recurrent mechanical contact and exploitable multidomain signal transduction. In particular, we illustrate the use of inductive coupling between electrical and mechanical degrees of freedom to actuate and interrogate changes in the mechanical response due to the onset of contact through the coupled behavior of the device current. It is shown that nonsmooth bifurcations associated with a transition from nonimpacting to impacting dynamics, induced by changes in critical parameter values, may result in detectable changes in the current amplitude and, significantly, frequency content. The device design and the observed results are argued to support a similar functionality in microscale devices, where multidomain coupling is common.

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.

The focus of this paper is on the possibility of formulating a consistent and unambiguous forward simulation model of planar rigid-body mechanical systems with isolated points of intermittent or sustained contact with rigid constraining surfaces in the presence of dry friction. In particular, the analysis considers paradoxical ambiguities associated with the coexistence of sustained contact and one or several alternative forward trajectories that include phases of free-flight motion. Special attention is paid to the so-called Painleve paradoxes where sustained contact is possible even if the contact-independent contribution to the normal acceleration would cause contact to cease. Here, through taking the infinite-stiffness limit of a compliant contact model, the ambiguity in the case of a condition of sustained stick is resolved in favour of sustained contact, whereas the ambiguity in the case of a condition of sustained slip is resolved by eliminating the possibility of reaching such a condition from an open set of initial conditions. A more significant challenge to the goal of an unambiguous forward simulation model is afforded by the discovery of open sets of initial conditions and parameter values associated with the possibility of a left accumulation point of impacts or reverse chatter—a transition to free flight through an infinite sequence of impacts with impact times accumulating from the right on a limit point and with impact velocities diverging exponentially away from the limit point, even where the contact-independent normal acceleration supports sustained contact. In this case, the infinite-stiffness limit of the compliant formulation establishes that, under a specific set of open conditions, the possibility of reverse chatter in the rigid-contact model is an irresolvable ambiguity in the forward dynamics based at the terminal point of a phase of sustained slip. Indeed, as the existence of a left accumulation point of impacts is associated with a one-parameter family of possible forward trajectories, the ambiguity is of infinite multiplicity. The conclusions of the theoretical analysis are illustrated and validated through numerical analysis of an example single-rigid-body mechanical model.

A model is developed to describe the flow of grain through a clean grain elevator system on a combine in order to facilitate accurate mass flow rate estimation. The relationship between mass flow rate and impact force described by the model depends upon machine operational characteristics, mechanical interactions of the grain and the machine geometry, and material properties of the grain. The model was designed to be adaptable to varying grain conditions, such as those influenced by moisture content, by allowing free parameters of the model to be estimated through a nonlinear regression algorithm. Simulations were performed using discrete element modeling software and data was obtained from experiments conducted on a clean grain elevator system at the University of Kentucky Combine Yield Monitor Test Facility to determine the ability of the model to accurately estimate mass flow rate. The model estimated mass flow rate with a normalized root mean squared residual (NRMSR) less than 2% for discrete element modeling simulations. For experiments involving machine components, NRMSR values were less than 3% for corn at 14% moisture, less than 3% for corn at 21% moisture, and less than 5% for corn at 26% moisture.

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.

This paper develops a rigorous control paradigm for regulating the near-grazing bifurcation behavior of limit cycles in piecewise-smooth dynamical systems. In particular, it is shown that a discrete-in-time linear feedback correction to a parameter governing a state-space discontinuity surface can suppress discontinuity-induced fold bifurcations of limit cycles that achieve near-tangential intersections with the discontinuity surface. The methodology ensures a persistent branch of limit cycles over an interval of parameter values near the critical condition of tangential contact that is an order of magnitude larger than that in the absence of control. The theoretical treatment is illustrated with a harmonically excited damped harmonic oscillator with a piecewise-linear spring stiffness as well as with a piecewise-nonlinear model of a capacitively excited mechanical oscillator. Copyright © 2009 John Wiley & Sons, Ltd.

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.