Petri T. Piiroinen

Bifurcations in Nonsmooth Dynamical Systems

A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of “normal form” or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

An event-driven method to simulate Filippov systems with accurate computing of sliding motions

This article describes how to use smooth solvers for simulation of a class of piecewise smooth systems of ordinary differential equations, called Filippov systems, with discontinuous vector fields. In these systems constrained motion along a discontinuity surface (so-called sliding) is possible and requires special treatment numerically. The introduced algorithms are based on an extension to Filippovs method to stabilise the sliding flow together with accurate detection of the entrance and exit of sliding regions. The methods are implemented in a general way in MATLAB and sufficient details are given to enable users to modify the code to run on arbitrary examples. Here, the method is used to compute the dynamics of three example systems, a dry-friction oscillator, a relay feedback system and a model of an oil well drill-string.

TWO-PARAMETER DISCONTINUITY-INDUCED BIFURCATIONS OF LIMIT CYCLES: CLASSIFICATION AND OPEN PROBLEMS

This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. Several such codimension-one events have recently been identified, causing for example, period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincare map from a neighborhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the grazing cycle is itself degenerate (e.g. nonhyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that with discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

Chaos and Period-Adding; Experimental and Numerical Verification of the Grazing Bifurcation

SummaryExperimental results are presented for a single-degree-of-freedom horizontally excited pendulum that is allowed to impact with a rigid stop at a fixed angle θ to the vertical. By inclining the apparatus, the pendulum is allowed to swing in an effectively reduced gravity, so that for each fixed θ less than a critical value, a forcing frequency is found such that a period-one limit cycle motion just grazes with the stop. Experimental measurements show the immediate onset of chaotic dynamics and a period-adding cascade for slightly higher frequencies. These results are compared with a numerical simulation and continuation of solutions to a mathematical model of the system, which shows the same qualitative effects. From the model, the theory of discontinuity mappings due to Nordmark is applied to derive the coefficients of the square-root normal form map of the grazing bifurcation for this system. The grazing periodic orbit and its linearisation are found using a numerical continuation method for hybrid systems. From this, the normal-form coefficients are computed, which in this case imply that a jump to chaos and period-adding cascade occurs. Excellent quantitative agreement is found between the model simulation and the map, even over wide parameter ranges. Qualitatively, both accurately predict the experimental results, and after a slight change in the effective damping value, a striking quantitative agreement is found too.

Low-velocity impacts of quasiperiodic oscillations

Abstract A method based on the idea of a discontinuity mapping is derived for predicting the characteristics of system attractors that occur following a grazing intersection of a two-frequency, quasiperiodic oscillation with a two-dimensional impact surface in a three-dimensional state space. Within certain restrictions, the correction to the non-impacting flow afforded by the discontinuity mapping is computable using quantities determined solely by the non-impacting flow and the properties of the impact surface and the associated impact mapping in the immediate vicinity of the initial grazing contact. A model example is discussed to illustrate the quantitative predictive power of the discontinuity-mapping approach even relatively far away in parameter space from the original grazing intersection. Finally, constraints on the applicability of the methodology are described in detail with suggestions for suitable modifications.

Exploiting discontinuities for stabilization of recurrent motions

A rigorous mathematical technique is presented for exploiting the presence of discontinuities in non-smooth dynamical systems in order to control the local stability of periodic or other recurrent motions. In particular, the formalism allows one to predict the effects of the control strategy based entirely on information about the uncontrolled system. The methodology is illustrated with examples from impacting systems, namely a model hopping robot and a Braille printer. It is shown how initially unstable motions can be successfully stabilized at negligible cost and without active energy injection.

Low-cost control of repetitive gait in passive bipedal walkers

Small, discrete, corrective adjustments to foot geometry in a class of bipedal passive walkers are employed to affect the local stability properties of a periodic reference gait. It is demonstrated that successful stabilization can be accomplished for otherwise strongly unstable motions of vertically constrained as well as entirely unconstrained model mechanisms. In particular, recent results by the authors on stabilization of repetitive motion in hybrid dynamical systems are implemented to formulate a rigorous analytical methodology for predicting the stability characteristics of the controlled system. It is demonstrated that these predictions can be explicitly obtained based entirely on knowledge of the local stability properties of the reference motion in the absence of control.

Basins of attraction in nonsmooth models of gear rattle

This paper is concerned with the computation of the basins of attraction of a simple one degree-of-freedom backlash oscillator using cell-to-cell mapping techniques. This analysis is motivated by the modeling of order vibration in geared systems. We consider both a piecewise-linear stiffness model and a simpler infinite stiffness impacting limit. The basins reveal rich and delicate dynamics, and we analyze some of the transitions in the systems behavior in terms of smooth and discontinuity-induced bifurcations. The stretching and folding of phase space are illustrated via computations of the grazing curve, and its preimages, and manifold computations of basin boundaries using DsTool (Dynamical Systems Toolkit).

Local dynamics in decision making: The evolution of preference within and across decisions

Within decisions, perceived alternatives compete until one is preferred. Across decisions, the playing field on which these alternatives compete evolves to favor certain alternatives. Mouse cursor trajectories provide rich continuous information related to such cognitive processes during decision making. In three experiments, participants learned to choose symbols to earn points in a discrimination learning paradigm and the cursor trajectories of their responses were recorded. Decisions between two choices that earned equally high-point rewards exhibited far less competition than decisions between choices that earned equally low-point rewards. Using positional coordinates in the trajectories, it was possible to infer a potential field in which the choice locations occupied areas of minimal potential. These decision spaces evolved through the experiments, as participants learned which options to choose. This visualisation approach provides a potential framework for the analysis of local dynamics in decision-making that could help mitigate both theoretical disputes and disparate empirical results.

Original article: Numerical analysis of codimension-one, -two and -three bifurcations in a periodically-forced impact oscillator with two discontinuity surfaces

We analyse a model of a periodically-forced impact oscillator with two discontinuity surfaces. This model describes a pair of meshing gears, where the discontinuities arise from impacts between the gear teeth. A classical approach of basin-of-attraction computations and bifurcation diagrams is used in conjunction with the recently developed discontinuity-geometry methodology to provide new insights into the extremely rich dynamical behaviour observed. In particular, we show that all periodic solutions with impacts emanate from a codimension-three bifurcation.