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Research Papers

An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints

[+] Author and Article Information
Harry Dankowicz

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801danko@illinois.edu

Frank Schilder

Department of Mathematics, Technical University of Denmark, 2800 Kongens Lyngby, Denmarkf.schilder@mat.dtu.dk

J. Comput. Nonlinear Dynam 6(3), 031003 (Dec 15, 2010) (8 pages) doi:10.1115/1.4002684 History: Received May 05, 2009; Revised August 23, 2010; Published December 15, 2010; Online December 15, 2010

Abstract

This paper presents an extended formulation of the basic continuation problem for implicitly defined, embedded manifolds in $Rn$. The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-codimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand, and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation core package COCO and auxiliary toolboxes, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as the detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.

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Figures

Figure 1

Cusp normal form equilibrium manifold. Highlighted are a branch of equilibria (thin) and a branch of saddle-node bifurcation points (thick).

Figure 2

Schematic of a mass m resting against a rough substrate and acted upon by a linear restoring force with stiffness k and collisional interactions with a harmonically oscillating unilateral constraint

Figure 3

Branch of periodic trajectories of the hybrid dynamical system of a given signature under variations in k. Only those points on the branch to the left of and including the point with k≈5.573 satisfy the compatibility conditions.

Figure 4

A five-segment state-space trajectory obtained when k≈5.573. Note in particular the termination of the second negative slip segment on the boundary between initial conditions that result in a subsequent phase of stick and initial conditions that result in a subsequent phase of positive slip.

Figure 5

Two-parameter switching-sliding (solid) and grazing (dashed) bifurcation curves under simultaneous variations in k and b. Of the points on the switching-sliding curve, only those above and including the point with k≈3.448 and b≈0.6614 satisfy the compatibility conditions.

Figure 6

A five-segment state-space trajectory obtained when k≈3.448 and b≈0.6614. Note in particular that the first negative slip segment terminates at a grazing intersection with the surface himpact=0, while the fourth segment again terminates on the boundary between initial conditions that result in a subsequent phase of stick and initial conditions that result in a subsequent phase of positive slip.

Figure 7

Three-parameter switching-sliding/grazing (thick solid) bifurcation curve and switching-sliding and grazing bifurcation surfaces under simultaneous variations in k, b, and ω. Here, the switching-sliding (solid) and grazing (dashed) bifurcation curves are the intersections of two-dimensional bifurcation surfaces with the ω=1 plane.

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