Impact dynamics of large dimensional systems

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Dynamics of symmetric dynamical systems with delayed switching

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincare map near the colliding periodic orbit. The Poincare map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision.The research of JS and PK was partially supported by EPSRC grant GR/R72020/0

Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. 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 Poincaré map from a neighbourhood 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 the grazing cycle is itself degenerate (e.g. non-hyperbolic) 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 have 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.

Submarine glacial-landform distribution along an Antarctic Peninsula palaeo-ice stream: A shelf-slope transect through the Marguerite Trough system (66-70° S)

The Antarctic Peninsula comprises a thin spine of mountains and islands presently covered by an ice sheet up to 500 m thick that drains eastwards and westwards via outlet glaciers (Davies et al. 2012). Recently, the Peninsula has undergone rapid warming, resulting in the collapse of fringing ice shelves and the retreat, thinning and acceleration of marine-terminating outlet glaciers (e.g. Pritchard & Vaughan 2007). At the Last Glacial Maximum (LGM), the ice sheet expanded to the continental-shelf break around the Peninsula, and was organised into a series of ice streams that drained along cross-shelf bathymetric troughs (Ó Cofaigh et al. 2014). Marguerite Bay is located on the west side of the Antarctic Peninsula, at about 66° to 70° S (Fig. 1). A 12–80 km wide and 370 km long trough extends across the bay from the northern terminus of George VI Ice Shelf to the continental shelf edge. Extensive marine-geophysical surveys of the trough reveal a suite of glacial landforms which record past flow of an ice stream, which extended to the shelf edge at, or shortly after, the LGM. Subsequent retreat of the ice stream was underway by ~14 kyr ago and proceeded rapidly to the mid-shelf, where it slowed before accelerating once again to the inner shelf at ~9 kyr (Kilfeather et al. 2011).This is the author accepted manuscript. The final version is available from the Geological Society of London via https://doi.org/10.1144/M46.18