Stephen J. Hogan

Euler’s disk is a heavy, polished steel disk that can spin on a concave base-plate. As it spins, the contact point of the disk with the base-plate describes a circle, and its center of mass slowly descends towards the base-plate, after which the disk falls flat to a sudden halt. We present an extended review of the experimental and theoretical literature. We describe a series of experiments on a commercially available Euler’s disk spinning on different base-plates (flat and concave). We show for the first time that the material properties (Poisson’s ratio and Young’s modulus) of flat base-plates affect the motion of the disk. We show that the disk can continue to rotate about a vertical axis for a short time after it falls flat, an effect (which has not been seen before) exaggerated when the base-plate is lubricated with general purpose oil. We solve the equations of motion numerically for the disk, subject to both Coulomb friction and contour friction. Qualitative agreement with our experimental data was found. An additional multiplicative factor applied to contour friction gave much better quantitative agreement. The discontinuity in contour friction provides us with a mechanism for the continued rotation of the disk after it falls flat, giving good agreement with experiments. This part of the motion has two phases for the flat base-plate: a contactless phase, where the disk is supported by a thin layer of fluid (air or oil), followed by a contact phase across the face of the disk where Coulomb friction brings it to a halt. For a concave base-plate, there is no contactless phase and the circumference of the disk contacts the base-plate and Coulomb friction brings it to a halt.

Sometimes, when a golfer attempts to putt a golf ball, it appears to enter the hole, only to re-emerge almost immediately, having undergone an angle of turn around the hole rim that can exceed 180 ∘ . We consider the problem from the point of view of mechanics. We show analytically that there are at least two distinct types of lip out : the rim lip out, where the centre of mass of the golf ball does not fall below the level of the green, and the hole lip out where it does. At the heart of both lip outs is a family of degenerate saddle equilibria of the dynamics on the rim ( the golf balls of death ). When perturbed one way, the golf ball executes a rim lip out. When perturbed another way, the golf ball enters the hole, only to re-emerge (provided it does not touch the base of the hole) if it is spinning about an axis perpendicular to the wall of the hole.

We classify Filippov’s type 3 singular points of planar bimodal piecewise smooth systems. These singular points consist of fold or cusp tangencies of the vector fields to both sides of a switching surface. For isolated analytic type 3 singular points there are 25 topological classes, up to time reversal. For isolated general type 3 singular points there are 40 topological classes, up to time reversal.

In this paper, we analyze the celebrated Haken–Kelso–Bunz model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behavior, and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and antiphase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and center manifold reduction. Moreover, we uncover further details on the global dynamic behavior by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.

The classical Painlevé paradox consists of a slender rigid rod slipping on a rigid rough surface. If the coefficient of friction μ is high enough, the governing equations predict that the rod would be driven into the surface. The paradox is well studied in two dimensions, in which the paradox is resolved via regularization, where the rod tip meets the surface. In this paper, we consider the three-dimensional problem. There are two significant differences in three dimensions. Firstly, sticking now occurs on a co-dimension 2 surface. This results in a non-smooth problem, even when the three-dimensional problem is regularized. Secondly, unlike the highly singular two-dimensional problem, trajectories can now enter the inconsistent region from slipping, requiring a completely new analysis. We use blowup to investigate the problem and show that a key part of the dynamics of the regularized three-dimensional Painlevé problem is governed by a type I Painlevé equation.

The rocking can problem [M. Srinivasan and A. Ruina, Phys. Rev. E, 78 (2008), 066609] consists of an empty drinks can standing upright on a horizontal plane which, when tipped back to a single contact point and released, rocks down towards the flat and level state. At the bottom of the motion, the contact point moves quickly around the rim of the can. The can then rises up again, having rotated through some finite angle of turn [Formula: see text]. We recast the problem as a second order ODE and find a Frobenius solution. We then use this Frobenius solution to derive a reduced equation of motion. The rocking can exhibits two distinct phenomena: behavior very similar to an inverted pendulum, and dynamics with the angle of turn. This distinction allows us to use matched asymptotic expansions to derive a uniformly valid solution that is in excellent agreement with numerical calculations of the reduced equation of motion. The solution of the inner problem was used to investigate of the angle of turn phenomenon. We also examine the motion of the contact locus [Formula: see text] and see a range of different trajectories, from circular to petaloid motion and even cusp-like behavior. Finally, we obtain an approximate lower bound for the required coefficient of friction to avoid slip.

In this paper, we analyse the celebrated Haken-Kelso-Bunz (HKB) model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behaviour and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and anti-phase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and centre manifold reduction. Moreover, we uncover further details on the global dynamic behaviour by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.

The sewed focus is one of the singularities of planar piecewise smooth dynamical systems. Defined by Filippov in his book 'Differential Equations with Discontinuous Righthand Sides' (Kluwer, 1988), it consists of two invisible tangencies either side of the switching manifold. In the case of analytic focus-like behaviour, Filippov showed that the approach to the singularity is in infinite time. For the case of non-analytic focus-like behaviour, we show that the approach to the singularity can be in finite time. For the non-analytic sewed centre-focus, we show that there are uncountably many different topological types of local dynamics, including cases with infinitely many stable periodic orbits, and show how to create systems with periodic orbits intersecting any bounded symmetric closed set.

The sewed focus is one of the singularities of planar piecewise smooth dynamical systems. Defined by Filippov in his book (Differential Equations with Discontinuous Righthand Sides, Kluwer, 1988), it consists of two invisible tangencies either side of the switching manifold. In the case of analytic focus-like behaviour, Filippov showed that the approach to the singularity is in infinite time. For the case of non-analytic focus-like behaviour, we show that the approach to the singularity can be in finite time. For the non-analytic sewed centre-focus, we show that there are uncountably many different topological types of local dynamics, including cases with infinitely many stable periodic orbits, and show how to create systems with periodic orbits intersecting any bounded symmetric closed set.