Piecewise Isometries: Applications in Engineering

Deane, Jonathan H. B.

doi:10.1007/s11012-005-5895-3

Cited by 23 publications

(21 citation statements)

References 10 publications

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“…From the viewpoint of nonlinear dynamics, herding systems belong to the class of piecewise isometries; the state space can be divided into regions by the symbol s that minimizes the energy function, and in each region, its dynamics is merely a translation by a constant vector μ−φ(s). It is known that piecewise isometries generally have complex dynamics and fractal attracting sets despite the Lyapunov exponents strictly equal to zero [3][4][5][6][7][8][9][10]. Herding systems also have fractal attracting sets, as shown in Fig.…”

Section: Herding Systemmentioning

confidence: 99%

“…More specifically, the dynamics of herding systems belongs to the class of piecewise isometries. As is the case for many piecewise isometries [3][4][5][6][7][8][9][10], the herding systems typically have complex dynamics with fractal attracting sets [1,2]. However, the Lyapunov exponents of the dynamics are strictly zero, which does not seem to properly characterize the complexity of the dynamics.…”

Section: Introductionmentioning

confidence: 99%

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Chaotic billiard dynamics for herding

Suzuki

2015

NOLTA

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Abstract:Herding systems are discrete-time nonlinear dynamical systems designed efficiently for statistical inference. In this paper, we introduce a continuous-time version of these systems, which we call herding billiard systems. In contrast to the weakly chaotic dynamics of the original version, the continuous-time version is shown to have chaotic pseudo-billiard dynamics, while inheriting the fundamental sampling functions. We also present the connection between these two versions. Thus, herding billiard systems provide a novel approach to the complexity of herding systems from the viewpoint of chaotic billiard dynamics.

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Section: Herding Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Chaotic billiard dynamics for herding

Suzuki

2015

NOLTA

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“…The induced isometries, which are rotations about points S 0 and S 1 determined by T and the line S 0 S 1 is perpendicular to the discontinuity line. T is also the simplified model for the buck converter [21,22]. The two discs centered at S 0 and S 1 are fixed by the map T and they constitute a global piecewise isometric attractor, which is called the 8-attractor [29].…”

Section: The Goetz Mapmentioning

confidence: 99%

On global attractors for a class of nonhyperbolic piecewise affine maps

Duan

2008

Physica D: Nonlinear Phenomena

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“…[3][4][5][6] This can be considered an example of complex dynamics without the "usual symptoms" of chaos and related 3,5,6 to the mathematical concept of a piecewise isometry (PWI). [7][8][9][10] Because such complex dynamics do not arise from the usual "stretching and folding" mechanism of chaotic mixing, the new mechanism has been termed 3 "cutting and shuffling," a tip to the similarity between PWIs and the "mixing" of a deck of cards. 11 However, many questions remain as to the extent of PWI dynamics in the vanishingflowing-layer limit and the implications for mixing and transport in granular (and related) flows.…”

Section: Introductionmentioning

confidence: 99%

From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows

2011

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Through a combined computational-experimental study of flow in a slowly rotating quasi-twodimensional container, we show several new aspects related to the kinematics of granular mixing. In the Lagrangian frame, for small numbers of revolutions, the mixing pattern is captured by a model termed "streamline jumping." This minimal model, arising at the limit of a vanishingly thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories. Meanwhile, for intermediate numbers of revolutions, we show the presence of naturally persistent granular mixing patterns, i.e., "strange" eigenmodes of the advection-diffusion operator governing the mixing process in Eulerian frame. Through a comparative analysis of the structure of eigenmodes and the corresponding Poincaré section and finite-time Lyapunov exponent field of the flow, the relationship between the Eulerian and Lagrangian descriptions of mixing is highlighted. Finally, we show how the mapping method for scalar transport can be modified to include diffusion. This allows us to examine (for the first time in a granular flow) the change in shape, lifespan, and eventual decay of eigenmodes due to diffusive effects at larger numbers of revolutions.

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Piecewise Isometries: Applications in Engineering

Cited by 23 publications

References 10 publications

Chaotic billiard dynamics for herding

Chaotic billiard dynamics for herding

On global attractors for a class of nonhyperbolic piecewise affine maps

From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows

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