Bifurcations of Stable Sets in Noninvertible Planar Maps

England, James P.; Krauskopf, Bernd; Osinga, Hinke M.

doi:10.1142/9789812774569_0011

Cited by 3 publications

(8 citation statements)

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“…Moreover, a preimage of part of the rightmost open-loop boundary forms another portion of the basin boundary, which ‘erodes’ the basin from the bottom. Consistent with these numerical results, the open-loop basin boundaries are, in fact, the stable set of the saddle [29,30] of the walker’s non-invertible map F (equation (2.1)). The disjoint basin boundaries, in between the leftmost and the rightmost boundary, form either ‘slits’ with open ends or ‘holes’ (due to closed loops) in the basin.…”

Section: Global Stability Of the Powered Compass Walkersupporting

confidence: 72%

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Task-level regulation enhances global stability of the simplest dynamic walker

Patil

Dingwell

Cusumano

2020

J. R. Soc. Interface.

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Much remains unknown about how considerations such as stability and energy minimization shape the way humans walk. While active neuromotor control keeps humans upright, they also need to choose from multiple stepping regulation strategies to achieve one or more task goals, such as maintaining a desired speed or direction. Experiments on human treadmill walking motivate an important question: why do humans prefer one task-level regulation strategy over another—perhaps to enhance their ability to reject large disturbances? Here, we study the relationship between task-level regulation and global stability in a powered compass walker on a treadmill, with added step-to-step speed and position regulators. For treadmill walking, we find that speed regulation greatly enlarges and regularizes the unregulated walker’s stability region, i.e. its basin of attraction, much more than position regulation. Thus, our results suggest a possible explanation for the experimental finding that humans strongly prioritize regulating speed from one stride to the next, even as they walk economically on average. Furthermore, our work suggests a functional connection between task-level motor regulation and global stability—and, thus, perhaps even fall risk.

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Section: Global Stability Of the Powered Compass Walkersupporting

confidence: 72%

“…a union of many disjoint simply and/or multiply connected regions). We suspect that the latter is more likely because of the non-invertibility [29,30] of the walker’s hybrid dynamics (appendix A). However, our numerical results do not allow us to resolve this subtle question here.…”

Section: Global Stability Of the Powered Compass Walkermentioning

confidence: 99%

Task-level regulation enhances global stability of the simplest dynamic walker

Patil

Dingwell

Cusumano

2020

J. R. Soc. Interface.

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“…Cusp points arise when the tangent to the torus at the point of intersection with the critical curve LC −1 coincides with the eigendirection corresponding to vanishing eigenvalue for the non-invertible map [19,23,22]. The formation of cusp points on the unstable manifold can be interpreted as a global bifurcation of the invariant torus [33]. This bifurcation leads to the creation of structural stable selfintersections and loops on the unstable manifold as seen, for example, in Fig.…”

Section: Multi-layered Tori In Other Non-invertible Mapsmentioning

confidence: 99%

Novel routes to chaos through torus breakdown in non-invertible maps

Zhusubaliyev

Mosekilde

2009

Physica D: Nonlinear Phenomena

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“…In the case of fold maps, bifurcations of the critical curves with the boundaries of basins of attractions have been investigated in [4,17,37,43,47]. The approach taken in [22,35,41] reveals that the myriad of exotic transformations of basin boundaries for fold maps are different global manifestations of local tangencies of a stable or unstable set with the critical curves. The bifurcations we find for system (1.1) share some overall mechanisms with the results presented in [22,35,41], such as changes in the connectivity of the stable set and loop forming of the unstable set.…”

Section: Introductionmentioning

confidence: 99%

“…The approach taken in [22,35,41] reveals that the myriad of exotic transformations of basin boundaries for fold maps are different global manifestations of local tangencies of a stable or unstable set with the critical curves. The bifurcations we find for system (1.1) share some overall mechanisms with the results presented in [22,35,41], such as changes in the connectivity of the stable set and loop forming of the unstable set. However, the map (1.1) maps the plane onto itself in a 2-to-1 fashion that is different from that of fold maps.…”

Section: Introductionmentioning

confidence: 99%

Interacting Global Invariant Sets in a Planar Map Model of Wild Chaos

Hittmeyer¹,

Krauskopf²,

Osinga³

2013

SIAM J. Appl. Dyn. Syst.

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Wild chaos is a type of chaotic dynamics that can arise in a continuous-time dynamical system of dimension at least four. We are interested in the possible bifurcations or sequence of bifurcations that generate this type of chaos in dynamical systems. We focus our investigation on a planar noninvertible map introduced by Bamón, Kiwi, and Rivera-Letelier [Wild Lorenz-Like Attractors, arXiv 0508045, 2006] to prove the existence of wild chaos in a five-dimensional Lorenz-like system. The map opens up the origin (the critical point) to a disk and wraps the plane twice around it; points inside the disk have no preimage. The bounding critical circle and its images, together with the critical point and its preimages, form the so-called critical set. This set interacts with the stable and unstable sets of a saddle fixed point. Advanced numerical techniques enable us to study how the stable and unstable sets change as a parameter is varied along a path towards the wild chaotic regime. We find four types of bifurcations: The stable and unstable sets interact with each other in homoclinic tangencies (which also occur in invertible maps), and they interact with the critical set in three types of bifurcations specific to noninvertible maps, which cause changes (such as selfintersections) of the topology of these global invariant sets. Overall, a consistent sequence of all four types of bifurcations emerges, which we present as a first attempt towards explaining the geometric nature of wild chaos. Using two-parameter bifurcation diagrams, we show that essentially the same sequences of bifurcations occur along different paths towards the wild chaotic regime, and we use this information to obtain an indication of the size of the parameter region where wild chaos exists. Introduction.Vector fields and diffeomorphisms with chaotic attractors arise in numerous fields of applications, including neuroscience [23], chemical reactions [24], and laser systems [45]. One of the first and best known examples of a chaotic three-dimensional vector field is the Lorenz system [44]. It was introduced in 1963 by meteorologist Lorenz as a simplified model of convection dynamics in the earth's atmosphere. At the classical parameter values, its chaotic attractor-the Lorenz attractor -has the shape of the wings of a butterfly. The dynamics on this attractor has been studied in great detail; see, for example, [2,3,32,62,63]. More recently, the focus has shifted to what this means for the global behavior on the entire phase space [19,20].The approach taken for studying the Lorenz system at the classical parameter values is

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Bifurcations of Stable Sets in Noninvertible Planar Maps

Cited by 3 publications

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Task-level regulation enhances global stability of the simplest dynamic walker

Task-level regulation enhances global stability of the simplest dynamic walker

Novel routes to chaos through torus breakdown in non-invertible maps

Interacting Global Invariant Sets in a Planar Map Model of Wild Chaos

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