Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

Simpson, David J. W.

doi:10.1080/10236198.2022.2070009

Cited by 5 publications

(2 citation statements)

References 32 publications

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“…Under some conditions on the system parameters, there exists a well-defined and invertible coordinate change that transforms the general 2d PWL map (1) into a 2d PWL normal form map [19,41]. For instance when T is generic in the sense that T (Σ) intersects Σ at a unique point which is not a fixed point of T , or equivalently c ̸ = 0 and (1 − d) h 1 + c h 2 ̸ = 0; see [19] for more details.…”

Section: General 2d Pwl Mapsmentioning

confidence: 99%

“…Afterwards, [19] illustrated a constructive approach to examine robust chaos, in the original parameter regime of [6], based on invariant manifolds and expanding cones. Very recently, Simpson [41] detected invariant expanding cones and presented a general method to identify robust chaotic attractors in 2d border-collision normal form maps. Here, we investigate the existence of chaos for the general PWL system (1) where T is an invertible map.…”

Section: Chaosmentioning

confidence: 99%

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Robust chaos and multi-stability in piecewise linear recurrent neural networks

Monfared

Patra

Durstewitz

2022

Preprint

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Recurrent neural networks (RNNs) are major machine learning tools for the processing of sequential data. Piecewise-linear RNNs (PLRNNs) in particular, which are formally piecewise linear (PWL) maps, have become popular recently as data-driven techniques for dynamical systems reconstructions from time-series observations. For a better understanding of the training process, performance, and behavior of trained PLRNNs, more thorough theoretical analysis is highly needed. Especially the presence of chaos strongly affects RNN training and expressivity. Here we show the existence of robust chaos in 2d PLRNNs. To this end, necessary and sufficient conditions for the occurrence of homoclinic intersections are derived by analyzing the interplay between stable and unstable manifolds of 2d PWL maps. Our analysis focuses on general PWL maps, like PLRNNs, since normal form PWL maps lack important characteristics that can occur in PLRNNs. We also explore some bifurcations and multi-stability involving chaos, since the co-existence of chaotic attractors with other attractor objects poses particular challenges for PLRNN training on the one hand, yet may endow trained PLRNNs with important computational properties on the other. Numerical simulations are performed to verify our results and are demonstrated to be in good agreement with the theoretical derivations. We discuss the implications of our results for PLRNN training, performance on machine learning tasks, and scientific applications. Mathematics Subject Classification (2020) 37F46 · 34H10

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Section: General 2d Pwl Mapsmentioning

confidence: 99%

Section: Chaosmentioning

confidence: 99%

Robust chaos and multi-stability in piecewise linear recurrent neural networks

Monfared

Patra

Durstewitz

2022

Preprint

View full text Add to dashboard Cite

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Inclusion of higher-order terms in the border-collision normal form: Persistence of chaos and applications to power converters

Simpson,

Glendinning

2024

Physica D: Nonlinear Phenomena

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Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications

Belykh

Kuske

Porfiri

et al. 2023

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic “Bristol book” [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.

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Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

Cited by 5 publications

References 32 publications

Robust chaos and multi-stability in piecewise linear recurrent neural networks

Robust chaos and multi-stability in piecewise linear recurrent neural networks

Inclusion of higher-order terms in the border-collision normal form: Persistence of chaos and applications to power converters

Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications

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