# gst: the solitary route to chimera states.

AA << show how solitary states in a system of globally coupled FitzHugh-Nagumo oscillators can lead to the emergence of chimera states. By a numerical bifurcation analysis of a suitable reduced system in the thermodynamic limit (they) demonstrate how solitary states, after emerging from the synchronous state, become chaotic in a period-doubling cascade. Subsequently, states with a single chaotic oscillator give rise to states with an increasing number of incoherent chaotic oscillators. In large systems, these chimera states show extensive chaos. (AA) demonstrate the coexistence of many of such chaotic attractors with different Lyapunov dimensions, due to different numbers of incoherent oscillators. >>

<<  While it is well known that self-organized wave patterns typically coexist within an interval of possible different wave numbers (..)(AA) show here the coexistence of coherence-incoherence patterns with different numbers of incoherent oscillators, which are in fact coexisting chaotic attractors with different Lyapunov dimensions. The incoherent oscillators in these coexisting attractors show extensive chaos of different dimensions. The total share of incoherent oscillators in a chimera state is a macroscopic quantity. Hence, within the range of such shares, where stable chimera states exist, (AA) find, for large systems, an increasing number of coexisting attractors with their numbers of incoherent oscillators increasing as well. (They) showed that, varying the coupling parameter, this extensive scenario is linked to the thermodynamic limit of the solitary regime, where the range of admissible numbers of incoherent oscillators shrinks down to one single oscillator in an infinitely large system. For this case, the emergence of the chaotic motion of the single incoherent oscillator could be shown in a period doubling cascade. >>

Leonhard Schulen, Alexander Gerdes, et al. The solitary route to chimera states. arXiv:2204.00385v1 [nlin.CD]. Apr 1, 2022.

https://arxiv.org/abs/2204.00385

Also

keyword 'FitzHugh-Nagumo oscillators' in APS | PubMed

https://journals.aps.org/search/

https://pubmed.ncbi.nlm.nih.gov/?term=FitzHugh-Nagumo+oscillators

keyword 'chaos' | 'chaotic' in Font

https://flashontrack.blogspot.com/search?q=chaos

https://flashontrack.blogspot.com/search?q=chaotic

keyword 'caos' | 'caotico' in Notes (quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=caos

https://inkpi.blogspot.com/search?q=caotico

keywords: gst, solitons, solitary states, period-doubling cascade, chaos, Lyapunov dimension, FitzHugh-Nagumo oscillator, chimera state, dynamical systems.

# gst: weird but not so weird dynamics, basins with tentacles could be common in high-dimensional systems.

<< Basins of attraction are fundamental to the analysis of dynamical systems (..). Over the years, many remarkable properties of basins have been discovered (..), most notably that their geometry can be wild, as exemplified by Wada basins (..), fractal basin boundaries (..), and riddled or intermingled basins (..). Yet despite these foundational studies, much remains to be learned about basins, especially in systems with many degrees of freedom. >>

AA show that for locally-coupled Kuramoto oscillators << high-dimensional basins tend to have convoluted geometries and cannot be approximated by simple shapes such as hypercubes. Although they are impossible to visualize precisely (because of their high dimensionality), (they) present evidence that these basins have long tentacles that reach far and wide and become tangled with each other. Yet sufficiently close to its own attractor, each basin becomes rounder and more simply structured, somewhat like the head of an octopus. >>

<< In terms of (AA) metaphor, almost all of a basin’s volume is in its tentacles, not its head. This finding is not limited to Kuramoto oscillators. (AA) provide a simple geometrical argument showing that, as long as the number of attractors in a system grows subexponentially with system size, the basins are expected to be octopus-like. As further evidence of their genericity, basins of this type were previously found in simulations of jammed sphere packings (..) where they were described as “branched” and “threadlike” away from a central core (..) and accurate methods were developed for computing their volumes (.,). There is also enticing evidence of octopus-like basins in neuronal networks (..), power grids (..), and photonic couplers (..). >>

<< Figure 4 is a further attempt to visualize the structure of high-dimensional basins, now by examining randomly oriented two-dimensional (2D) slices of state space, either far from a twisted state or close to one. (..) Despite the fact that each basin is connected (..)  the basins look fragmented in this 2D slice. >>

 Fig. 4(a): << Perhaps another metaphor than tentacles—a ball of tangled yarn—better captures the essence of the basin structure in this regime, far from any attractor, in which differently colored threads (representing different basins) are interwoven together in an irregular fashion. >>

Fig. 4(b): << The basin structure near an attractor is strikingly different. (..) the basins near an attractor are organized like an onion. >>

Yuanzhao Zhang, Steven H. Strogatz. Basins with tentacles. arXiv: 2106.05709v3 [nlin.AO]. Nov 2, 2021. 

https://arxiv.org/abs/2106.05709

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.194101

Also

Reshaping Kuramoto model, when a collective dynamics becomes chaotic, with a surprisingly weak coupling. Dec 27, 2021.

https://flashontrack.blogspot.com/2021/12/gst-reshaping-kuramoto-model-when.html

Keywords: gst, dynamical systems, high-dimensional systems, Kuramoto oscillator, attractors, basin of attraction