# life: the pervasive "soft power" of ancient India.

<< One of the big contentions of popular historian William Dalrymple’s latest book “The Golden Road: How Ancient India Transformed the World,” which came out in the United States a few weeks ago, is that the Indian subcontinent’s connections to the West, especially via the Roman Empire, were far richer than those of China (i.e. the “Silk Road” cited). Once the might of Rome reached Egypt and the maritime routes of the Red Sea, it brought the customers of the Mediterranean to India’s doorstep. It also saw Indian philosophy and mathematics travel west and east. >>

<< Once their economic links to the West thinned with the collapse of the Roman Empire, South Indian merchant guilds turned east, embarking on trade and contacts that spread Indian religion and ideas across a wide expanse of Asia and underlay the grandeur of centuries-old temple complexes like Angkor Wat in Cambodia or Borubudur in Indonesia. >>

<< it’s one of the great soft power miracles of world history, because unlike Islam and unlike quite a lot of Christianity, no one took Buddhism at the point of a sword. No one imposed Buddhism at any point. It was the sophistication of its ideas and particularly its attractiveness to the merchant classes, bizarrely. The Buddhist monasteries act as banks, as factories and as caravanserais. >>

Ishaan Tharoor with Rachel Pannett. How ancient India changed the world. WorldView (by mail). washingtonpost.com. May 19, 2025.

https://s2.washingtonpost.com/camp-rw/?trackId=660de34bb906a33aae6b8103&s=682aadf6f062960b82a2f3e6

Also:  forms of power, waves, attractor, Zen, compassion, transition, in https://www.inkgmr.net/kwrds.html 

Keywords: life, forms of power, soft power, waves, attractors, Zen, compassion, transition.

# gst: strange attractors in complex networks

<< Disorder and noise in physical systems often disrupt spatial and temporal regularity, yet chaotic systems reveal how order can emerge from unpredictable behavior. Complex networks, spatial analogs of chaos, exhibit disordered, non-Euclidean architectures with hidden symmetries, hinting at spontaneous order. Finding low-dimensional embeddings that reveal network patterns and link them to dimensionality that governs universal behavior remains a fundamental open challenge, as it needs to bridge the gap between microscopic disorder and macroscopic regularities. >>

<< Here, the minimal space revealing key network properties is introduced, showing that non-integer dimensions produce chaotic-like attractors. >>

Pablo Villegas. Strange attractors in complex networks. Phys. Rev. E 111, L042301. Apr 15, 2025. 

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.111.L042301.  

arXiv: 2504.08629v1 [cond-mat.stat-mech] . Apr 11, 2025.

https://arxiv.org/abs/2504.08629

Also: disorder, disorder & fluctuations, noise, network, attractor, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, disorder, disorder & fluctuations, noise, networks, attractors, self-similarity, chaos 

# gst: weird quasiperiodic attractors

AA << consider a class of n-dimensional, n≥2, piecewise linear discontinuous maps that can exhibit a new type of attractor, called a weird quasiperiodic attractor. While the dynamics associated with these attractors may appear chaotic, (They)  prove that chaos cannot occur. The considered class of n-dimensional maps allows for any finite number of partitions, separated by various types of discontinuity sets. The key characteristic, beyond discontinuity, is that all functions defining the map have the same real fixed point. These maps cannot have hyperbolic cycles other than the fixed point itself. >>

Laura Gardini, Davide Radi, et al. Abundance of weird quasiperiodic attractors in piecewise linear discontinuous maps. arXiv: 2504.04778v1 [math.DS]. Apr 7, 2025.

https://arxiv.org/abs/2504.04778

Also: attractor, chaos, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, attractors, weird attractors, chaos

# 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