chaos

Lenny Sirivong | Notes | What It Is Actually Like to Be in the Engine Room...
My Blog: What It Is Actually Like to Be in the Engine Room of the Start-Up Economy
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19 days ago by vntn
[math/0702073] Unbounded Orbits for Outer Billiards
Outer billiards is a basic dynamical system, defined relative to a planar convex shape. This system was introduced in the 1950's by B.H. Neumann and later popularized in the 1970's by J. Moser. All along, one of the central questions has been: is there an outer billiards system with an unbounded orbit. We answer this question by proving that outer billiards defined relative to the Penrose Kite has an unbounded orbit. The Penrose kite is the quadrilateral that appears in the famous Penrose tiling. We also analyze some of the finer orbit structure of outer billiards on the penrose kite. This analysis shows that there is an uncountable set of unbounded orbits. Our method of proof relates the problem to self-similar tilings, polygon exchange maps, and arithmetic dynamics.
billiards  mathematical-recreations  generative-models  geometry  dynamical-systems  rather-interesting  chaos  to-simulate  to-write-about
5 weeks ago by Vaguery
[1206.5223] Outer Billiards, Digital Filters and Kicked Hamiltonians
In 1978 Jurgen Moser suggested the outer billiards map (Tangent map) as a discontinuous model of Hamiltonian dynamics. A decade earlier, J.B. Jackson and his colleagues at Bell Labs were trying to understand the source of self-sustaining oscillations in digital filters. Some of the discrete mappings used to describe these filters show a remarkable ability to 'shadow' the Tangent map when the polygon in question is regular.
In this paper we describe a specific digital filter map (Df) that appears to have dynamics which are conjugate to the Tangent map for a regular N-gon with N even. When N is odd, there is evidence of another conjugacy between the Tangent map dynamics of N and the matching 2N-gon, so a case like N = 7 can be studied with the Df map in the context of N = 14. This provides a many-fold increase in efficiency, and also allows us to generalize the Tangent map to obtain 'step-k' versions - which have dynamics that are unexplored.
We also present some related maps, including Chua and Lin's 3-dimensional version of Df, an Analog to Digital Converter from Orla Feely, a sawtooth version of the Standard Map by Peter Ashwin and various kicked harmonic oscillators. All of these seem to shadow the Tangent map in some form. Mathematica code is provided for all mappings both here and at this http URL.
billiards  dynamical-systems  chaos  rather-interesting  purdy-pitchers  to-do  to-write-about  physics!
5 weeks ago by Vaguery
Chiron: Initiation and the process of Individuation – Jessica Davidson
Superb: Chiron as gateway to #darknight and transformation + Link to global chaos and how to transcend thru awareness, acceptance #kudos
chiron  astrology  spirituality  god  ego  spirit  dark  night  soul  myth  wound  chaos  healing  global  acceptance  grace
5 weeks ago by csrollyson
Scientists reveal for first time the exact process by which chaotic systems synchronize
Via John Helm
"In a new study published in Physical Review E, physicists from Bar-Ilan University in Israel, along with colleagues from Spain, India and Italy, analyzed the Rossler system, a well-known chaotic system which physicists have studied thoroughly for almost 40 years. Looking at this system from a fresh perspective, they discovered new phenomena that have been overlooked until now. For the first time the researchers were able to measure the fine grain process that leads from disorder to synchrony, discovering a new kind of synchronization between chaotic systems. "

"The discovery of Topological Synchronization reveals that, in contrast to what was previously assumed, chaotic systems synchronize gradually through local structures that, surprisingly, kick off in the sparse areas of the system and only then spread to the more populated areas. In these sparse areas the activity is less chaotic than in other areas and, as a result, it is easier for these areas to sync relative to those that are much more erratic."

"This conceptual novelty pertains not only to our fundamental understanding of synchronization, but also carries direct practical implications on the predictability limits of chaotic systems. Indeed, thanks to this newly-defined local synchronization, the researchers show that the state of one system can be inferred from measurements of the other, even in the absence of global synchrony. "
chaos  complexity  synchronization  emergent-behavior
5 weeks ago by pierredv

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