#chaotic system

Explanation and Some Extras:

The basic idea is that these are created from simulations of a desk toy called a magnetic decision maker that happens to exhibit chaotic behavior. This is what the toy looks like:

The way it works is: a magnetic ball is at the end of a rigid wire and is allowed to pivot freely on its sphere of possible positions. Gravity pulls the ball down while magnets hidden under the answers attract the magnetic ball. The magnets are strong enough that the ball will never settle in the middle but rather will “choose” one of the answers, giving a seemingly random answer thanks to previously mentioned chaotic behavior. For rendering simplicity, the system will be rendered as viewed from directly above. Here’s a gif of the simple simulation. For simplicity and clarity, the ball is rendered as a white circle and the answers (attractors) are replaced with red circles. The ball’s path is traced by a white line.

So that alone is some pretty cool behavior. We could start the point anywhere, but the point with the wire parallel to the ground just gets the most interesting path. I wonder if we could somehow show where each initial condition would end up, in one convenient image... We can! Simply using color to indicate position is the easiest way to do this, as it allows us to indicate at each point, a different position. Let’s start by making a “legend”, assigning each point in the set of possibilities a color. The easiest way to do this is to use a hue wheel. If we color each point based on its own assigned color, we get this image:

It would be normal hue wheel, but I’m lazy with my programming and real hue cycles are a pain. In this specific case, we’re only interested in seeing which attractor the ball ends up pointing to. This means as long as the colors in the white circles (which attracted points will have) are distinct enough, we should get the information we’re looking for. So what happens if we let the system evolve, and color each point based on where a ball starting there would be at each time? The answer is some cool psychedelic gifs:

The bad quality is thanks to the astronomical size of gif images that have continuous rgb colors and the length of this one. Also tumblr’s gif upload size limits. Anyway though,

It’s pretty easy to tell what happens: as the system evolves, certain areas all converge to a solid color while the outer points turn into static. This actually makes perfect sense: if you drop the ball right next to the attractor, the ball will simply stay there, giving you an extremely predictable answer. However, if the ball is dropped from closer to the equator of the sphere, it will swing around for a long time before settling on which attractor it chooses, taking a complicated path. The chaos is evident in the random static of colors near the edge of the circle.

This chaos can be seen in the simple simulation as well by tracking two points that start very close to each other: the white ball and path from before will now be rendered as coral and cyan, with the coral and cyan balls slightly offset from each other initially, precisely starting 0.001 units apart when the wire is a single unit in length.

I would love feedback if you read this and have any suggestions or thoughts.

Were awre 👀

Design of State Estimator for a Class of Generalized Chaotic Systems

by Yeong-Jeu Sun ""Design of State Estimator for a Class of Generalized Chaotic Systems""

Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019,

URL: https://www.ijtsrd.com/papers/ijtsrd29270.pdf

Paper URL: https://www.ijtsrd.com/engineering/electrical-engineering/29270/design-of-state-estimator-for-a-class-of-generalized-chaotic-systems/yeong-jeu-sun

international journals in engineering, call for paper science, ugc list of journals

Deterministic chaos can exhibit robust dynamic behaviors such as sensitive dependence on initial conditions. The behaviors have warranted diverse engineering uses, which usually entail electronic hardware implementation. In this study, the circuit realization and its corresponding implementation by means of analog electronic components are presented for the polynomial chaotic Sun system. The system has twelve terms, twelve parameters and six nonlinear terms. A procedure is detailed for converting the chaotic parameters into corresponding electronic parameters such as the circuital resistances. Circuit realization of the system is simulated by PSPice-A/D. Next, the circuit is implemented by means of analog electronic components such as operational amplifiers and multipliers. Signals from electronic experiments are compared with numerical simulations.

Author(s) Details

Christian Nwachioma CIDETEC, Instituto Politecnico Nacional, UPALM, Mexico City, 07700, Mexico.

J. Humberto Perez-Cruz ESIME-Azcapotzalco, Instituto Politecnico Nacional, Mexico City 02250, Mexico.