Chaos Theory. Chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. These phenomena are often described by Fractal Mathematics, which captures the Infinite Complexity of Nature. Many natural objects exhibit Fractal Properties, including landscapes, clouds, trees, organs, rivers etc, and many of the systems in which we live exhibit Complex, Chaotic behavior. Recognizing the Chaotic, Fractal Nature of our world can give us new insight, power, and wisdom.  Principles of Chaos The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle.  Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc) in the world, accurate long-range weather prediction will always remain impossible. Order / Disorder: Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways. Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed.  Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

Chaos and the Double Pendulum

A chaotic system is one in which infinitesimal differences in the starting conditions lead to drastically different results as the system evolves.

Summarized by mathematician Edward Lorenz, "Chaos [is] when the present determines the future, but the approximate present does not approximately determine the future.“

There’s an important distinction to make between a chaotic system and a random system. Given the starting conditions, a chaotic system is entirely deterministic. A random system, on the other hand, is entirely non-deterministic, even when the starting conditions are known. That is, with enough information, the evolution of a chaotic system is entirely predictable, but in a random system there’s no amount of information that would be enough to predict the system’s evolution.

The simulations above show two slightly different initial conditions for a double pendulum — an example of a chaotic system. In the left animation both pendulums begin horizontally, and in the right animation the red pendulum begins horizontally and the blue is rotated by 0.1 radians (≈ 5.73°) above the positive x-axis. In both simulations, all of the pendulums begin from rest.

Mathematica code posted here.

[For more information on how to solve for the motion of a double pendulum, check out my video here.]

Doodles from my stream tonight! Thank you to everyone that came and kept me company! I will prob do more with the yohane later cause i really liked how it was going. 

Diana is @primalpantheon ‘s character!

A DA point commission design for “HieraticDragonLord”/ @primalpantheon from DeviantART.

Red Ant Space Queen

My art status is closed due to time restraints

I need your help

I’m doing something important in January,, and I need at least 3k USD to be able to do it,, and this is my only way to get money. I’m not asking for donations, but they’re appreciated. I am asking for you to buy stuff from me.

I offer commissions that cost from 7USD to 30USD, you can ask for whichever you like, every dollar helps!

If you’d like to help me, you can check the links below

COMMISSIONS

you can also check my shops

REDBUBBLE 

PAOM

If you do buy something from me, I would be forever thankful. If you cant, please reblog this post, that helps too.

Thank you all so much for yor help!!

Some art from my stream (still in session)

Character belongs to @primalpantheon, huge thank you for sticking with me through the whole process!

50 follower request stream!

send me gijinka or dragons and get a free sketch/doodle! reblogs are appreciated.

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An anthro wolf design commission for DA user “HieraticDragonLord”

My art status is closed due to time constraints