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Butterfly Effect Illustration

2010-11-07 19:31阅读：

http://blog.sina.cn/dpool/blog/u/1642221824

On studying optimal dynamics, I had to cease to review differential equations. Every time in retrospect in canons, there are always discoveries. This time it's Mathematical in Economics.
I found butterfly effect and chaos theory fascinating, for illuminating. And math provide a precise explanation, making it convincing.
And I decided to deduce them through programming.VB is good for providing a free environment that enables one to create his own product, to realize. The BASIC language is simple to use, and the visual version is convenient, especially for UI design.

On expressing the butterfly effect there's an idea that there's no need for a new programme but upgrade the Multiplier-Accelerator Model Illustrator(MAMI 3.51) only. Thus MAMI's existence raises from simply a MA Model illustrator to an more universal level that can approach a wider range of differential equation, or dynamic problems.
r> Butterfly Effect Illustration
A simple first-order non-linear autonomous differential equation is introduced, whose expression is Yt+1=rYt(1-Yt). There would be many interesting discoveries on exploring. When the parameter r escalates, route to time fluctuates following different patterns. It's better to present with phase images，which is intuitive.

Here only chaos is discussed.
When r exceeds 3.5, chaos occurs. Yt route apparently fluctuate randomly with time, without following any motion in pattern.
Let r=3.7, when small change that could even be ignored happens to Y0(Initial Yt), suppose Y0 change from 0.21 to 0.20008, route would go totally different from a point on the time axis.

r=3.7, Y0=2.0000, Blue Route
Butterfly <wbr>Effect <wbr>Illustration
r=3.7, Y0=2.0008, Green Route
Butterfly <wbr>Effect <wbr>Illustration

The result is illuminating. In this model Yt route behaves sensitive to the initial condition, which is called butterfly effect. When it comes to the business cycle, butterfly effect is worth cogitate.

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