Entry
Math: Chaos theory: Differential equation: How to convert a 2nd order dv to 3 first order dv's?
Nov 29th, 2003 12:28
Knud van Eeden,
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--- Knud van Eeden --- 29 November 2003 - 08:40 pm -------------------
Math: Chaos theory: Differential equation: How to convert a 2nd order
dv to 3 first order dv's?
---
The idea is that you convert the given non-linear (differential)
equation to a system of at least 3 first order differential equations
in 3 variables. And you know that in this systems chaotic conditions
can occur.
e.g.
+--------------------------------+
|. . 3 |
| x + k1 x + x = k2 cos( k3 t )| (1)
+--------------------------------+
this non-linear 2nd order differential equation in 1 independent
variable can be rewritten as a system of 3 linear differential
equations in 3 variables:
+-----+
|. | (2.1)
|x = y|
+-----+
. 3
and from (1): y + k1 y + x = k2 cos( k3 t )
by moving the non-differentials to the right side of the equation:
+--------------------------------+
|. 3 |
|y = - k1 y - x + k2 cos( k3 t )| (2.2)
+--------------------------------+
and from the identity
t = t
follows
+-----+
|. |
|t = 1| (2.3)
+-----+
In other words, equation (1) can be converted in 3 first order
differential
equations, non-linear, in 3 variables:
+--------------+
| . |
| x = f(x,y,z) |
+--------------+
| . |
| y = g(x,y,z) |
+--------------+
| . |
| z = h(x,y,z) | (3)
+--------------+
and thus chaos can occur.
---
Conclusion:
So, I believe this will work for a minimal non-linear 2nd order
differential equations or higher.
PS So, I believe, it will never work for single 1st order non-linear
differential equations, because you can not transform this equations
to the form (3).
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Book: see also:
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[book: source: Acheson, David - from Calculus to Chaos, an
Introduction to Dynamics - 1997 - p. 157]
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Internet: see also:
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Math: Chaos theory: Overview: Can you give me an overview of chaos
theory?
http://www.faqts.com/knowledge_base/view.phtml/aid/26929/fid/867
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