chaos theory and the lorenz equation
TRANSCRIPT
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dx
dt
= X=(y x)
dy
dt = Y =x y xz
dz
dt = Z=xy z
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0 0 0
X Y Z
x2
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( 1)
1 ( 1) ( 1) 2 ( 1) ( 1)
1
( 1)
( 1) 2
( 1)
( 1)
X Y Z
X
Y
Z
=
0
Z 1 X
Y X
X
Y
Z
det
0
Z 1 X
Y X
= 0
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3 + (+ + 1)2 + (+ + + Z+ X2
) + (1 ) + (XY + X2
+ Z) = 0(4)
X Y Z
3 + (+ + 1)2 + (+ + ) + (1 ) = 0
( + )(2 + (+ 1) + (1 )) = 0
1, 2= 1
(+ 1)2 + 4( 1)
2 , 3=
1 2 3
X Y Z 1 2
3 + (+ + 1)2 + (+ ) + (1 )2= 0
1 2 3
< (+ + 3)
1 =c
47019 c
1 2
1
2
3
1 2
c 1 2
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1 2
0 0 0
0 0 0
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http://%3Chttp//www.telegraph.co.uk/news/obituaries/1895916/Professor-Edward-Lorenz.html%3Ehttp://%3Chttp//www.telegraph.co.uk/news/obituaries/1895916/Professor-Edward-Lorenz.html%3Ehttp://%3Chttp//www.math.uni.wroc.pl/~karch/B2/lorenz-system.pdf.http://%3Chttp//www.math.uni.wroc.pl/~karch/B2/lorenz-system.pdf.http://%3Chttp//www.hypertextbook.com/chaos/21.shtml%3Ehttp://%3Chttp//www.hypertextbook.com/chaos/21.shtml%3Ehttp://%3Chttp//physics.brown.edu/physics/undergradpages/theses/matthew_carriuolo_thesis.pdf.http://%3Chttp//physics.brown.edu/physics/undergradpages/theses/matthew_carriuolo_thesis.pdf.http://%3Chttp//www.stsci.edu/~lbradley/seminar/attractors.html%3Ehttp://%3Chttp//www.stsci.edu/~lbradley/seminar/attractors.html%3E -
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http://%3Chttp//www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demo8.html%3Ehttp://%3Chttp//www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson1/Demo8.html%3E