Chaos Theory and the Lorenz Equation

8/13/2019 Chaos Theory and the Lorenz Equation

1/15


2/15


3/15


4/15

dx

dt

= X=(y x)

dy

dt = Y =x y xz

dz

dt = Z=xy z


5/15

0 0 0

X Y Z

x2


6/15

( 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


7/15

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


8/15

1 2

0 0 0

0 0 0


9/15


10/15


11/15


12/15


13/15


14/15
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


15/15
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

Chaos Theory and the Lorenz Equation

Documents