Lorenz equation

Lorenz equation

0.1

The history

The

Lorenz equation

was published in 1963 by a meteorologist and mathematician from MIT called Edward N. Lorenz. The paper containing the equation was titled “Deterministic non-periodic flows” and was published in the Journal of Atmospheric Science. What drove Lorenz to find the set of three dimensional

ordinary differential equations

was the search for an equation that would “model some of the unpredictable behavior which we normally

associate

with the weather”

[

SC

]

. The Lorenz equation represent the convective motion of fluid cell which is warmed from below and cooled from above.

[

SC

]

The same system can also apply to dynamos and laser. In

addition

some of its popularity can be attributed to the beauty of its solution. It is also important to state that the Lorenz equation has enough properties and interesting behavior that whole books are written analyzing results.

0.2

The equation

The Lorenz equation is commonly defined as three coupled ordinary differential equation like

d

⁢

x

d

⁢

t

=

σ

⁢

(

y

-

x

)

d

⁢

y

d

⁢

t

=

x

⁢

(

τ

-

z

)

-

y

d

⁢

z

d

⁢

t

=

x

⁢

y

-

β

⁢

z

where the three parameter

σ

,

τ

,

β

are

positive

and are called the Prandtl number, the Rayleigh number, and a physical proportion, respectively. It is important to note that the

x

,

y

,

z

are not spacial

coordinate

. The ”

x

is proportional to the intensity of the convective motion, while

y

is proportional to the temperature

difference

between the ascending and descending currents,

similar

signs of

x

and

y

denoting that warm fluid is rising and cold fluid is descending. The variable

z

is proportional to the distortion of vertical temperature profile from linearity, a positive value indicating that the strongest

gradients

occur near the boundaries.”

[

LNE

]

0.3

Properties of the Lorenz equations

•

The Lorenz equation has the following

symmetry of ordinary differential equation

:

(

x

,

y

,

z

)

→

(

-

x

,

-

y

,

z

)

This

symmetry

is present for all parameters of the Lorenz equation

(see natural symmetry of the Lorenz equation)

.

•

The

z

-axis is

invariant

, meaning that a solution that starts on the

z

-axis

(i.e.

x

=

y

=

0

)

will remain on the

z

-axis. In addition the solution will tend toward the origin if the

initial condition

are on the

z

-axis.

•

To solve for the

equilibrium points

we let

𝐱

˙

=

f

⁢

(

𝐱

)

=

[

σ

⁢

(

y

-

x

)

x

⁢

(

τ

-

z

)

-

y

x

⁢

y

-

β

⁢

z

]

and we solve

f

⁢

(

𝐱

)

=

0

. It is clear that one of those equilibrium point is

𝐱

0

=

(

0

,

0

,

0

)

and with some

algebraic

manipulation we detemine that

𝐱

C

1

=

(

β

⁢

(

τ

-

1

)

,

β

⁢

(

τ

-

1

)

,

τ

-

1

)

and

𝐱

C

2

=

(

-

β

⁢

(

τ

-

1

)

,

-

β

⁢

(

τ

-

1

)

,

τ

-

1

)

are equilibrium points and real when

τ

>

1

.

•

If

σ

,

τ

,

β

>

0

then all solution of the Lorenz equation will
enter an ellipsoid centered at

(

0

,

0

,

2

⁢

τ

)

in finite time. In addition
the solution will remain inside the ellipsoid once it has entered. It follows
by definition that the ellipsoid is an

attracting set

.

(see all solution of the Lorenz equation enter an ellipsoid)

0.4

An example

(The

x

solution with respect to time.)

(The

y

solution with respect to time.)

(The

z

solution with respect to time.)

the above is the solution of the Lorenz equation with parameters

σ

=

10

,

τ

=

28

and

β

=

8

/

3

(which is the classical example). The inital condition of the system is

(

x

0

,

y

0

,

z

0

)

=

(

3

,

15

,

1

)

.

0.5

Experimenting with octave

By changing the parameters and initial condition one can observe that some solution will be drastically different.

(This is in no way rigorous but can give an idea of the qualitative property of the Lorenz equation.)

\PMlinkescapetext{function} y = lorenz (x, t)
y = [10*(x(2) - x(1));
x(1)*(28 - x(3)) - x(2);
x(1)*x(2) - 8/3*x(3)];
endfunction
solution = lsode ("lorenz", [3; 15; 1], (0:0.01:50)');

gset parametric
gset xlabel "x"
gset ylabel "y"
gset zlabel "\PMlinkescapetext{z}"
gset nokey
gsplot solution

References

LNE

Lorenz, N. Edward: Deterministic non-periodic flows. Journal of Atmospheric Science, 1963.

MM

Marsden, E. J. McCracken, M.: The

Hopf Bifurcation

and Its Applications. Springer-Verlag, New York, 1976.

SC

Sparow, Colin: The Lorenz Equations:

Bifurcations

, Chaos and Strange Attractors. Springer-Verlag, New York, 1982.

.6

See also

•

Paul Bourke,

http://astronomy.swin.edu.au/p̃bourke/fractals/lorenz/

The

Lorenz Attractor

in 3D

Title

Lorenz equation

Canonical name

LorenzEquation

Date of creation

2013-03-22 13:42:21

Last modified on

2013-03-22 13:42:21

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

19

Author

Daume (40)

Entry type

Definition

Classification

msc 34-00

Classification

msc 65P40

Classification

msc 65P30

Classification

msc 65P20

Classification

msc 65P99

Synonym

Lorenz attractor

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