The Lorentz Strange Attractors-Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation

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The Lorentz Strange Attractors

Mathematical meteorologist E. N. Lorentz came across three-dimensional nonlinear system which showed exotic behaviors. The model equations are

( x ( t ) y ( t )), dt

dx  

x(t)z(t) rx(t) y(t), dt

dy   

x ( t ) y ( t ) bz ( t )

dt

dz

Where , , r & b are constants. These equations are found in number of processes including motion of water wheel, lasers, dynamos and simple convection part of models for atmosphere.

Lorentz simulated three-dimensional trajectories for above system and found that trajectory is a set of complicated shape and its not fixed points or limit cycles.

The Lorentz Strange Attractors

dx/dt x -10x

y 10y

dx/dt = -10x + 10y

z

x (^) xz -xz

28x

y

dy/dt

dy/dt = y + 28x - xz

x y xy z -8z/ dz/dt = -8z/3 + xy

XY Graph

z To Workspace

y To Workspace

x To Workspace

Product

Product

1 s Integrator

1 s Integrator

1 s Integrator

-8/ Gain

28 Gain

- Gain

10 Gain

- Gain

10 x ( t ) 10 y ( t ), dt

dx  

x ( t ) z ( t ) 28 x ( t ) y ( t ), dt

dy   

x ( t ) y ( t ) ( 8 / 3 ) z ( t ) dt

dz (^)  

This model is in simulink model file as lorentz

The Lorentz Strange Attractors

10 x ( t ) 10 y ( t ), dt

dx  

x ( t ) z ( t ) 28 x ( t ) y ( t ), dt

dy   

x ( t ) y ( t ) ( 8 / 3 ) z ( t ) dt

dz (^)  

-20 -15 -10 -5 0 5 10 15 20

5

10

15

20

25

30

35

40

45

50

Z-values

x-values

Result of the model from simulink model file lorentz is shown here..

3D Lorentz Attractor

3d Lorentz Attractor is shown here as a result of the program shown in previous slide.

The attractor is indicating folds.

This is result of the Matlab file plotlrnz which can be rotated in three-D

3D Lorentz Attractor

% program uses euler method for lorentz attractor global sig r b f = @(x,y,z) 10.(y - x); g = @(x,y,z) 28.x - y - xz; p = @(x,y,z) xy - (8/3)z; %sig = 10.0; r = 28.0; b = 8/3; n = 2000; h = 0.005; % initial conditions t(1)=0.0; x(1)=0.0; y(1)=5.0; z(1)= 25.0; for i=1:n t(i+1) = t(i) + h; x(i+1) = x(i) + hf(x(i), y(i), z(i)); y(i+1) = y(i) + hg(x(i), y(i), z(i)); z(i+1) = z(i) + hp(x(i), y(i), z(i)); %plot(x(i+1),z(i+1),'.r','LineWidth',1) %plot3(x(i+1),z(i+1), y(i+1),'.b') hp=plot3(x(i+1),z(i+1), y(i+1),'.b'); set(hp,'LineWidth',12); box on; xlabel('x','FontSize',12); ylabel('y','FontSize',12); zlabel('z','FontSize',12); axis([-50 30 -30 60 0 60]); set(gca,'CameraPosition',[100 200 -100],'FontSize',12); hold on drawnow end**

This program is a Matlab file as eulerlrnz

dx/dt x

y y

dx/dt = - y - z

z

x xz

x

0.2y

dy/dt

dy/dt = x + ay

z

-5.7z

dz/dt = 0.2 + zx - cz

z

Rossoler attactor

-5. c

0. a

XY Graph z To Workspace

y To Workspace

x To Workspace

Scope

Scope

Product

1 s Integrator

1 s Integrator

1 s Integrator

Constant

This program is a Simulink model file as rossoler

Simulink model for Rossoler Attractor

This is result of the Simulink model file called rossoler

Simulink model for Rossoler Attractor

X and z vs time

Phase diagram ( z vs x)

3D Rossoler Attractor

This is result of Matlab file eulerlrnz It can be rotated in 3D.

Lorenz Equations

• Chaotic effects arises when
  • At least one of the functions f i contains a nonlinear term (e.g., x 12 , x 12 x 2 , x 1 x 23 )
  • The dimension of the system of equations is 3 or greater  s, r , and b are all constants  Lorenz attractor – when  s = 10, b = 8/3, and r = 28  The initial values are x(0) = 0., y(0) = 5., and z(0) = 25.

( x ( t ) y ( t )), dt

dx  

x(t)z(t) rx(t) y(t), dt

dy   

x ( t ) y ( t ) bz ( t )

dt

dz

Flows vs. Maps

Flows Maps

Dimension N  3 N  1

Method for solving

system of differential equations

iteration s

The simple maps includes following:

• The logistic map
• The Hénon attractor
• Chaos esthétique
• The Standard Map

The Hénon Attractor

• 2-D map given by

the equations:

• xn +1 = yn + 1 – a xn^2
• yn +1 = B xn

• General form of

the attractor does

not depend on

initial x and y

values

The Chaos Esthétique

• 2-D mapping for modeling the dynamics of

a particle accelerator

– xn +1 = yn + f ( xn )

– yn +1 = - bxn + f( xn +1)

– where a and b are constants and

– f ( x ) = ax + [2(1 – a ) x^2 / (1 + x^2 )]

Conservative Mapping