11

Nonlinear Differential EquationsNonlinear Differential Equations

and The Beauty of Chaos

2

Examples of nonlinear equations

)()(2

tkxdt

txdm =Simple harmonic oscillator (linear ODE)

More complicated motion (nonlinear ODE)

))(1)(()(2

txtkxdt

txdm =Other examples: weather patters, the turbulent motion of fluidsMost natural phenomena are essentially nonlinear.

3

What is special about nonlinear ODE?

For solving nonlinear ODE we can use the same methods we use for solving linear differential equations

What is the difference? Solutions of nonlinear ODE may be simple, complicated,

or chaotic

Nonlinear ODE is a tool to study nonlinear dynamic: chaos, fractals, solitons, attractors

4

A simple pendulum

Model: 3 forces

gravitational force

frictional force is proportional to velocity

periodic external force

)cos(,.),sin(

2

2

tFdtdmgL

dtdI

extfg

extfg

===

++=

5

Equations

2220

202

2

,,

)cos()sin(

mLFf

mLLg

ImgL

tfdtd

dtd

====

+=

Computer simulation: there are very many web sites there are very many web sites with Java animation for the with Java animation for the simple pendulumsimple pendulum

6

Case 1: A very simple pendulum

)sin(2022

=dtd

code

Mohd Fairuz - PhDTypewriterLink

27

0 10 20 30 40 50 60-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2 (0)=1.0

(t

)

time

0.8

8

)sin(2022

=dtd

2022

=dtd

Is there any difference between the nonlinear pendulum

and the linear pendulum?

9

0 10 20 30 40 50 60-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2 (0)=1.0 (0)=0.2

(t)

time

0.8

10

Amplitude dependence of frequency

For small oscillations the solution for the nonlinear pendulum is periodic with

For large oscillations the solution is still periodic but with frequency

explanation:

Lg== 0