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1

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 α−−=

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

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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

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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

ωτθβτθτ

τττθ

=−=−=

++=

5

Equations

2220

202

2

,,

)cos()sin(

mLFf

mLLg

ImgL

tfdtd

dtd

====

+−−=

βαω

ωθαθωθ

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

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Case 1: A very simple pendulum

)sin(202

2

θωθ −=dtd

code

2

7

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 θ(0)=1.0

θ(

t)

time

0.8

8

)sin(202

2

θωθ −=dtd

θωθ 202

2

−=dtd

Is there any difference between the nonlinear pendulum

and the linear pendulum?

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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 θ(0)=1.0 θ(0)=0.2

θ(t)

time

0.8

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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== 0ωω

θϑ

θθθ

<

+−≈

)sin(21)sin( 2 K

Lg=< 0ωω

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Phase-Space Plotvelocity versus position

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0 θ(0)=1.0 θ(0)=0.2

dθ/d

t

θ

E1

E2

phase-space plot is a very good way to explore the dynamic of oscillations

0.8

12

Case 2: The pendulum with dissipation

dtd

dtd θαθωθ −−= )sin(2

02

2

code

0 10 20 30 40 50-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

θ(0)=1.0, α=0.1

θ(t)

time

How about frequency in this case?

3

13

Phase-space plot for the pendulum with dissipation

0 10 20 30 40 50-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

θ(0)=1.0, α=0.1

θ(t)

time

-0.5 0.0 0.5-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

dθ/d

t

θ

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Case 3: Resonance and beats

)cos()sin(202

2

tfdtd ωθωθ +−=

code

When the magnitude of the force is very large – the system is overwhelmed by the driven force (mode locking) and the are no beats

When the magnitude of the force is comparable with the magnitude of the natural restoring force the beats may occur

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Beats

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ −=+≈ tttt

2sin

2cos2)sin()sin( 00

0000ωωωωθωθωθθ

code

In beating, the natural response and the driven response add:

mass is oscillating at the average frequency and an amplitude is varying at the slow

frequency 2)( 0ωω −2)( 0ωω +

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Example: beats

0 20 40 60 80 100 120-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

θ(0)=0.2, α=0.0, f=0.2, ω=1.1

θ(t)

time

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.61.82.0

dθ/d

t

θ

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Resonance

0 20 40 60 80 100 120-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

θ(0)=0.2, α=0.0, f=0.1, ω=1.0

θ(t)

time

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.61.82.0

dθ/d

t

θ

That is not true for the nonlinear oscillator

For a simple harmonic oscillator the amplitude of oscillations increases without bound

code

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Case 4: Complex Motion

)cos()sin(202

2

tfdtd

dtd ωθαθωθ +−−=

code

We have to compare the relative magnitude of the natural restoring force, the driven force and the frictional force

The most complex motion one would expect when the three forces are comparable

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Case 4: Chaotic Motion

0 20 40 60 80 100120140160180200220240-25

-20

-15

-10

-5

0

5

10

15

θ(0)=1.0, α=0.2, f=0.7, ω=0.66

θ(t)

time

-24-22-20-18-16-14-12-10-8 -6 -4 -2 0 2 4 6 8 10-3

-2

-1

0

1

2

3

dθ/d

t

θ

Chaotic motion is not random!

Chaos is the deterministic behavior of a system displaying no discernable regularity

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Case 4: Chaotic Motion

0 20 40 60 80 100120140160180200220240-25

-20

-15

-10

-5

0

5

10

15 θ(0)=1.0, α=0.2, f=0.7, ω=0.66 θ(0)=1.00001, α=0.2, f=0.7, ω=0.66

θ(t)

time

A chaotic system is one with an extremely high sensitivity to parameters or initial conditions

The sensitivity to even miniscule changes is so high that, in practice, it is impossible to predict the long range behavior unless the parameters are known to infinite precision (which they never are in practice)

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Measuring Chaos

How do we know if a system is chaotic?

The most important characteristic of chaos is sensitivity to initial conditions.

Sensitivity to initial conditions implies that our ability to make numerical predictions of its trajectory is limited.

-24-22-20-18-16-14-12-10-8 -6 -4 -2 0 2 4 6 8 10-3

-2

-1

0

1

2

3

dθ/d

t

θ

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How can we quantify this lack of predictably?

This divergence of the trajectories can be described by the Lyapunov exponent λ, which is defined by the relation:

where Δxn is the difference between the trajectories at time n.

If the Lyapunov exponent λ is positive, then nearby trajectories diverge exponentially.

Chaotic behavior is characterized by the exponential divergence of nearby trajectories.

nexxnλ

0Δ=Δ

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nexxnλ

0Δ=Δ

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Chaotic structure in phase space1. Limit cycles: ellipse-like figures with

frequencies greater then

2. Strange attractors: well-defined, yet complicated semi-periodic behavior. Those are highly sensitive to initial conditions. Even after millions of observations, the motion remains attracted to those paths

3. Predictable attractors: well-defined, yet fairy simple periodic behaviors that not particularly sensitive to initial conditions

4. Chaotic paths: regions of phase space that appear as filled-in bands rather then lines

0ω-24-22-20-18-16-14-12-10-8 -6 -4 -2 0 2 4 6 8 10

-3

-2

-1

0

1

2

3

dθ/d

t

θ

5

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The Lorenz Model & the butterfly effectIn 1962 Lorenz was looking for a simple model for weather predictions and simplified the heat-transport equations to the three equations

The solution of these simple nonlinear equations gave the complicated behavior that has led to the modern interest in chaos

zxydtdz

yxxzdtdy

xydtdx

38

28

)(10

−=

−+−=

−=

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Example

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Hamiltonian ChaosThe Hamiltonian for a particle in a potential

for N particles – 3N degrees of freedom

Examples: the solar system, particles in EM fields, ...more specific example: the rings of Saturn

Attention: no dissipation!

Constants of motion: Energy, Momentum (linear, angular)

When a number of degrees of freedom becomes large, the possibility of chaotic behavior becomes more likely.

),,()(21 222 zyxVpppm

H zyx +++=

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Summary

The simple systems can exhibit complex behavior Chaotic systems exhibit extreme sensitivity to initial conditions.

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PracticeDuffing Oscillator

Write a program to solve the Duffing model. Is there a parametric region in where the system is chaotic

),,( ωα f

)cos()1(21 2

2

2

tfxxdtdx

dtxd ωα =−−+

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Fourier Analysis of Nonlinear Oscillations

The traditional tool for decomposing both periodic and non-periodic motions into an infinite number of harmonic functions

It has the distinguishing characteristic of generating a periodic approximations

6

31

Fourier series

For a periodic function

one may write

The Fourier series is a “best fit” in the least square sense of data fitting

)()( tyTty =+

( ),)sin()cos(2

)(1

0 ∑∞

=

++=n

nn tnbtnaaty ωω

A general function may contain infinite number of components. In practice a good approximation is possible with about 10 harmonics

Tπω 2=

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Coefficients:

the coefficients are determined by the standard technique for orthogonal function expansion

T

dttytnT

b

dttytnT

a

T

n

T

n

πω

ω

ω

2

,)()sin(2

,)()cos(2

0

0

=

=

=

∫

∫

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Fourier transform

The right tool for non-periodic functions

and the inverse transform is

a plot of versus is called the power spectrum

∫+∞

∞−

= ωωπ

ω deYty ti)(21)(

∫+∞

∞−

−= dtetyY tiω

πω )(

21)(

2)(ωY ω

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Spectral function

If represent the response of some system as a function of time, is a spectral function that measures the amount of frequency making up this response

)(ty)(ωY

ω

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Methods to calculate Fourier transform

Analytically

Direct numerical integration

Discrete Fourier transform (for functions that are known only for a finite number of times tkFast Fourier transform (FFT)

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Discrete Fourier transform

Assume that a function y(t) is sampled at a discrete number of N+1 points, and these times are evenly spaced

Let T is the time period for the sampling:a function y(t) is periodic with T, y(t+T)=y(t)

The largest frequency for this time interval isand T/21 πω = )/(2/21 NhnTnnn ππωω ===

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Discrete Fourier transform

The discrete Fourier transform, after applying a trapezoid rule

k

N

k

Nkniti

n yehdttyeY n ∑∫=

−∞

∞−

− ==1

2

2)(

21)(

πω

ππω

)(2)(21)(

1

2

n

N

n

hNntiti Ye

hNdYety n ωπωω

π

πω ∑∫

=

∞

∞−

==

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DFT in terms of separate real and imaginary parts

))]Re()/2sin()Im()/2(cos(

))Im()/2sin(

)Re()/2[(cos(2

)(1

k

k

k

k

N

kn

yNknyNkni

yNkn

yNknhY

ππ

π

ππ

ω

−++

= ∑=

)sin()cos( xixeix +=

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Practice for the simple pendulum

Decompose your numerical solutions into a Fourier series. Evaluate contribution from the first 10 terms

Evaluate the power spectrum from your numerical solutions

Solve the simple pendulum for harmonic motion, beats, and chaotic motion (the dissipation and driven forces are included)