Nonlinear Physics

• Textbook:– R.C.Hilborn, “Chaos & Nonlinear Dynamics”, 2nd ed., Oxf

ord Univ Press (94,00)• References:

– R.H.Enns, G.C.McGuire, “Nonlinear Physics with Mathematica for Scientists & Engineers”, Birhauser (01)

– H.G.Schuster, “Deterministic Chaos”, Physik-Verlag (84)• Extra Readings:

– I.Prigogine, “Order from Chaos”, Bantam (84)

Website: http://ckw.phys.ncku.edu.tw (shuts down on Sundays)

Home work submission: class@ckw.phys.ncku.edu.tw

Linear & Nonlinear Systems

• Linear System:– Equation of motion is linear. X’’ + ω2x = 0– linear superposition holds:

f, g solutions → αf + βg solution– Response is linear

• Nonlinear System:– Equation of motion is not linear. X’’ + ω2x2 = 0– Projection of a linear equation is often nonlinear.

• Linear Liouville eq → Nonlinear thermodynamics• Linear Schrodinger eq → Quantum chaos ?

– Sudden change of behavior as parameter changes continuously, cf., 2nd order phase transition.

• Two Main Branches of Nonlinear Physics:– Chaos– Solitons

• What is chaos ?– Unpredictable behavior of a simple, deterministic system

• Necessary Conditions of Chaotic behavior– Equations of motion are nonlinear with DOF 3.– Certain parameter is greater than a critical value.

• Why study chaos ?– Ubiquity– Universality– Relation with Complexity

1. Examples of Chaotic Sytems.2. Universality of Chaos.3. State spaces

• Fixed points analysis• Poincare section• Bifurcation

4. Routes to Chaos5. Iterated Maps6. Quasi-periodicity7. Intermittency & Crises8. Hamiltonian Systems

Plan of Study

Ubiquity

• Some Systems known to exhibit chaos:– Mechanical Oscillators– Electrical Cicuits– Lasers– Optical Systems– Chemical Reactions– Nerve Cells, Heart Cells, …– Heated Fluid– Josephson Junctions (Superconductor)– 3-Body Problem– Particle Accelerators– Nonlinear waves in Plasma– Quantum Chaos ?

Three Chaotic Systems

• Diode Circuit• Population Growth• Lorenz Model

R.H.Enns, G.C.McGuire,

“Nonlinear Physics with Mathematica for Scientists & Engineers”,

Birhauser (01)

Specification of a Deterministic Dynamical System

• Time-evolution eqs ( eqs of motion )• Values of parameters.• Initial conditions.

Deterministic Chaos

Questions

• Criteria for chaos ?• Transition to chaos ?• Quantification of chaos ?• Universality of chaos ?• Classification of chaos ?• Applications ?• Philosophy ?

• Becomes capacitor when reverse biased. • Becomes voltage source -Vd = Vf when forward biased.

R.W.Rollins, E.R.Hunt,Phys. Rev. Lett. 49, 1295 (82)

Diode Circuit

1 exp m

r mc

I

I

Cause of bifurcation:After a large forward bias current Im , the diode will remain conducting for time τ r after bias is reversed, i.e.,there’s current flowing in the reverse bias direction so that the diode voltage is lower than usual.

Reverse recovery time =

Bifurcation

Period 4

period 4

period 8

Divergence of evolution in chaotic regime

Period 4

I(t) sampled at period of V(t)

Bifurcation diagram

In

Larger signal

Period 3 in window

Summary

• Sudden change ( bifurcation ) as parameter ( V0 ) changes continuously.

• Changes ( periodic → choatic ) reproducible.• Evolution seemingly unrelated to external forces.• Chaos is distinguishable from noises by its diverg

ence of nearby trajectories.

Population Growth

max1k

AN N

B

21 1 0k k kN AN BN

maxk

k

Nx

N

R.M.May M.Feigenbaum

max 21 1k k kx Ax BN x 1 11k kAx x Logistic eq.

1f x Ax x Iteration function

Iterated Map

' 1 2f x A x Maximum:1

2mx

4m

Af x

0 4 & 0 1 0 1A x f x

Fixed point

* *A A Ax f x * * *1A A Ax A x x

* 11Ax A

* 0Ax

* 0Ax

4 1A

0 1A →if

if

A = 0.9

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

A = 1.5

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X0=0.1

X0=0.8

0.0002 0.0004 0.0006 0.0008 0.001

0.0002

0.0004

0.0006

0.0008

0.001

A = 1.0

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

N=5000

A = 3.1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

1-D iterated map ~ 3-D state space

Dimension of state space = number of 1st order autonomous differential eqs.

Autonomous = Not explicitly dependent on the independent variable.

Diode circuit is 3-D.

Poincare section

0

0

cos

1sin

fIR LI V V t

IR LI I V tC

0

0

1sin

c

1

osf

LJ JR I V z

I J

z

CIR LJ V V z

Lorenz Model

Navier-Stokes eqs. + Entropy Balance eq.L.E.Reichl, ”A Modern Course in Statistical Physics”, 2nd e

d., §10.B, Wiley (98).

X p Y X

Y rX XZ Y

Z XY bZ

3

Kinetic viscosityPrandtl number

Thermal diffusion coef

Rayleigh number

Coefficient of Thermal Expansion

T

T

pD

ghr R T

D

8

310 ( )

:

convection begins for smallest rb

p coldwater

r control parameter

X ~ ψ(t) Stream function (fluid flow)Y ~ T between ↑↓ fluid within cell.

Z ~ T from linear variation as function of z.

Derivation of the Lorenz eqs.: Appendix C

r < rC : conduction

r > rC : convection

Dynamic Phenomena found in Lorenz Model

• Stable & unstable fixed points.• Attractors (periodic).• Strange attractors (aperiodic).• Homoclinic orbits (embedded in 2-D manifold ).• Heteroclinic orbits ( connecting unstable fixed po

int & limit cycle ).• Intermittency (almost periodic, bursts of chaos)• Bistability.• Hysteresis.• Coexistence of stable limit cycles & chaotic regio

ns.• Various cascading bifurcations.

3 fixed points at (0,0,0) & 1 , 1 , 1b r b r r

r = 1 is bifurcation pointr < 1 attractive repulsive

r > 1 attractiverepulsive

r > 14 repulsive regions outside atractive ones,

complicated behavior.

repulsive

r = 160 : periodic.

X oscillates around 0 → fluid convecting clockwise, then anti-clockwise, …

r = 150 : period 2.

r = 146 : period 4.

…

r < 144 : chaos

2 4 6 8 10

0.2

0.4

0.6

0.8

1

z

y

x

0.2 0.4 0.6 0.8

0.025

0.05

0.075

0.1

0.125

0.15

0, , 0, 1, 0

8, , 010, ,

3.5

tx y z

p br

Back

5 10 15 20

0.2

0.4

0.6

0.8

1

z

y

x

0.2 0.4 0.6 0.8

0.05

0.1

0.15

0.2

0, , 0, 1, 0

8, , 10, ,

31

t

r

x y z

p b

Back

-2 -1.5 -1 -0.5 0.5 1 1.5 2

-0.5

-0.25

0.25

0.5

0.75

1

1.25

1.5

0, , 0, 1,0

8, , 10, ,

32

t

r

x y z

p b

5 10 15 20

0.25

0.5

0.75

1

1.25

1.5

1.75

z

y

x

Back

0, , 0, 5, 15

8, , 10, ,25

3

t

r

x y z

p b

5 10 15 20

-20

-10

10

20

30

40

z

y

x

-10

0

10

-20

-10

0

10

0

10

20

30

-10

0

10

-20

-10

0

10 Intermittence

Back

0, , 0, 1, 0

8, , 10, 160,

3

tx y z

p br

Period 1

-40-20

020

40

-50

0

50

125

150

175

200

-40-20

020

40

-50

0

50

26 27 28 29 30-50

50

100

150

200

z

y

x

Back

0, , 0, 1, 0

8, , 10, 150,

3

tx y z

p br

Period 2

-200

2040

-50

0

50

100

125

150

175

200

-200

2040

-50

0

50

26 27 28 29 30

-50

50

100

150

200

z

y

x

Back

-20

020

40

-50

0

50

100

125

150

175

200

-200

2040

-50

0

50

0, , 0, 1, 0

8, , 10, 146,

3

tx y z

p br

Period 4

26 27 28 29 30

-50

50

100

150

200

z

y

x

26 27 28 29 30

-50

50

100

150

200

z

y

x

Back

21 22 23 24 25

-50

50

100

150

200

z

y

x

-40

-20

0

20

40

-50

0

50

100

150

200

-40

-20

0

20

40

-50

0

50

0, , 20, 1, 163

8, , 10, 143,

3

tx y z

p br

Chaos

21 22 23 24 25-50

50

100

150

200

z

y

x

0, , 20, 1,

8, , 10, 1

163

43,3

tx y z

p r b

0, , 20, 1,

8, , 10, 1

166

43,3

tx y z

p r b

Divergence of nearby orbits

21 22 23 24 25-50

50

100

150

200

z

y

x

Determinism vs Butterfly Effect

• Divergence of nearby trajectories → Chaos → Unpredictability

– Butterfly Effect• Unpredictability ~ Lack of solution in closed form • Worst case: attractors with riddled basins.• Laplace: God = Calculating super-intelligent → determinism (no free will).• Quantum mechanics: Prediction probabilistic. Mult

iverse? Free will?• Unpredictability: Free will?