nonlinear physics textbook: –r.c.hilborn, “chaos & nonlinear dynamics”, 2 nd ed., oxford...
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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: [email protected]
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?