Nonlinear Physics Textbook R C Hilborn Chaos Nonlinear
Nonlinear Physics • Textbook: – R. C. Hilborn, “Chaos & Nonlinear Dynamics”, 2 nd ed. , Oxford Univ Press (94, 00) • References: – R. H. Enns, G. C. Mc. Guire, “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. – linear superposition holds: f, g solutions → X’’ + ω2 x = 0 αf + βg solution – Response is linear • Nonlinear System: – Equation of motion is not linear. X’’ + ω2 x 2 = 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. , 2 nd 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
Plan of Study 1. Examples of Chaotic Sytems. 2. Universality of Chaos. 3. State spaces • Fixed points analysis • Poincare section • Bifurcation 4. 5. 6. 7. 8. Routes to Chaos Iterated Maps Quasi-periodicity Intermittency & Crises Hamiltonian Systems
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. Mc. Guire, “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 ?
Diode Circuit R. W. Rollins, E. R. Hunt, Phys. Rev. Lett. 49, 1295 (82) • Becomes capacitor when reverse biased. • Becomes voltage source -Vd = Vf when forward biased.
Cause of bifurcation: After a large forward bias current Im , the diode will remain conducting for time τr 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 8 Period 4
Divergence of evolution in chaotic regime
I(t) sampled at period of V(t) Period 4 Bifurcation diagram
Larger signal In Period 3 in window
Summary • Sudden change ( bifurcation ) as parameter ( V 0 ) changes continuously. • Changes ( periodic → choatic ) reproducible. • Evolution seemingly unrelated to external forces. • Chaos is distinguishable from noises by its divergence of nearby trajectories.
Population Growth R. M. May M. Feigenbaum Logistic eq. Iterated Map Iteration function
Maximum: Fixed point → if if
A = 0. 9
X 0=0. 1 X 0=0. 8 A = 1. 5
N=5000 A = 1. 0
A = 3. 1
Poincare section 1 -D iterated map ~ 3 -D state space Dimension of state space = number of 1 st order autonomous differential eqs. Autonomous = Not explicitly dependent on the independent variable. Diode circuit is 3 -D.
Lorenz Model
Derivation of the Lorenz eqs. : Appendix C Navier-Stokes eqs. + Entropy Balance eq. L. E. Reichl, ”A Modern Course in Statistical Physics”, 2 nd ed. , § 10. B, Wiley (98). X ~ ψ(t) Stream function (fluid flow) Y ~ T between ↑↓ fluid within cell. Z ~ T from linear variation as function of z. r < r. C : conduction r > r. C : 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 point & limit cycle ). • Intermittency (almost periodic, bursts of chaos) • Bistability. • Hysteresis. • Coexistence of stable limit cycles & chaotic regions. • Various cascading bifurcations.
3 fixed points at (0, 0, 0) & attractive repulsive r>1 repulsive attractive r > 14 repulsive r<1 r = 1 is bifurcation point repulsive regions outside atractive ones, complicated behavior. r = 160 : periodic. X oscillates around 0 → fluid convecting clockwise, then anti-clockwise, … r = 150 : period 2. r = 146 : period 4. … r < 144 : chaos
Back
Back
Back
Intermittence Back
Period 1 Back
Period 2 Back
Period 4 Back
Chaos
Divergence of nearby orbits
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. Multiverse? Free will? • Unpredictability: Free will?
- Slides: 38
