Nonlinear Physics Textbook: –R.C.Hilborn, “Chaos & Nonlinear Dynamics”, 2 nd ed., Oxford 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: down on Sundays) Home work submission:

Linear & Nonlinear Systems Linear System: –Equation of motion is linear. X’’ + ω 2 x = 0 –linear superposition holds: f, g solutions → αf + βg solution –Response is linear Nonlinear System: –Equation of motion is not linear. X’’ + ω 2 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

1.Examples of Chaotic Sytems. 2.Universality of Chaos. 3.State spaces Fixed points analysis Poincare section Bifurcation 4.Routes to Chaos 5.Iterated Maps 6.Quasi-periodicity 7.Intermittency & Crises 8.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 -V d = V f when forward biased. R.W.Rollins, E.R.Hunt, Phys. Rev. Lett. 49, 1295 (82) Diode Circuit

Cause of bifurcation: After a large forward bias current I m, 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

InIn Larger signal 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.MayM.Feigenbaum Logistic eq. Iteration function Iterated Map

Maximum: Fixed point → if

A = 0.9

A = 1.5 X 0 =0.1 X 0 =0.8

A = 1.0 N=5000

A = 3.1

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

Lorenz Model

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 ~  Tbetween ↑↓ fluid within cell. Z ~  Tfrom linear variation as function of z. Derivation of the Lorenz eqs.: Appendix C 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) & r = 1 r = 1 is bifurcation point r < 1attractive repulsive r > 1 attractive repulsive r > 14 repulsive regions outside atractive ones, complicated behavior. repulsive r = 160 r = 160 : periodic. X oscillates around 0 → fluid convecting clockwise, then anti-clockwise, … r = 150 r = 150 : period 2. r = 146 r = 146 : period 4. … r < 144 r < 144 : chaos

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?