Lecture title table LectTable.pdf

Story Line StoryLine.pdf  (Dec 7) This summarizes the course. However, to use the hypertext features, you must use the whole set of lecture notes PDynamicsLectures2018Dec7.pdf.  Story Line is at the end.

Video: Chaos  is recommended in 1.10 

I found these videos after designing the outline of the course, but they give almost a parallel line of the whole story.

Some portions (maybe 70% or more) may be too elementary, but watch while eating or drinking.

    Chapter 1: Motion and determinism 𝜋𝛼𝜈𝜏𝛼 𝜌𝜀𝜄.

            https://www.youtube.com/watch?v=c0gDLEHbYCk&t=4s&frags=pl%2Cwn

    Chapter 2: The vector fields|The lego race               

        https://www.youtube.com/watch?v=_Y68GX2UpQ0&frags=wn

    Chapter 3: Mechanics|The apple and the Moon (Newton, universal law of gravity, etc.)

        https://www.youtube.com/watch?v=ZwTGAW0b_bo&frags=wn

    Chapter 4: Oscillations|the swing (including Lotka-Volterra; Poincare-Bendixson theorem including the idea to prove it)

        https://www.youtube.com/watch?v=uEfB5DG9x9M&frags=wn

    Chapter 5: Billiards|Duhem's bull (geodesics on negative curvature surface included, symbolic dynamics a bit)

        https://www.youtube.com/watch?v=3u2SJKxJhh8&frags=wn

    Chapter 6: Chaos and the horseshoe|Smale in Copacabana (Poincare map, Smale's horseshoe, symbolic dynamics, structural stability)

        https://www.youtube.com/watch?v=ItZLb5xI_1U

    Chapter 7: Strange attractors|the buttery effect (Lorenz system, Lorenz template, symbolic dynamics

        https://www.youtube.com/watch?v=aAJkLh76QnM&frags=wn

    Chapter 8: Statistics|Lorenz' mill (measure-theoretical aspect, physical model of Lorenz system, sensitivity to initial conditions, SRB measure)

        https://www.youtube.com/watch?v=SlwEt5QhAGY&frags=wn

    Chapter 9: Chaotic or not|research today (bifurcation diagram, heteroclinic con- nection, non-generic case, Palis conjecture)

        https://www.youtube.com/watch?v=_xfi0NwoqX8

Lecture 44 Palis conjecture Lect44Palis.pdf (Dec 5)  Stability conditions, Palis conjecture(s)

Lecture 43 Heterodimensional cycles Lect43Heterocycle.pdf (Dec 5 typo corrected) Cycles, heterodimensional cycles

Lecture 42 Newhouse phenomenon Lect42Newhouse.pdf (Dec 1)  Homoclinic tangency and its persistence, cascade, Newhouse phenomenon

Lecture 41 1D Quantum lattice spectrum Lect41QLattice.pdf (Nov 28)  1D almost periodic potential and Cantor spectrum, discrete cat maps

Lecture 40 Anosov systems  Lect40Anosov.pdf (Nov 21) Anosov systems, Thom’s diffeomorphism, Markov partition

Lecture 39 Axiom A systems  Lect39AxiomA.pdf (Nov 26 figures added) Axiom A, hyperbolic set, stability manifold theorem, canonical coordinate, spectral decomposition, approximation and shadowing, Markov partition (proof illustrated) and coding

Lecture 38 Peixoto’s  theorem Lect38Peixoto.pdf (Nov 15 Considerably improved) Morse-Smale system, Peixoto’s theorem (original proof outline)

Lecture 37 Takens embedding theorem (draft) TakensTheorem.pdf (Nov 12)  Grossly incomplete (a better proof for practical cases will be posted hopefully soon)

Lecture 36 Thermodynamic formalism  Lect36Thermodynamics.pdf (Nov 25) Thermodynamic formalism, free energy, entropy and temperature, `ultimate conjecture’ corresponding to Palis’

Lecture 35 Large deviation approach  Lect35LDApproach.pdf (Nov 12) Level 1-3 large deviations, Pesin revisited, Gibbs measure

Lecture 34 Isomorphism invariant  Lect34IsoInvariant.pdf  (Nov 2 ) Ornstein-Weiss theorem, Bernoulli system, Sinai’s theorem, Ornstein’s theorem [I have given up to introduce the modern soft proof]

Lecture 33 Lyapunov characteristic number  Lect33LyapunovCN.pdf (Nov 6 proofs illustrated) LCN, Oseledec’s multiplicative ergodic theorem, Pesin’s equality, Ruelle’s inequality, Kingman’s subadditive ergodic theorem (not complete)

Lecture 32 Kolmogorov-Sinai entropy Lect32KSEntropy.pdf (Nov 8 Top-ent added) Partition, steady-state information loss, Kolmogorov-Sinai entropy, generator, Krieger’s theorem, Shannon-McMillan-Breiman’s  theorem, Brin-Katok’s theorem,  noise-induced order, topological entropy

Lecture 31 Information loss rate Lect31InformationLoss.pdf (Nov 6 Perron cleaned) Entropy loss by endomorphisms, Rokhlin’s formula, Markov chain entropy, Perron-Frobenius equation, Information formula/Sanov’s theorem

Lecture 30 Ergodic theorems Lect30Ergodic.pdf (Nov 1 made more readable) Poincare’s recurrence theorem, maximal ergodic theorem, Birkhoff’s theorem, Weyl’s theorem, von Neumann’s theorem, tragicomical history of ergodicity

Lecture 29 Measure theoretical dynamical systems Lect29Measure.pdf (Nov 1 slight cleaning) Invariant measure, absolute continuity, ergodicity, mixing, Appendix: What is a measure?

Lecture 28 Horseshoe dynamical systems Lect28Horseshoe.pdf (Oct 27) Horseshoe, symbolic dynamics

Lecture 27 Baker’s transformation Lect27Baker.pdf (Oct 29 figures cleaned) Baker’s  transformation, symbolic dynamics

Lecture 26 Symbolic dynamics Lect26SymbolicDynamics.pdf (Nov 1 slight changes) Shift, shift dynamical system, cylinder set, Markov subshift, Perron-Frobenius theorem, zeta function

Lecture 25 Computable analysis  Lect25ComputableAnalysis.pdf (Oct29 slight augmentation) Decision problem, halting problem, RENR set, nondeterministic TM, computable real numbers, computable functions, Myhill’s theorem, Physics implications

Lecture 24 Characterization of chaos  Lect24ChaosCharacterization.pdf (Oct 24 slightly kinder) Randomness of trajectories, Brudno’s theorem, what is chaos?

Lecture 23 Randomness Lect23Randomness.pdf (Oct 17 slightly cleaned) Random numbers, featurelessness, compressibility, Church-Turing thesis, recursiveness, algorithmic random number

Lecture 22 Interval maps Lect22IntervalMaps.pdf  (Oct 24 figures added) Interval maps, Li-Yorke’s theorem, chaos, period \neq 2^n implies chaos, time correlation, Wiener-Khinchin theorem, Bochner theorem, FFT, Sharkovskii’s theorem

Lecture 21 Strange attractors Lect21StrangeAttractors.pdf (Oct 15 slightly augmented) Landau theory of turbulence, Ruelle-Takens theorem, Plykin map, strange attractor

Lecture 20 Lorenz system: advanced topics Lect20LorenzAdv.pdf (Oct 18 Lorenz-Ising correspondence added) Knotted and linked orbits, geometrical Lorenz model, existence of Lorenz attractor, pseudo-orbit tracing, Lorenz template, Ising representation by Shimada, Galerkin method

Lecture 19 Lorenz system: introduction  Lect19LorenzIntro.pdf (Oct 16 fine tuned) Motivation, Saltzman’s equation, Lorenz’s observations, Lorenz map, Magnetic stripes, Earth dynamo (Rikitake model)

            [Some demo sites may not work, depending on your OS]

Lecture 18  Lect18IntroChaos.pdf (Oct 7 version) Coupled relaxation oscillators, why unpredictability?, discretization, randomization

Lecture 17 Billiards Lect17Billiard.pdf (Oct 11 Baldwin’s soft billiard added) Billiards, Ambrose-Kakutani representation, Abramov formula, Sinai billiard, Bunimovich stadium, KS entropy, Soft billiards

Lecture 16 General picture of Hamiltonian systems Lect16GeneralPicture.pdf (Oct 8 version) Near integrable systems, Henon system, resonance, Poincare’s picture and King Oscar II prize paper, area preserving maps, twist map, Poincare-Birkhoff theorem, Chirikov-Taylor or standard map, Fermi acceleration problem, Arnold diffusion

Lecture 15 Siegel and KAM theorems Lect15SiegelKAM.pdf (Oct 4 version)  Stability around center, Poincare’s lemma, Diophantine condition, Siegel’s theorem, KAM demonstration outline

Lecture 14 Celestial mechanics Lect14Celestial.pdf (Oct 3 details added + some corrections)  Bertrand’s theorem. collisions, restricted three-body problem, Bruns and Poincare’s theorems, Asteroids: Lagrange points and Trojan group, asteroids and mass extinction?

Lecture 13 Canonical transformation Lect13Canonical.pdf (Sept 28 correction added)  Canonical transformation, generator, infinitesimal canonical transformation, Lagrange bracket, integral invariants, Liouville’s theorem, Jacobi’s method, separation of variables

Lecture 12 LaxToda Lect12LaxToda.pdf (Sept 28 Burgers more detail added)  Lax Pair, Toda lattice, Fermi-Pasta-Ulam(-Tsingou) problem, Korteweg-de Vries equation, Burgers equation, Cole-Hopf transformation, solitons

Lecture 11 Determination of motion Lect11DetMotion.pdf (Sept 26 slight corrections)  Determination of motion,  (complete) integrability, Liouville-Arnold’s theorem, action-angle variables, adiabatic invariant

Lecture 10 Classical mechanics review Lect10CMReview.pdf (Sept 25 slight modification)  Newton-Laplace determinacy, Newton’s equation of motion, Vainberg’s theorem, minimum action principle, Hamilton’s principle,  Poisson bracket, Hamilton-Jacobi’s equation, Schrodinger’s Paper I.

 

Lecture 9 Singular perturbation and renormalization Lect9RGSingular.pdf (Nov 1 RG calculations corrected)  singular perturbation, renormalization (CGO), resonance, Chiba theory

Lecture 8 Bifurcations 2  Lect8Bifurcation2.pdf (Sept 17 typos corrected)   versal unfolding, pitchfork bifurcation,  Feigenbaum critical phenomenon

Lecture 7 Bifurcations 1    Lect7Bifurcation1.pdf (Sept 17 typos corrected) Bifurcation, versal unfolding, normal forms, saddle-node bifurcation, Hopf bifurcation,

Lecture 6 Periodic motions   Lect6Periodic.pdf (Sept 12 version ) Periodic orbits, Poincare map. Floquet’s theorem, Poincare-Bendixson’s theorem, limit cycle, chemical oscillator, nullcline

Lecture 5 Hyperbolicity  Lect5Hyperbolicity.pdf (Sept 11 typo corrected)   limit sets, hyperbolicity, stable manifold (with RG flow example for critical phenomena), Hartman’s theorem, center manifold, Lyapunov stability, asymptotic stability

Lecture 4 Singularity   Lect4Singularity.pdf (Sept 6 example corrected and dgree theory simplified) Linearization, e^A, Jordan normal form, Classification of singular points,     

    Classification of singularity

        https://media.pearsoncmg.com/aw/ide/idefiles/media/JavaTools/lnclppan.html

Lecture 3 ODE 1  Lect3ODE1.pdf  (Sept 11  ODE history added) ODE, Peano, Cauchy-Lipshitz, nondeterministic case

    Vector field and phase flow

        https://media.pearsoncmg. com/aw/ide/idefiles/media/JavaTools/twoddfeq.html

    Direction field and solution curve

        https://media.pearsoncmg.com/aw/ide/idefiles/media/JavaTools/ exunqtrg.html

Lecture 2 Setting the stage Lect2Stage.pdf   (Sept 3 cleaned) manifold, vector field, Cantor set, dimension

Lecture 1 Introduction   Lect1Intro.pdf (Aug 27 version)