Math 53: Resources - FALL 2011

Theory

• M. Pilant's course at Texas A&M, using our textbook.
• Jim Wiseman's dynamical systems course Math 85 at Swarthmore, using our textbook.
• Sutherland's course Math 351 at SUNY Stony Brook, using the same textbook.
• Logistic map for a>4 notes based on Devaney book.
• Wilson's course Math 312 at St Louis, using same textbook.
• Applied Math 3010 at UC Boulder, similar course, with excellent project ideas.
• R. Clark Robinson's Math 354 at Northwestern, with list of exams, and excellent project ideas.
• Liz Bradley's course at UC Boulder, CS. She also does chaotic dance algorithms
• Chris King's course at Auckland; some excellent links and applets, project ideas.
• The Cantor Set notes by Neal Carothers.
• Chaotic maps web notes collected by E. Demidov.
• Devaney notes on Mandelbrot set: I and II, and Mandelbrot interactive explorer notes (Devaney / BU). Mandelbrot and Julia sets anatomy web notes collected by E. Demidov.
• Fatou Theorem that basin of periodic attractor of rational complex map must contain a critical point of the map is Thm. 8.6 in Milnor, Dynamics in one complex variable (2006). It is proved in Chapter 8.
• Dynamical Systems Web tutorials from SIAM working group.
• Kanamaru and Thompson's Chaos course page including double pendulum applet.
• Physics 123 course introduction to fractals and chaos, has good experiments.
• Double pendulum in Hamiltonian formalism from Eric Weisstein.
• Jackson Pollock paintings are not fractals after all, study from Case Western Reserve.
• General notes on exam and study technique (geared mainly towards final exams in intermediate-level math courses).
• How to do and write proofs (crucial for this course): notes by Larry Cusick, William Turner, and Craig Silverstein.
• Proof that determinant is multiplicative, ie det(AB) = detA.detB. For elementary row operation matrices E_i see here.
• Holzmann's notes proving every square matrix can be reduced to Jordan normal form (even the ones that can't be diagonalized; that's the point).
• Big Eckmann-Ruelle review (1985) on ergodic theory of chaos. Discusses numerical measurement of Lyapunov exponents in Secton V.C.

Tools

• Bifurcation diagram applet by Takashi Kanamaru and J. Michael T. Thompson.
• Beautiful animation of cobweb plots for logistic map as parameter a (which they call r) is varied.
• Interacitve Mandelbrot explorer applet, aka Fractal Microscope, from Shodor Educational Foundation.
• Mandelbrot and corresponding Julia set applet by Devaney / James Denvir (BU).
• Chaos for Java great applets from Brian Davies, ANU, Canberra.
• MAPLE tutorials, useful for symbolic calculation.
• List of computational resources from Boulder, CO.
• Box-counting dimension estimation in matlab. See the following demos, for which you'll need to download code by Frederic Moisy.

Example systems

• Stephen Strogatz demo videos (local copy; originally from eCommons@Cornell
• Diana Dabby's chaotic musical variations and interview and 1996 paper

MATLAB

• Download the software from Dartmouth we have 100 or so on-campus licenses (to work you must be online in dartmouth.edu domain).
• Susan A. Schwarz (email her if you have Mac OS 10.2 or earlier for install CDs) can help with installation issues.
• Dartmouth's Academic Computing Center's own Matlab tutorial sessions on Oct 4, followed by more advanced on Oct 17 (scroll down; sign up reqd).
• M. Pilant's Matlab examples
• Bent Petersen's Matlab starter page
• Robert Higdon's nice introductory notes
• My 1-page intro53.m code, and 1-page intro code from Linear Algebra (shows more matrix stuff)
• Guide from Cambridge University Engineering Department.
• Simple intro from Utah, Hany Farid's intro reference, and Gilbert Strang's intro at MIT.
• Self-guided courses from Dartmouth academic computing: Introduction to Matlab, Programming in Matlab, and Introduction to Matlab Graphics
• Codes from our class: iter_hw1, iter_hw1_sol, beetle, explormap2d, explorbasin2d, iterdisc2d, lyap1d.m, cantorifs, julia, juliacolor, explormandel, henon_boxdim, henon_cordim, vectorize, potential1d, lyap2d, lyapflow (needs lorenz_time1map.m), henon_timedelay. henon_autoc. test_poincare (needs zerocross as the event function)
• ode45 tricks:
  • if you call a function f(t,x) defined in a separate file, use ode45(@functionname, ...). Don't use ode45('functionname',...)
  • If you want output at regularly spaced t values (useful for animations, comparisons for lyapunov meas), do this:

    tmax = 10;
    sol = ode45(@functionname, [0 tmax], ...);
    t = 0:0.01:tmax;         % here you choose the t values
    x = deval(sol,t);        % x now contains the soln's at these t values (cols)
    

  • If you want to increase accuracy (you should always check that your accuracy is high enough that your results don't depend on it), do this:

    ode45(@functionname, [0 tmax], xo, odeset('abstol', 1e-12))
    

  • For a Poincare section (catching intersections with a coordinate plane), see the above codes test_poincare and zerocross.