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Your suggestions for program

  1. Aug 6, 2010 #1
    I need a computer program I can use for studying chaotic systems. I should be more particular. I need something for drawing bifurcation diagrams or whatever one does in an introductory chaos and non-linear dynamics class. Here is a link to the text being used :

    https://www.amazon.com/Chaos-Kathleen-T-Alligood/dp/0387946772

    I suppose I'll be covering chapters 1-7 or something along those lines.

    I know of two programs - Matlab and Mathematica. Having never worked with either of this seriously before, (I've tinkered around with Mathematica) I was wondering how I'd go about doing the computations required by me.

    I checked the previous threads on these topics, but they were 'old'. At the rate at which computer software is growing, a thread from a year back is history. There might have been a billion changes to both programs.

    I also see that both programs are phenomenally expensive. If you guys have suggestions for any alternative programs that might do the trick, please recommend.
     
    Last edited by a moderator: Apr 25, 2017
  2. jcsd
  3. Aug 6, 2010 #2
    Here are the contents of the book:

    Table of Contents

    CHAPTER 1. One-Dimensional Maps
    1.1 One-dimensional maps
    1.2 Cobweb plot: Graphical representation of an orbit
    1.3 Stability of fixed points
    1.4 Periodic points
    1.5 The family of logistic maps
    1.6 The logistic map G(x)=4x(1-x)
    1.7 Sensitive dependence on initial conditions
    1.8 Itineraries
    Challenge 1: Period three implies chaos
    Exercises
    Lab Visit 1: Boom, bust, and chaos in the beetle census


    CHAPTER 2. Two-Dimensional Maps
    2.1 Mathematical models
    2.2 Sinks, sources, and saddles
    2.3 Linear maps
    2.4 Coordinate changes
    2.5 Nonlinear maps and the Jacobian matrix
    2.6 Stable and unstable manifolds
    2.7 Matrix times circle equals ellipse
    Challenge 2: Counting the periodic orbits of linear maps on a torus
    Exercises
    Lab Visit 2: Is the solar system stable?


    CHAPTER 3. Chaos
    3.1 Lyapunov Exponents
    3.2 Chaotic orbits
    3.3 Conjugacy and the logistic map
    3.4 Transition graphs and fixed points
    3.5 Basins of attraction
    Challenge 3: Sharkovsky's Theorem
    Exercises
    Lab Visit 3: Periodicity and chaos in a chemical reaction


    CHAPTER 4. Fractals
    4.1 Cantor sets
    4.2 Probabilistic constructions of fractals
    4.3 Fractals from deterministic systems
    4.4 Fractal basin boundaries
    4.5 Fractal dimension
    4.6 Computing the box-counting dimension
    4.7 Correlation dimension
    Challenge 4: Fractal basin boundaries and the uncertainty exponent
    Exercises
    Lab Visit 4: Fractal dimension in experiments


    CHAPTER 5. Chaos in Two-Dimensional Maps
    5.1 Lyapunov exponents
    5.2 Numerical calculation of Lyapunov exponents
    5.3 Lyapunov dimension
    5.4 A two-dimensional fixed-point theorem
    5.5 Markov partitions
    5.6 The horseshoe map
    Challenge 5: Computer calculations and shadowing
    Exercises
    Lab Visit 5: Chaos in simple mechanical devices


    CHAPTER 6. Chaotic Attractors
    6.1 Forward limit sets
    6.2 Chaotic attractors
    6.3 Chaotic attractors of expanding interval maps
    6.4 Measure
    6.5 Natural measure
    6.6 Invariant measure for one-dimensional maps
    Challenge 6: Invariant measure for the logistic map
    Exercises
    Lab Visit 6: Fractal scum


    CHAPTER 7. Differential Equations
    7.1 One-dimensional linear differential equations
    7.2 One-dimensional nonlinear differential equations
    7.3 Linear differential equations in more than one dimension
    7.4 Nonlinear systems
    7.5 Motion in a potential field
    7.6 Lyapunov functions
    7.7 Lotka-Volterra models
    Challenge 7: A limit cycle in the Van der Pol system
    Exercises
    Lab Visit 7: Fly vs. fly


    CHAPTER 8. Periodic Orbits and Limit Sets
    8.1 Limit sets for planar differential equations
    8.2 Properties of omega-limit sets
    8.3 Proof of the Poincare-Bendixson Theorem
    Challenge 8: Two incommensurate frequencies form a torus
    Exercises
    Lab Visit 8: Steady states and periodicity in a squid neuron


    CHAPTER 9. Chaos in Differential Equations
    9.1 The Lorenz attractor
    9.2 Stability in the large, instability in the small
    9.3 The Rossler attractor
    9.4 Chua's circuit
    9.5 Forced oscillators
    9.6 Lyapunov exponents in flows
    Challenge 9: Synchronization of chaotic orbits
    Exercises
    Lab Visit 9: Lasers in synchronization


    CHAPTER 10. Stable Manifolds and Crises
    10.1 The Stable Manifold Theorem
    10.2 Homoclinic and heteroclinic points
    10.3 Crises
    10.4 Proof of the Stable Manifold Theorem
    10.5 Stable and unstable manifolds for higher dimensional maps
    Challenge 10: The lakes of Wada
    Exercises
    Lab Visit 10: The leaky faucet: minor irritation or crisis?


    CHAPTER 11. Bifurcations
    11.1 Saddle-node and period-doubling bifurcations
    11.2 Bifurcation diagrams
    11.3 Continuability
    11.4 Bifurcations of one-dimensional maps
    11.5 Bifurcations in plane maps: Area-contracting case
    11.6 Bifurcations in plane maps: Area-preserving case
    11.7 Bifurcations in differential equations
    11.8 Hopf bifurcations
    Challenge 11: Hamiltonian systems and the Lyapunov Center Theorem
    Exercises
    Lab Visit 11: Iron + sulfuric acid = Hopf bifurcation


    CHAPTER 12. Cascades
    12.1 Cascades and 4.66920169...
    12.2 Schematic bifurcation diagrams
    12.3 Generic bifurcations
    12.4 The cascade theorem
    Challenge 12: Universality in bifurcation diagrams
    Exercises
    Lab Visit 12: Experimental cascades


    CHAPTER 13. State reconstruction from data
    13.1 Delay plots and time series
    13.2 Delay coordinates
    13.3 Embedology
    Challenge 13: Box-counting dimension and intersection

    APPENDIX A. Matrix Algebra
    A.1 Eigenvalues and eigenvectors
    A.2 Coordinate changes
    A.3 Matrix times circle equals ellipse

    APPENDIX B. Computer Solution of ODEs
    B.1 ODE solvers
    B.2 Error in numerical integration
    B.3 Adaptive step-size methods

    HINTS AND ANSWERS TO SELECTED EXERCISES

    BIBLIOGRAPHY
     
  4. Sep 26, 2011 #3
    here is a simple HTML5 program (javascript) used for solving Chua's circuit equations:
    http://www.chuacircuits.com/sim.php" [Broken]

    you have to view the source code and the attached file www.chuacircuits.com/canvas3DGraph.js

    But the page itself is great an introductory look at the strange attractor/double scroll.
     
    Last edited by a moderator: May 5, 2017
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