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12.006

Nonlinear Dynamics: Chaos

3-0-9 units · Undergraduate

Introduction to nonlinear dynamics and chaos in dissipative systems. Forced and parametric oscillators. Phase space. Periodic, quasiperiodic, and aperiodic flows. Sensitivity to initial conditions and strange attractors. Lorenz attractor. Period doubling, intermittency, and quasiperiodicity. Scaling and universality. Analysis of experimental data: Fourier transforms, Poincare sections, fractal dimension, and Lyapunov exponents. Applications to mechanical systems, fluid dynamics, physics, geophysics, and chemistry. See 12.207[J] / 18.354[J] for Nonlinear Dynamics: Continuum Systems.

Fall 2022

Topics

  • Introduction to Nonlinear Dynamics: Chaos
  • Flows and Bifurcations in One Dimension
  • Flows and Bifurcations in One Dimension (cont.)
  • Pendulum and Stability in Two Dimensional Systems
  • Pendulum and Stability in Two Dimensional Systems (cont.)
  • Dissipation and Contraction of Volumes in Phase Space
  • Dissipation and Contraction of Volumes in Phase Space (cont.)
  • Forced Oscillators and Limit Cycles
  • Forced Oscillators and Limit Cycles (cont.)
  • Bifurcations in Two Dimensions
  • Bifurcations in Two Dimensions (cont.)
  • Spectral Analysis for Dynamical Systems
  • Spectral Analysis for Dynamical Systems (cont.)
  • Poincaré Sections
  • Poincaré Sections (cont.)
  • Fluid Dynamics and Rayleigh-Benard Convection
  • Fluid Dynamics and Rayleigh-Benard Convection (cont.)
  • Introduction to Strange Attractors
  • Lorenz Equations
  • Lorenz Equations (cont.)
  • Hénon Attractor
  • Fractal Dimension
  • Lyapunov Exponents
  • Period Doubling Route to Chaos
  • Period Doubling Route to Chaos (cont.)
  • Intermittency (and Quasiperiodicity)
  • Final Project Oral Presentation

Readings

  • Strogatz, S. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, 2020. ISBN: 9780738204536.
    Listed reading
  • Berge, P., Y. Pomeau, and C. Vidal. Order within Chaos: Towards a Deterministic Approach to Turbulence. Wiley-VCH, 1987. ISBN: 9780471849674. (An undergraduate-level physical introduction to the subject.)
    Listed reading
  • Cross, M. and H. Greenside. Pattern Formation and Dynamics in Nonequilibrium Systems . Cambridge University Press, 2009. ISBN: 9780521770507.
    Listed reading
  • Cvitanovic, P. Universality in Chaos. Adam Hilger, Ltd., 1989. ISBN: 9780852747650. (Contains reprints of a number of original research papers in the field.)
    Listed reading
  • Cvitanovic, P., R. Artuso, R. Mainieri, G. Tanner, and G. Vattay. Chaos: Classical and Quantum (PDF - 8.4 MB) ↗
    Listed reading
  • Gleick, J. Chaos . Viking Books, 1987. ISBN: 9780749386061. (An excellent popular introduction.)
    Listed reading
  • Guckenheimer, J. and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 1983. ISBN: 9780387908199. (A graduate-level applied mathematics textbook.)
    Listed reading
  • Schuster, H. and W. Just. Deterministic Chaos: An Introduction , 4 th edition. Wiley-VCH, 2005. ISBN: 9783527404155. (An advanced book of interest to physicists.)
    Listed reading
  • Turcotte, D. Fractals and Chaos in Geology and Geophysics , 2 nd edition. Cambridge University Press, New York, 1997. ISBN: 9780521567336.
    Listed reading
Fall 2012 · Nonlinear Dynamics I: Chaos

Topics

  • Free oscillators: Linear and nonlinear pendulum. Phase space and phase portraits. Fixed points. Stability. Liouville’s theorem and conservation of areas in phase space. Damped pendulum: At-tractors. Contraction of areas in phase space.
  • Forced oscillators: Van der Pol equation. Limit cycles. Forced pendulum. Resonance. Stability.
  • Introduction to bifurcation theory.
  • Methods for analyzing periodic, quasiperiodic, and aperiodic systems. Poincare sections. Floquet matrices and stability. Maps. Reduction of flows to maps. Reconstruction of phase space from one-dimensional signals.
  • Strange attractors. Dissipation, attraction, and reduction of dimensionality. Minimum dimension of deterministic aperiodic systems. Sensitivity to initial conditions. Stretching and folding. Rossler attractor. Derivation of Lorenz attractor. Stability of Lorenz equations. Henon attractor.
  • Quantitative analysis of strange attractors. Lyaponov exponents. Fractal dimension.
  • Transitions to chaos. Period doubling: logistic map, Feigenbaum numbers, scaling, and universality. Quasiperiodicity. Intermittency. Illustrations from experiments.
  • If time permits, various applications of the course material in geosciences, math biology, etc.

Readings

  • Strogatz, S. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering . Westview Press, 2001. ISBN: 9780738204536. Strogatz will be especially valuable for our discussion of stability of ordinary differential equations.
    Listed reading
  • The following book may also be useful: Berge, P., Y. Pomeau, and C. Vidal. Order within Chaos . Wiley-VCH, 1987. ISBN: 9780471849674.
    Listed reading
  • Beltrami, E. Mathematics for Dynamic Modeling . Academic Press, 1987. ISBN: 9780120855551. An elementary presentation of many of the mathematical techniques useful in studying dynamics. Especially useful for a review of ordinary differential equations, stability, fixed points, and phase space representations.
    Listed reading
  • Cvitanovic, P., ed. Universality in Chaos: A Reprint Selection . Adam Hilger, 1984. Contains reprints of a number of original research papers in the field.
    Listed reading
  • Gleick, J. Chaos: The Making of a New Science . Viking Adult, 1987. ISBN: 9780670811786. An excellent popular introduction to the subject for the lay reader.
    Listed reading
  • Guckenheimer, J., and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifur- Cations of Vector Fields . Springer-Verlag, 1983. ISBN: 9780387908199. A graduate-level applied mathematics text.
    Listed reading
  • Hilborn, Robert C. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers . Oxford University Press, 1994. ISBN: 9780195057607. One of the more elementary books on the subject, oriented more for the physicist than for the mathematician. Big (over 600 pages). Lots of good exercises.
    Listed reading
  • Kaplan, D., and L. Glass. Understanding Nonlinear Dynamics . Springer-Verlag, 1995. ISBN: 9780387944401. A nice undergraduate-level introduction, oriented towards scientists, and especially biologists.
    Listed reading
  • Manneville, P. Dissipative Structures and Weak Turbulence . Academic Press, 1990. ISBN: 9780124692602. Covers the low-dimensional theory discussed in the course but also includes considerable discussion of research on spatially-extended systems. Advanced.
    Listed reading
  • Schuster, H. G. Deterministic Chaos: An Introduction . 2nd ed. Vch Publisher, 1988. ISBN: 9780895736116. An advanced book of interest to physicists.
    Listed reading
  • Thompson, J. M. T., and H. B. Stewart. Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists . John Wiley and Sons, 1986. ISBN: 9780471909606. A textbook of moderate difficulty with lots of nice pictures and a fair amount of material of engineering interest.
    Listed reading
  • Turcotte, D. Fractals and Chaos in Geology and Geophysics . 2nd ed. Cambridge University Press, 1997. ISBN: 9780521567336. [Preview with Google Books ]. Provides a number of nice examples of nonlinear phenomena in the earth sciences. ↗
    Listed reading

Prerequisites & course information

Same subject as 2.050[J] , 18.353[J] Prereq: Physics II (GIR) and ( 18.03 or 18.032 ) U (Fall) 3-0-9 units

Sources