Nonlinear dynamics and chaos

(Fall 2003)

Instructors: Eli Tziperman; Jeremie Charles Korta.

Day and time of course: Mon, Wed, Fri, 10:00-11:00. Location: Pierce 307

Announcements Last updated: Jan 10, 2003.
Final exam will be a one day take home, Jan 13. Please come take the exam from my office at 10am, and return the following day at 16:00. Allowed material: course notes (yours and from the web) and HW plus their solutions. Date is flexible, so that if you have conflicts with other exams, you can do the take home a few days earlier or later. Please contact me regarding any such conflicts or other difficulties with this.
Matlab may be used but wont be necessary for the final. You'll need to plot one or two things, but can also do this by hand and using a pocket calculator.

Feel free to write or call me with any questions:
Eli Tziperman; eli AT eps.harvard.edu

Teaching notes online:
intro1, intro2, bif1d1, bif1d2, bif2d1, bif2d2, bif2d3, chaos1, chaos2, chaos3, chaos4, fract, Ham1, Ham2,

Sample Matlab programs: logistic_map.m, pendulum.m, pendulum_self_sustained.m, euler _course.m, circle_map.m, lorenz.m, henon.m, lorenz2.m, standard_map.m, standard_map_interactive.m, shilnikov.m, my_quiver.m, bakers_map.m, cobweb.m, feigenbaum.m,

Homework: 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11

Jeremie's homework solutions: 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11

What's the point about optional/ extra credit problems: apart from the fun of doing them, they will count against homework problems in which you may have missed an answer. . .

# Textbooks:

• (St) Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz
• (Sc) Deterministic Chaos: An Introduction; Heinz Georg Schuster, [VCH, 2nd edition, 1989]
• (Ott) Chaos in dynamical systems, 1993. Edward Ott, Cambridge University Press.
• (GH) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Guckenheimer, J and P. Holmes, Springer-Verlag, 1983.
• (W) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Stephen Wiggins, 1990. (Texts in Applied Mathematics, Vol 2).
• (JS) Classical Dynamics, a contemporary approach. Jorge V. Jose and Eugene J. Saletan. 1993 Cambridge University Press.
• (G) Classical Mechanics, Herbert Goldstein, 2nd edition, 1981. Addison Wesley.

# Outline

The course will introduce the students to the basic concepts of nonlinear physics, dynamical system theory, and chaos. These concepts will be demonstrated using simple fundamental model systems based on ordinary differential equations and some discrete maps. Additional examples will be given from physics, engineering, biology and major earth systems. The aim of this course is to provide the students with analytical methods, concrete approaches and examples, and geometrical intuition so as to provide them with working ability with non-linear systems.

## Introduction

(1 week) (St 1-37,+)
• A bit of history (Lorentz and the ``butterfly effect'')
• Modeling - defining phase space, dimension, parameters, deterministic versus stochastic modeling finite vs infinite dimensional (PDE's, integral eq.) models, linear vs non-linear, autonomous vs non-autonomous systems
• Examples: population dynamics, pendulum, Lorenz eq., ...
• The geometric approach to dynamical systems
• Fixed points, linearization, and stability
• Non-dimensionalization, the Buckingham Pi theorem (see notes here), small parameters, scales.
• Dynamical systems - continuous vs discrete time (ODEs vs maps; St 348), conservative vs dissipative (St 312).
• Existence, uniqueness and smooth dependence of solutions of ODE's on initial conditions and parameters.
• The role of computers in nonlinear dynamics, a simple example of a numerical solution method for ODEs (improved Euler scheme).
• Outline of rest of course.

## Bifurcations in one dimensional systems

(3 weeks)
• What's a bifurcation, local vs global bifurcations (GH §3.1). Implicit function theorem, classification of bifurcations by number and type (real/ complex) eigenvalues that cross the imaginary axis.
• saddle-node bifurcation (St §3.1; GH §3.4)
• Transcritical bifurcation, super critical and sub critical (St §3.2; GH §3.4).
• Pitchfork, super-critical and sub-critical. bead on a rotating hoop, higher order nonlinear terms and hysteresis (St § 3.4; GH §3.4)
• Some generalities: center manifold and normal form. (GH §3.2-3.3).
• Role of symmetry and symmetry breaking (imperfect bifurcations), relation to catastrophes and sudden transitions. (St §3.6)
• Flows on a circle - oscillators, synchronization (fireflies flashing, Josephson junctions) (St §4)

## Two-dimensional systems and some more basics

(4 weeks)

• Linear systems: classifications, fixed points, stable and unstable spaces (St §5)
• Non-linear systems: phase portrait (St §6.1), fixed points and linearization(St §6.3), stable and unstable manifolds (St §6.4), conservative systems (St §6.5), reversible systems (St §6.6), Solution of the (fully non-linear) damped pendulum equation (St §6.7), index theory (St §6.8).
• Limit cycles: Ruling out and finding out closed orbits (Lyapunov functions, Poincare Bendixon theorem) (St §7.2 and §7.3)
• relaxation oscillations (relation to glacial cycles) (St §7.5), weakly non-linear oscillators (Duffing eq) (St §7.6), Averaging method and two time-scales (St §7.6)
• Hopf bifurcation and oscillating chemical reactions (St §8.2),
• Global bifurcations of cycles: saddle-node infinite period, and homoclinic bifurcations, examples in Josephson Junction and driven pendulum in 2D (St §8.4 and§8.5)
• Quasi periodicity, coupled oscillators, nonlinear resonance/ frequency locking (Frequency locking of glacial cycles to earth orbital variations), (St §8.6)

## Chaos, transition to chaos

(4 weeks)

The Lorentz model as an introduction to chaotic systems (examples briefly motivating it from atmospheric dynamics and as a model of Magnetic field reversals of the Earth); and then a more systematic characterization of chaotic systems (examples from fluid dynamics and mantle convection) (St §9). Some preliminaries: Poincare maps.

Universal routes to chaos:

• Period doubling: logistic map, chaos, periodic windows, renormalization, quantitative and qualitative universality. (Sc §3)
• Intermittency: in Lorenz system, in logistic map. Length of laminar intervals from renormalization and simpler approaches. Categories of intermittency (types I,II,III), (Sc §4).
• Quasi-periodicity/ 1-2-chaos/ Ruelle-Takens-Newhouse; breakdown of 2d torus; in experimental systems; 1D circle map and overlapping of resonances; reconstructed circle map from a time series; damped-forced pendulum and El Nino's chaos (Sc §6)
More:
• Characterizing chaotic systems: Delay coordinates, embedding, Lyapunov exponents (Ott §4.4 p. 129); Kolmogorov entropy (Sc Appendix F and p 113; Greiner, Neise and Stocker ``thermodynamics and statistical mechanics'', p. 150); fractals and fractal dimensions, dimension spectrum (St § 11, p. 398-412; Ott §3, p69-71, 78-79, 89-92); Multi-fractals: dissipation in a turbulent flow, relation to dimension spectrum. (Ott §9, p 305-309).
• The horseshoe map and symbolic dynamics (Ott 108-114); Heteroclinic and homoclinic tangles and creation of a horseshoe from a homoclinic intersection (Ott §4.3). Shilnilov's phenomenon and chaos due to a 3d homoclinic orbit (GH, §6.5, p 318-323; and p 12-14 in Vered Rom-Kedar's notes).

## Chaos in Hamiltonian systems

(2 weeks)
• Examples (Pendulum, The n-body problem)
• Basics: Hamiltonian systems; Liouville theorem/ symplectic condition; (Ott §7.1.1-7.1.2 p 208-215).
• Motivation: the kicked rotor and chaos in the standard map (Ott, p 216-217, 235-237; JS §7.5.1 p. 453-459).
• More Basics: integrable vs non-integrable Hamiltonian systems; motion of integrable on N-torus; Canonical change of coordinates and generating functions; (G, §9-1, p. 378-385, Ott §7.1.1-7.1.2 p 208-215).
• Perturbations to integrable systems; averaging; resonant and non-resonant tori (G, §11-5, p 519-523); destruction of resonance tori and arising of chaos, KAM theory (Ott § 7.2).
• ``diffusion'' (Ott §7.3.3), fluid mixing (Ott p 246-249).

## Misc

(time permitting)
• Controlling chaos

# Course requirements

Homeworks will be given throughout the course. The best 80% of the assignments will constitute 50% of the final grade. A final take home exam will constitute another 50%.

# Misc

1. A nice on-line chaos course is at http://www.cmp.caltech.edu/~mcc/Chaos_Course/
2. Also nice: an interactive on line demo of a driven pendulum.
3. Devil's staircase in circle map and Farey tree:
4. For some interesting details about the KAM theorem, check here.
5. Links to some climate related papers using misc nonlinear dynamics tools (of mine at this stage, will try to add more later): El Nino's (quasi-periodicity route to) chaos, here and here. Glacial cycles and a climate bifurcation that happened one million years ago: here. Controlling El Nino's chaos (don't take this seriously): here. El Nino as a weakly nonlinear oscillation and its amplitude-period relation: here. Is the oceanic circulation/ climate close to an instability threshold? here and here.