Nonlinear dynamics and chaos

Instructors: Einat Aharonov, Vered Rom-Kedar and Eli Tziperman.

Day, time & location of course: Tuesday, 11:15-13:00, Feinberg room 3. A few special session will be held on Thursdays, 16:15-18:00, check here for details.

Announcements Last updated: Apr 7.
Finally, here is the exam and its solution: here and here
Feel free to write or call me with any questions:
Eli Tziperman; eli@beach.weizmann.ac.il; x2544

Teaching notes online:
Lecture 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12, 13, 14, 15, 16,

Sample Matlab programs: logistic_map.m, pendulum.m, pendulum_self_sustained.m, euler_course.m, circle_map.m, lorenz.m, henon.m, lorenz2.m,

Homework: 01, 02, 03, 04, 05 (optional!), 06, 07, 08, 09, 10, 11

Homework solutions: 09, 10, 11

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]
(GH) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Guckenheimer, J and P. Holmes, Springer-Verlag, 1983.

Additional reading:

(W) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Stephen Wiggins, 1990. (Texts in Applied Mathematics, Vol 2).
(PR) Introduction to Dynamics. Ian Percival and Derek Richards 1982 Cambridge University Press.
(Ott) Chaos in dynamical systems. Edward Ott. 1993 Cambridge University Press.

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

(2 lectures, Eli) (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.
Perturbation theory - regular vs singular perturbations
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 ( $dx/dt=f(x)$ )

(2 lectures, Eli)

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 ( $d\vec\mathbf{x}/dt=\vec\mathbf{f}(\vec\mathbf{x})$ ; $\vec\mathbf{x}=(x,y)$ ) and some more basics

(3 lectures, Einat)

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 "stick-slip", friction) (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)
Poincare maps (St §8.7)

Chaos, transitions to chaos

(3 lectures, Eli)

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, delay coordinates, embedding, fractal dimensions (St § 8.7, 10.5, 11.4-5). Lyapunov exponents, Kolmogorov entropy. Possibility of short-time predictability (Gronwall's Lemma). The problem in relying on simulations, the shadowing lemma.

Universal routes to chaos:

Period doubling: logistic map, chaos, periodic windows, renormalization, quantitative and qualitative universality. ( $x_{n+1}=r x_n(1-x_n)$ ) (Sc § 3)
Intermittency: in Lorenz system, in logistic map. Length of laminar intervals from renormalization and simpler approaches. Categories of intermittency (types I: $x_{n+1}=\varepsilon + x_n +u x_n^2$ ; II: $r_{n+1}=(1+\varepsilon)r_n+ur_n^3$ , $\theta_{n+1}=\theta_n+\Omega$ ; III: $x_{n+1}=-(1+\varepsilon)x_n-u x_n^3$ ). (Sc § 4)
Quasi-periodicity/ 1-2-chaos/ Ruelle-Takens-Newhouse; breakdown of 2d torus; in experimental systems; 1D circle map ( $\theta_{n+1}=\theta_n+\Omega-\frac{K}{2\pi}\sin(2\pi\theta_n)$ ) and overlapping of resonances; reconstructed circle map from a time series; damped-forced pendulum and El Nino's chaos (Sc § 6)

More: (2 lectures, Vered)

The horseshoe map and homoclinic tangles (GH 230-255, W 420-443, 482-483, 519-552)
Shilnikov's model and revisiting the Lorenz attractor. (GH 318-340 & 312-318, W 552-591)

Hamiltonian systems

(3 lectures, Vered)

Examples (Pendulum, Duffing, The n-body problem, point vortices, Lagrangian advection) (GH, W etc)
Basics (canonical coordinates, Linear systems, generating functions,1 d.o.f. systems) (e.g. PR 42-60, 84-116)
On near integrable (chaotic) systems: (GH 166-193, W 106-175,483-513)
- 1.5 d.o.f.
  - Averaging and KAM theory.
  - Melnikov theory and resonances.
  - Transport theory and applications to fluid flows
  - Influence of small dissipation.
On n2 d.o.f. systems: (Encyclopedia of mathematical sciences, vol. 3, papers)
- Integrable systems (Liouville-Arnold theorem)
- Near integrable systems in Higher dimensions
- The resonance web and instabilities in phase space

Course requirements

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

Misc

See Mathematical Analysis and Applications Seminar
A complementary course emphasizing a more mathematical point of view: Dynamical Systems, Differential and Difference Equations
A nice on line chaos course is at http://www.cmp.caltech.edu/~mcc/Chaos_Course/
Also nice: an interactive on line demo of a driven pendulum: http://monet.physik.unibas.ch/~elmer/pendulum/spend.htm.
home pages of people at Weizmann doing work related to nonlinear-dynamics and chaos: Eli Tziperman, Einat Aharonov, Vered Rom-Kedar, Uzi Smilansky,

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