Nonlinear dynamics and chaos by Steven Strogatz (Westview Press).
Paperback can be purchased at Tapir bookstore (426 NOK).
Lectures:
Monday 14.15-16.00 and
Thursday 12.15-15.00 in E5-103. First lecture Monday August 22. Last
lecture Thursday November 24. Exercises will be treated in class
when it fits.
Office hours:
Thursday 11.00-1200. Generally, I have an open-office policy so feel free
to drop by anytime.
Course description:
Graphical solution methods for non-linear differential equations. Phase
portraits, fixed point analysis, bifurcations, limit cycles, strange
attractors, Poincare and Lorenz maps, multiscale perturbation theory. Iterative
maps. Period doubling, chaos, scaling and universality. Fractals. Examples
from physics, chemistry, and biology.
Final curriculum:
Chapter 2, 3.0-3.2, 3.4, 3.6, 4.0-4.4, 5.0-5.2.
chapter 6, 7.0-7.4, 7.6. 8.0-8.4, 9.2-9-5,
chapter 10, 11.0-11.4, 12.0-12.2.
All problem sets.
Assignments:
The students will be assigned one exercise set each to work out on the
blackboard. In addition, a numerical exercise including some programming
must be handed in. These assignments must be completed in order to take
the exam. Assignments are presented on thursdays.
If you find typos or have suggestions for improvements on the solutions,
please let me know. The students taking the course next year will be thankful.
Week 34:
Systems of differential equations and difference equations. Linear versus
nonlinear systems. Graphical techniques. Fixed points and their stability.
Population dynamics. Existence and uniqueness theorem. mechanical analog
and potentials.
Saddle-node bifurcation. Stability of fixed points.
Normal forms.
Week 35:
Euler's and Runge-Kutta method for solving differential equations numerically.
Transcritical bifurcations.
Supercritical and subcritical pitchfork bifurcations.
Week 36:
Flows on the circle. Nonuniform oscillator: Fixed points and their stability.
Period T, ghosts and bottlenecks. Linear systems. Vector fields and
periodic phenomena (harmonic oscillator).
Straight-line solutions and stable and unstable manifolds.
Classification of linear systems: saddle point, stable and unstable nodes,
star nodes, degenerated nodes, unisolated fixed points, stable and
unstable spirals.
Week 37:
1D code
and
2D code
Nonlinear systems: Existence and uniqueness of solutions.
Topological consequences (curves cannot cross, approach to closed orbit
when solution is inside and no fixed point inside).
Fixed points and linearization. Borderline fixed points and nonlinear effects.
Sheep vs rabbits population dynamics (Lotka-Volterra model) and the
principle of mutual exclusion. Basin of attraction.
Week 38:
Conservative systems and nonlinear centers. Reversible systems and
closed trajectories around linear centers.
Pendulum revisited. Index theory: Smooth
velocity field and index of a simple closed curve.
Week 39:
Index of a point (saddle, center, star...) and of a trajectory.
Ruling out various closed curves.
Limit cycles, potential functions and Liapunov functions.
Dulac's criterion. Poincare-Bendixon theorem for closed orbits.
Trapping regions and
(repelling) fixed points.
Week 40:
Lienard systems. Weakly nonlinear oscillators and regular perturbation theory.
Bifurcations revisited: saddle-node bifurcations.
Hopf Bifurcations: supercritical, subcritical, and degenerate.
Change of stability of fixed point and birth/death of limit cycles.
Week 41:
Åsmund will present the
numerical exercise monday October 10th.
Here
.
Example 8.3.1 Infinite-period bifurcations. Poincare maps and
closed orbits for driven pendulum with damping.
Plot
.
Week 42:
Discrete maps. Fixed points and their stability. Cobwebs.
Numerical study of the logistic map.
Week 43:
Logistic map: fixed points, instability of fixed points
and fliop bifurcations. Tangent bifurcations. Periodic windows and ghosts.
Liapunov exponents. Tent map and its orbit diagram. Universality of
unimodal maps, alpha and delta.
Week 44:
Renormalization of f(x). Universal functions g_i(x) and g(x).
Renormalization of parameter at period-doubling bifurcation.
Countable and uncountable sets.
Fractals: Cantor set and von Koch curve. Cantor set: closed and uncountable
with zero measure. Fractal dimensions.
Week 45:
Box dimensions. Fat fractals. Strange repellors.
Stretching and folding. Baker's map and strange attractors.
Henon map: area contraction, fixed points and 2-cycles.
Week 46:
Summary chapter 12. Lorenz equations. Fixed points and their stability.
Unstable limit cycle. Hopf bifurcation for r=r_H.
Chaos and (strange) attractors. Homoclinic bifurcation at r=13.926
Week 47:
No lectures.
Exercise sets:
Week 35:Exercises 2.2.3, 2.4.6, 2.5.3, 2.6.1, 2.6.2, 2.7.6.
Solutions set1
.
Exercises Example 2.2.+2.2.11, 2.3.3, 2.4.8, and 2.5.1 (Glesaaen).
Solutions set2
Week 36:
Exercise 2.4.7, 2.7.5, 3.1.3, 3.2.2 (Berge).
Solutions set3
Week 37: Exercises 3.4.5, 3.4.6, 3.6.2, 4.1.2, and 4.3.8 (Grav).
Solutions set4
Week 38:
Exercise 3.7.4 and 5.1.9 (Bauer).
Solutions set5
Week 39: 5.2.12, and Exam fall 2007
problem 1
(Ofstad).
Solution
and
Solution 5.2.12
Week 40:
Exercises 6.3.9, 6.3.10, 6.3.13, 6.3.14, 6.5.1, and 6.5.12
(Sjøstrøm).
Solution set7
Week 41: Exercise 6.5.6, 6.5.11, 6.5.19, and 6.6.10
(Sivertsen Bergslid).
Solution set8
Week 42:
Exercises
6.7.1, 6.7.2, 6.8.7, 7.1.4, 7.1.8, 7.2.5, and 7.2.16 (Bjørklund).
Solution set9
.
Exercises 7.3.1, 7.3.4, 7.3.6, 7.4.1, 7.6.2, 7.6.13, and 7.6.16.
(Andresen).
Solution set10
.
Week 43: 8.1.9, 8.2.1, 8.2.8, 8.2.11, and 8.4.2
(Hansen).
Solutions set11
Week 44:
10.1.9, 10.1.10, 10.1.12 (a-c) 10.1.13, and 10.3.2,
(Lisø)
Solutions set12
and
10.3.6, 10.3.11, 10.4.3, and 10.5.4.
(Kittang).
Solutions set13
Week 45:
10.7.1, 10.7.5, 10.7.6, 11.1.6.11.2.4
(Hauge).
Solutions set14
Week 46:
11.3.2, 11.3.4, 11.3.7, 11.3.8, 11.4.1, and 11.4.2 (Ellingsen).
Solutions set15
Week 47:
No set.
