Nonlinear dynamics and chaos by Steven Strogatz (Westview Press).
Paperback can be purchased at Tapir bookstore (426 NOK).
Dictionary Norwegian-English
Lectures:
Tuesdays 10.15-12.00 in R93 and
Thursdays 08.15-10.00 in E5-103. First lecture Tuesday August 19. Last
lecture Thursday November 20. Exercises will be treated in class
when it fits.
Office hours:
Wedensday 11.00-1200 (E5-145).
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:
1) Numerical exercise. Exercises from textbook.
2) From textbook: 2.0-2.4, 2.6-2.7, 3.0-3.2, 3.4-3.6
(3.5: untill dimensional analysis and scaling), 4.0-4.4, 5.0-5.2,
6.0-6.8 (not cylindrical phase space in 6.7), 7.0-7.3. 8.0-8.3,
8.5 (only existence of a closed orbit for driven pendulum with damping).
10.0-10.7 (only p373 from 10.6. To middle of p384 in 10.7). 11.0-11.4.
12.1.
The students will be assigned one exercise set each to work out on the
blackboard (30-45 min duration).
In addition, a numerical exercise must be solved and a report
handed in. These assignments must be completed in order to take
the exam. The report must be handed in no later than Friday November 7.
(electronically to fredrkro@stud.ntnu.no
or in my mail box).
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 greatful.
Week 34: Linear versus nonlinear systems. 2nd order equation as a coupled
system of two first-order equations (damped oscillator).
Pendulum. Fixed points and their
stability. Graphical techniques. One-dimensional flow and phase portraits.
Logistic model for growth. Linear stability analysis.
Impossibilty of oscillations and mechanical analogy (overdamped systems).
Potentials V(x), stable and unstable fixed as min/max of V(x)-
Week 35:
Saddle-node bifurcation (collision and annihilation of fixed point).
Trancritical bifurcation (exchange of stability), and sub/supercritical
pitchfork
bifurcation (birth of pair of fixed point+change of stability of origin).
Normal form of bifurcations.Potential V(x) depending on parameter r.
Overdamped bead on a rotating hoop. Imperfect bifurcation and imperfection
parameter h.
Week 36:
Number of fixed points depending on sign of r and critical function h_c(r).
Division of hr-plane into regions with different number of fixed points.
Fixed points as functions of r for fixed h.
Flow on the line and periodicity of f(theta).
Uniform and nonuniform oscillator. Fixed points, ghost, and
period T(w/a).Overdamped pendulum. Two-dimensional flow. Harmonic
oscillator. Example 5.1.2 and different types of fixed points:
nodes, stars, non-isolated fixed points, and saddle points.
Week 37:
Tutorials on Tuesday.
Week 38:
(Globally) attracting, unstable, and
Liapunov stable (neutrally stable) fixed points.
Stable and unstable manifold, straight-line solutions.
Classification of fixed points: stable and unstable nodes, stable and
unstable spirals, saddles, borderline cases: star nodes, degenerate
nodes, centers, and nonisolated fixed points.
Nonlinear systems and linearization. Jacobian matrix. Nullclines and
fixed points.
Uniqueness of solutions
611phase.
Linearization works for saddles, nodes, and
spirals, but not for stars, centers, degenerate nodes, and nonisolated
fixed points. Lotka-Volterra (LV) model for sheep and rabbits.
Week 39:
Fixed points for the LV model. Basin of attraction and
principle of mutual exclusion.
Conservative systems, linearization and centers.
Reversible systems, linearization and centers.
667phase.
No lectures on Thursday. Tutorials on Thursday (Krohg).
Week 40:
Fixed points and phase portrait of pendulum with and without damping.
Index theory: Index of a curve and index of a fixed point.
Properties of the index. Index theory to rule out closed curves.
Limit cycles as isolated closed curves. Van der Pol oscillator:
fixed point and change of stability+existence of closed orbit for mu>0.
Gradient systems and non-existence of closed trajectories.
Week 41:
Presentation of numerical exercise Tuesday (Krohg).
Numerical assignment.
(Ignore times and dates in document:)).
Liapunov functions and nonexistence of closed orbits.
Dulac's criterion for nonexistence of closed orbits.
Week 42:
Poincare-Bendixon theorem. Trapping region (exclude unstable fixed points).
Saddle-node bifurcations and ghosts.
Transcritical and pitchfork bifurcations.
Supercritical Hopf bifurcations: A fixed becomes unstable and a stable
limit cycle is born.
Week 43:
Subcritical Hopf bifurcations. Degenerate Hopf bifurcations.
Infinite-period bifurcations. Driven pendulum with damping. Fixed points
and limit cycles. Poincare maps and existence of fixed points (implies
limit cycle). One-dimensional maps and chaos. Fixed points and their
stability. Cobwebs. The logistic map: fixed points and their stability.
Transcritical bifurcations.
Birth of periodic cycles via flip bifurcations (supercritical).
Periodic windows and chaos. Universal numbers.
Week 44: Logistic map: fixed points and p-cycles. Stability and
flip bifurcations. Birth of period-3 cycles and periodic windows.
Liapunov exponents and chaos. Liapunov exponents for stable p-cycles.
Week 45:
Tent map and Liapunov exponent. Unimodal maps and universal numbers.
Renormalization (iterates and scaling). Universal functions.
Fractals: structure and self-similarity (Cauliflower=Bloemkool=Blumenkohl).
Countable and uncountable sets. Cantor set as a fractal.
Fractal food.
Week 46:
More properties of fractals (totally disconnected and no isolated points).
Similarity - and box dimensions as generalizations of ideas from smooth sets.
Fractals that are not self-similar.
Problem 11.4.6. Baker's map, fixed points, periodic cycles, and
strange attractors. Problem 12.1.5.
Lozi map, fixed points and their stability.
Week 47:
Tutorial on Tuesday in E5-103. No lecture on Thursday.
