5A1352 Complex Systems - Log 2004

My lecture notes follow approximately the same order as this log.
Additional references are to sections or pages in Hilborn's book.

Lecture  1 
Introduction. Mathematical models, dynamical systems (DS). Applications to
many different fields of science and technology.  Structure of DSs - phase
space and deterministic dynamics defined by systems of ODEs. 
Initial values, orbits. Autonomous and driven (non-autonomous) systems,
control parameters.  Example of a linear system - the simple harmonic
oscillator - integration by matrix exponential. 
Iterative solution of ODEs by Picard method, Lipschitz conditions for
uniqueness of solutions.  Non-crossing rule for autonomous systems.
Flows, orbits as flow lines with the vector field as tangent field. 
Discretization of time varaible, iterated maps. 
Constants of motion.   
Example - the planar pendulum, its energy as constant of motion. 
Transformation to dimensionless variables and parameters.
The periodic orbits and the limiting aperiodic separatrix solution. Stable
and unstable fixed points. Time reversed orbits.
Read Chapter 1 of the lecture notes.
You can read Hilborn Chapter 1 as an introduction, then 3.1 - 3.8. 

Lecture  2
Definition of fixed points (FPs) and cycles (periodic solutions). 
Analysis of stability of fixed points. Verhulst model, an example in 1-D.
Dimensionless variables. Lotka-Volterra model an example in 2-D
of a model with a constant of motion and periodic orbits. 
Glycolytic oscillator as a 2-D example of a limit cycle (LC). 
Analysis  by linear approximation around the fixed point. The Jacobian
matrix and its eigenvalues decide the type of stability/instability.  A
repetition of some linear algebra, characteristic equation, the  roots
(eigenvalues) and eigenvectors. Similarity transformations, diagonal
and Jordan normal forms. The existence of n linearly independent solutions
of the linear system in n dimensions. 
Read: 2.1 - 3.3 of the lecture notes,  Hilborn 3.10 - 3.15 

Lecture  3
Stability of of FPs in the linear approximation. The Hartman-Grobman 
theorem. The different stability types in 2-D: stable and  unstable nodes
and spiral nodes, saddles, and centers.  Conservative systems (eg
pendulum) in 2-D can only have saddles and centers. The Poincare index. 
Stability for iterated maps, again from eigenvalues of Jacobian. Limit
cycles in the example of the van der Pol oscillator.
Read: 3.4 - 4.4 of the lecture notes.  Hilborn 3.15 - 3.18, 
Appendix I about the van der Pol oscillator

Lecture  4
The Poincare map for the van der Pol oscillator and the stability of the
LCs. Poincare-Bendixson theorem for finding limit cycles in 2-D. 
Application to the glycolytic oscillator - finding a trapping region.
Saddle connections, homoclinic and heteroclinic cycles.
Bifurcation theory for  for vector fields in R^2. The simplest types of
bifurcations - saddle-node, transcritical, pitchfork bifurcations.
Andronov-Hopf bifurcations of limit cycles, application to the
glycolytic oscillator.  Global bifurcations through saddle connections. 
Read rest of Chapter 4 and Chapter 5 of lecture notes.  Hilborn, start
on Ch 4, Appendix B. You had  better calculate some examples on you
computer. 

Lecture  5
Differentiable manifolds, the Stable Manifold theorem. Local  and global
stable and unstable manifolds. A stable manifold cannot  intersect itself
or a stable manifold belonging to another FP, the same holds for unstable
manifolds. A stable manifold can intersect an unstable one, from the same
FP (homoclinic) or other FP (heteroclinic intersection). The FPs of the
Lorenz model and some of their bifurcations. A sketch of what happens when
the unstable manifold of the origin returns to  intersect the stable
manifold of the origin. Showed some pictures  of the strange deformations
of the stable manifold.
Read 6.1-6.3 of lecture notes.  Hilborn, continue on Ch 4.  You had 
better calculate some examples on your computer. 

Lecture  6
Distributed the last set of lecture notes - available on the home page.
Distributed some copies of figures shown last time - not on the web. 
The construction of a Poincare section and the Poincare iterated map
as a method of finding periodic orbits (cycles) in a flow.
A  discussion of the Rössler model as an example of a system with
stable cycles, and period-doubling pitchfork bifurcations of cycles as you
change a parameter. There also appears to be aperiodic motion for some
parameter values. Run the Matlab file rossler.m!
The homoclinic and heteroclinic tangles created by  the intersection of
stable and unstable manifolds for iterated maps (incl the Poincare map).
The  Henon iterated map is an important example of a 2-D iterated map
where there  is an apparent strange attractor. 
Play around with the Matlab filed henon1.m.
Read rest of Chapter 6 of lecture notes. Hilborn, rest of Chapter 4
and 5.1-2.

Lecture  7
The basic ideas of chaos. Sensitive dependence on initial values and 
possible exponential divergence of nearby orbits. There are 'regular'
systems with at most a slow divergence of orbits, like the pendulum. The
definition of the maximal Liapunov exponent, as a measure of the
exponential divergence. The Arnold cat map as an example of a system 
where the Liapunov exponent is well defined, positive, and calculable. The
idea of a sequence of symbols as a characteristic of chaos - the  random
sequence of heads and tails created by the flipping of a coin. This is a
Bernoulli process, the most random of all. 
The Smale horseshoe map and its properties.  Construction of a  Cantor set
invariant under the dynamics. The equivalence between  the geometrically
defined horseshoe and a symbolic dynamics defined on all doubly infinite
sequences of two symbols.  This is a hyperbolic limit set where action of
the map is defined by two multipliers, there is then an exponential
sensitivity  on the initial values. The existence of periodic, aperiodic,
and dense orbits. There exist horseshoe maps  near homoclinic
intersections. 
Read Chapter 7 of lecture notes

Lecture  8
Some aspects of 'chaos' - positive Liapunov exponent - symbolic dynamics -
stretching-folding action in phase space. Iterated maps on the unit
interval. The tent map as an extreme example of chaos, how to find FPs and
cycles, all unstable.  The cycle points are dense in the interval.
Exponential instability, the Lyapunov exponent can be calculated.
Generating a symbolic sequence from the map.  Invariant measure.  Symbolic
dynamics as random a 'coin-flip process'. 
The logistic map, also a 2 --> 1 map. The period-doubling bifurcation
cascade of stable cycles. Superstable cycles in each stability interval.
Read the beginning of Chapter 8 of lecture notes, Chapter 5 of Hilborn.
.
Lecture  9. 
Distributed the 2nd set of homework problems. 
Logistic map continues. How new primary cycles are created by  saddle-node
(tangent) bifurcations in pairs of stable and unstable cycles. Thy form
the starting point of new period-dubling bifurcation cascades. Just before
such a tangent bifurcation there is intermittency. Invariant intervals and
reverse bifurcations.  Periodic chaos. The logistic map at lambda = 4 is
conjugate to the tent map and has same properties of exponential
instability. An introduction to Feigenbaum universality, the unversal
numbers alpha and delta, and Feigenbaums universal function. 
Read Chapter 8 of lecture notes,Chapter 2 and Appendix F of Hilborn.

Lecture  10
Forced (non-autonomous) dynamical systems, periodic forcing.  Mathieu's
equation as an example of a periodically forced linear  system. Floquet's
theorem. Stable and unstable solutions. Stability regions in parameter
space. An example: the 2-D quadrupole ion trap.
Forced nonlinear systems and mode locking. Some history - Huygens and 
frequency locking of clocks, van der Pol and forced oscillators as models
of heart arrhythmia. Applications in electronics, biorhythms, musical
instruments and noise control, planetary dynamics. A forced system
in n-D is equivalent to an autonomous systems in dimension n+1. 
Example of forced pendulum with friction. Harmonic, subharmonic,  
frequency-locked, and quasiperiodic solutions. The winding number.
The return plot and the return map.  
Read Chapter 9 of lecture notes, Chapter 6 of Hilborn

Lecture  11
The sine-circle map as an iterated map model of a periodically driven
pendulum. Fixed points and cycles and their relation to the frequency 
locked orbits. The Devil's staircase and Arnold's tongues. Bifurcation
diagrams and Lyapunov exponents. The quasiperiodic  orbits, and possible
chaotic solutions in certain regions of parameter space. 
Run the Matlab files devils.m and  circlebif.m
for some different choices of the parameters.
Hamiltonian dynamics. Coordinates and momenta. The canonical equations
of motion. Conservation of the energy and the phase space volume.
Other constants of motion. Completely integrable systems.
Read Chapter  10 in lecture notes, Chapters 6 and 8 in Hilborn.