8.0-8.2

Main Results

This section introduces bifurcations in two-dimensional systems.  It turns out that nothing really new happens in 2D.  All the action is confined to a one-dimensional subspace along which the bifurcations occur while in the extra dimensions the flow is either simple attraction or repulsion.  The situation in which two complex conjugate eigenvalues simultaneously cross the imaginary axis is a Hopf bifurcation, of which they are supercritical cases(decay slows down and eventually becomes growth), subcritical cases (when an unstable cycle shrinks and engulfs the origin, and degenerate cases (where the sign of the parameter affects the stability of the fixed point).

Challenges

I am a little unclear on how to determine which type of bifurcation occurs after determining that a system has a Hopf bifurcation.

Relfections

Bifurcations made a lot of sense in one-dimension, so I'm confident about studying them in 2-D.

6.5-6.6

Main Results

The first concept introduced is conservative systems.  Definition: a conserved quantity is a real-valued continuous function E(x) that is constant on trajectories and E(x) must be nonconstant on every open set.  A conservative system cannot have attracting fixed points.  A trajectory that starts and ends at the same fixed point is called a homoclinic orbit, and are usually only found on conservative systems.  Next Strogatz discusses reversible systems, which are defined as a system that has time-reversal symmetry.  If xdot = f(x,y) and ydot = g(x,y) , then a reversible system is one in which f is odd in y and g is even in y.  In reversible systems if the origin is a center, then all trajectories sufficiently close to the origin are closed curves.  Centers usually are fragile, but in reversible and conservative systems they are more robust.

Challenges

In the proof that all trajectories close to the origin are closed orbits in reversible systems, it seemed like in the first step the trajectory would create a closed orbit with using reversibility because it was stated that the flow swirls around the origin.

Reflections

Conservative systems made a lot of sense and they coincided with the small amount of physics I remember from high school.  Reversible systems seem manageable, but the concept was definitely more shaky for me.

6.0-6.3

Main Results

Nonlinear two-dimensional systems are introduced in this reading.  In general there is no way to study them analytically, so their qualitative behavior is determined.  The phase portraits are entirely filled with trajectories and there is a enormous varieties of phase portraits that arise.  Vector fields become cluttered for nonlinear systems, so direction fields are used, which are usually made by computer.  When quantitive aspects are needed, the Runge-Kutta method is used.  Trajectories never intersect due to the uniqueness theorem.  Linearization can be used to approximate the phase portrait near a fixed and the Jacobian matrix of a system is used to to analyze the dynamics of fixed points and phase portraits.  

Challenges

Example 6.3.2 moved a little fast for me.  It jumped through a lot of steps because we either did them earlier in the book, or we will do them in the exercises.  I wish it would have walked through the linearization process more in depth.

Reflection

I like the phase portraits that get generated by nonlinear systems.  It is clear how they arise in nature and have scientific applications.  Its also exciting that we are finally into nonlinear systems.

5.0-5.3

Main Results

This section introduced linear systems.  The solutions of a linear systems are trajectories moving in the (x,y) plane, which is called the phase plane in this context.  The graph of the trajectories forms the phase portrait of a system.  In these systems there are several types of fixed points.  When the trajectories approach the fixed point it is labeled a stable node or a star.  When there are infinite fixed points along a line it is called a line of fixed points.  When the solutions are asymptotic away from the fixed point it is a saddle point, but there is a line that is stable called the stable manifold, and conversely there is an unstable manifold.  You can analyze and classify linear systems by drawing its trajectories and manifolds.

Challenges

I wasn't sure how to graph the phase portrait of the system with complex eigenvalues(5.2.4 I believe).  I tried to convert it into terms of sin and cos, and I think it is a clockwise spiral that grows away from the origin, but do I just draw an arbitrary spiral with these characteristics, or is there a more specific way?

Reflections

Love affairs.  How cool is that application of linear systems?  Determining how Romeo and Juliet will end up feeling about each other is definitely one of the most interesting things so far in the book.  I like this chapter overall, too.

Linear Systems Reading

Main Results

These section were trying to find a solution for y' = Ay, where A is a matrix made up of constants.  For a system of dimension n there will be a set of n linearly independent solutions.  In dealing with these solutions, the idea of an eigenvalue was introduced.  w is an eigenvalue of A if there is a nonzero vector v such that Av = wv.  v is called an eigenvector.  The eigenvalues of A are the roots of its characteristic polynomial p(w) = det(A - wI), where I is the identity matrix.  The next section moves into 2-D systems are theorems for the general solutions to the systems are given.

Challenges

Many.  From the beginning I was lost as to why we were dealing with the equation y' = Ay, with A being a matrix.  Then in the second section with the complicated general solutions I was even more lost.

Reflections

It was interesting material but I think I got a little scared off by the eigenvalues and eigenvectors since I lack linear algebra experience.

Fireflies!

Main Results

It was discovered that some fireflies are able to adjust their flashing so as to synchronize with the flashing of those around them.  They will also respond to artificial flashing.  The dynamics of the phase difference between the firefly's frequency and the stimuli's frequency can be expressed with the nonuniform oscillator equation : pdot = q - w - Asin(p), where q is the frequency of the stimulus and w is the frequency of the firefly and A measures the firefly's ability to modify and is >0.  Analyzing this equation shows that the firefly can only match the stimulus, or become entrained, for a certain range of entrainment.

Challenges

I didn't understand equation 6 in section 4.5 because I thought the phase difference during entrainment was zero, because the firefly is entrained to flash at the same time.  But if it's zero, why do we need a model to predict the phase difference.

Reflections

This sure is a cool biological application of what we're studying.  I really want to see one of those videos of fireflies synchronized.

4.0-4.3

Main Results

This reading introduced the idea of flows on the circle.  The circle is one-dimensional, like the line, but now periodic solutions are possible.  If we let o=theta, the equation looks like odot=f(o) where o is a point on the circle and odot is its velocity.  Vector fields are sketched in a similar way.  If odot=w, where w is a constant then there is uniform motion around the circle and the solution is periodic with the period, T, equal to 2pi/w.  A nonuniform oscillator like odot=w-a(sin(o)) has a saddle-node bifurcation and a "bottleneck" near o=pi/2.  A bottleneck is an area where it takes the phase point most of its time to pass through.  In fact, when calculating periods, we can approximate the entire time as just the time spent in the bottleneck.

Challenges

Ghosts.  The idea seems cool, but I don't quite follow why ghosts exist and how you determine the time spent in a bottleneck because of a ghost.

Reflections

I like the fact that equations like odot=sin(o) no longer have infinite fixed points, which I found to be a hassle.  But so far it is slightly harder to visualize flows and vector fields in a circular manor.