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.