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.

3.4-3.5

Main Results

This reading introduced pitchfork bifurcations.  Instead of going from zero to two fixed points as a parameter is varied (like S.N. and transcritical), pitchfork bifurcations go from one fixed point to three as a parameter is varied.  The name pitchfork is fitting because the bifurcation diagram of these equations look exactly like a pitchfork.  There are two different kinds of pitchfork diagrams: supercritical and subcritical.  Supercritical pitchfork bifurcations have a cubic term that stabilizes the dependent variable whereas the cubic term is subcritical bifurcations is destabilizing.  The bifurcation diagram of subcritical pitchfork equation is just an inverted version of the supercritical one.

Challenges

The section on phase planes went over my head for the most part.  I did not understand the idea of phase fluids and I also didn't follow how the final trajectories were derived.

Reflection

The reading started out really on a good note and I really understood the pitchfork bifurcations but by the end I was frustrated because I had no grasp on the ideas that were popping up like rapid transient and singular limits.

Transcritical Bifurcations

Main Results

A transcritical bifurcation is one in which the fixed points never disappear (as was the case with saddle-node) but instead they change their stability.  For example, if xdot=rx-x^2, then for r<0>0, the fixed point at the origin is unstable and the one at x*=r is now stable.  An exchange of stabilities has occured.

Challenges

I had no trouble understanding the idea of a transcritical bifurcations but example 3.2.1 lost me when it was showing that the system undergoes a transcritical bifurcation.

Reflections

I noticed that the bifurcation diagram for the normal form of a transcritical bifurcation is linear, and I was wondering if all diagrams of transcritical bifurcations were linear.  Just a thought.  Thanks for the help during office hours today.

3.0-3.1

Main Results

The topic in these sections is bifurcations.  Bifurcations are points where vector fields make a qualitative shift.  This shift occurs due to a change in the value of a parameter.  For example, if you have the function r+x^2, then when r<0, r="0">0 you have zero fixed points.  Hence, for different values of the parameter, r, the vector fields are qualitatively different.  In this example the bifurcation occurred at r=0 because that is the transition point.

Challenges

I do not have any questions right now because the prototypical saddle-node bifurcations are pretty straightforward, but I'm sure I will have challenges as the bifurcations and applications of the bifurcations become more complex.

Reflection

I like how the author presented bifurcations, and I really feel like I understood everything he put forth, but I don't quite see the applications of bifurcations, because it just seems like a new name for something I knew or at least could have figured out before.

Dimensional Analysis

Main Results

The idea of dimensional analysis is a way to form a model of something when you know very little about it.  You determine all of the factors that influence a system, determine the the product of variables that make up these factors (a combination of M (mass), L (length), and T (time)).  Then we make these variables dimensionless by setting the exponents equal to zero, and a general model is then derived.

Challenges

So far I definitely have trouble remembering all of the steps that go into dimensional analysis.  The spot that specifically slows me down is on page 298 of the Weir and Fox book (example 1) when you go from equation 8.3 to 8.4.  I don't always remember how to set up the exponents.

Reflections

I am amazed how this technique gives accurate general laws so easily and with so little information.  From a physical sciences stand point these laws seem much more intricate and harder to derive.

2.8

Main Results

The main idea of this section is numerical approximations of integrals using computers.  The first technique presented is Euler's method.  This method entails calculation x and the velocity of x multiple times over very small steps.  This is the simplest form of numerical integration, and is the least accurate.  The methods increase in order up to the fourth-order Runge-Kutta method which is a good balance of accuracy and non-tediousness.  This method involves finding 4 values and then x(n+1) is a number based on these values.  Avoiding excessively small step-sizes is important because computers round-off at every calculation.

Challenges

My main question from this section is: how do you get a computer to do this for you?  The book mentions writing your own numerical integration routine but I'm sure there is a program already available.  Are we using numerical integration to check answers, or to find solutions when other methods don't work?

Reflections

I like the idea of numerical integration because it brings back fond memories of checking my answers in Calculus using the fnInt function on my calculator.  I have a definite interest in computer science, also, so I am looking forward to class tomorrow and to working on this section.

2.4-2.7

Main Results

The idea in these sections is to provide techniques that compensate for the geometric and qualitative approach that the book has taken so far.  The use of linearization about x* gives a quantitative method for determining if a point is stable or not.  Determining uniqueness addresses the issue that sometimes an equation appears to a unique solution when graphed, but in fact there can be multiple (even infinite) solutions.  Potentials is another great way to visualize dynamics because you simply have to picture how a ball would act if it were placed on the graph of the potential.

Challenges

I had trouble understanding the Existence and Uniqueness Theorem.  Example 2.5.1 was clear to me but the wording of the theorem itself went over my head for the most part.

Reflections

The most interesting part of this reading was the idea of potentials.  This concept really makes sense to me for some reason and I'm excited to use it.  Overall, I am enjoying the writing style and approach of the author.  As he said himself, the book has an informal style and I like it.

1.0-2.3

Main Results:

One of the main ideas in the reading was that, even in a diff eq can be solved algebraically, sometimes it is better to analyze it graphically.  The technique that comes out of this idea is interpreting a diff eq's as vector fields.  If you plot the diff eq on a graph and then put arrows along the x axis representing the velocity (arrow points to the right if velocity if positive; to the left if it's negative), it becomes easy to see what happens from different initial conditions as t approaches infinity.

Challenges:

In 2.2.2 and 2.3 I didn't quite follow how the author plotted the graphs from equations with several variables.  Perhaps I need to review my physics equations and perhaps it's not crucial that I know how he derived the graph but he seemed to jump straight from the equation to the graph, and I got lost.

Reflections:

Overall in this reading I liked how the techniques made complicated problems simple.  Specifically, in example 2.2.3 instead of trying to graph f(x)=x-cosx, the author simply broke it into 2 seperate, easily-graphable parts, and analyzed it from there.  I'm used to doing ugly problems the hard way, so this chapter looks like a nice alternative.

First Post

My name is Mike Snavely.  I am a freshman at Mac and I'm planning on a Major in Biology with at least a minor in Math.  I have taken Calc 1 and Calc 2 and Multivariable Calc (at the University of Minnesota).  A weak spot in my math is geometry and I also have not yet taken linear algebra.  The strongest part of my math is calculus; I took Calc 2 twice because of IB regulations.  I am taking this class mostly because I love math and I wouldn't want to go a semester without it.  Also, I very well may major or minor in Math.  What I hope to get out of this class is a broader math background and some experience with applied math.

My interests outside of school are almost 100% sports.  I am on the Mac football team and I am going to be on the club hockey team in the winter.  Then in my free time I'm usually playing basketball or disc or ping-pong.

The worst Math teacher I ever had was the teacher I had during my senior year in high school.  He would just lecture in a monotone voice for the first part of class and then just go sit at his desk, while we were expected to work.  If anyone talked or took out a bag of chips, or did anything that kids normally do in class he would yell and scream about how we were disrespectful.  And by the end of the year kids had stopped trying to ask him questions on the homework because he just didn't care.

The best math teacher I had was my Calc 2 teacher during my sophomore year in high school.  His genuine love of math rubbed off on everyone and he was one of the funniest people I've ever met.  In the middle of one of his interactive lectures on Taylor and Maclaurin series, he made up a limerick of the top of head about Taylor and Maclaurin traveling together.  It was just a relaxed atmosphere.  On days when we all felt burnt out on math he would take out his guitar and play songs he had wrote, and one time during the winter we spent the entire day learning how to make huge, ornate paper snowflakes.  I'll never forget that class.