MAIN POINTS
Pitchfork bifurcation is a third type of bifurcation that arises in problems with a symmetry. The supercritical pitchfork bifurcation has the form x'=rx-x^3. This equation has a single, central stable fixed point for all values of r ≤ 0, and three fixed points for positive r. In this case, the two symmetric fixed points on either side are stable, and the central fixed point is unstable. Plotting this on a bifurcation diagram shows a pitchfork shape. Example 3.4.2 shows how this idea conveys onto a diagram of Potential. As x increases past 0, two pits form on either side of the origin. The subcritical bifurcation has two symmetric FP's that are unstable and converge towards the origin as r is increased to zero from a negative value.
3.5 discusses the overdamped bead on a rotating hoop. They find a way to simplify the second order Newton equation to a first order.
CHALLENGES
The bead/hoop example was kind of impenetrable to me. I expect that we'll go over this in class. On p.59, the book says that the detailed analysis of (3) is left in exercises... perhaps it would be good to go over how the bifurcation diagram for that equation with the 5th degree term is derived.
REFLECTION
What are some examples of situations where the symmetry causes a pitcfork bifurcation? Are there any simple ones? Does it arise in chemistry or physics?
Monday, September 29, 2008
Sunday, September 21, 2008
tues sep 23
MAIN POINTS
Some fixed points must exist for all values of a given parameter— however, the nature of their stability may change as the parameter changes. Transcritical bifurcation is the model for this. This is given by the equation x'=rx-x^2. As r varies the parabola defined by x vs. x' shifts from the left to the right, but always crosses the axis at x=0. Thus, there is no disappearance of the fixed point as there is with saddle bifurcation.
The next chapter discusses an application— lasers. Energy excites a material, which emits light. After a certain threshold, the energy is released in phase as a laser. The text shows that the differential equation for the number of photons, n'=gain-loss, can be rewritten in a form similar to the transcritical bifurcation. As pump strength increases, the fixed point goes from being a stable fixed point at 0 to an unstable fixed point at 0. This means that the number of photons increases given any slight prodding. A new stable fixed point appears for some positive value of n. The point at which the origin fixed point becomes unstable is called the laser threshold.
CHALLENGES
Some practice drawing bifurcation diagrams would be nice... it seems like you just have to draw a vector field for many values of a parameter, and pretend that they're cross-sections of the bifurcation diagram.
Is there a graphical way to envision Fig. 3.2.1 on the computer?
REFLECTION
3.3 provided a way about thinking about physics that I never thought about. It really is a new dimension of physics when you can state behaviors in terms of their derivatives rather than explicitly...
Also, the emphasizes that first order systems are interesting because of the way we can change parameters... Wouldn't this be similar to simply adding another variable, and making the system second-order?
Some fixed points must exist for all values of a given parameter— however, the nature of their stability may change as the parameter changes. Transcritical bifurcation is the model for this. This is given by the equation x'=rx-x^2. As r varies the parabola defined by x vs. x' shifts from the left to the right, but always crosses the axis at x=0. Thus, there is no disappearance of the fixed point as there is with saddle bifurcation.
The next chapter discusses an application— lasers. Energy excites a material, which emits light. After a certain threshold, the energy is released in phase as a laser. The text shows that the differential equation for the number of photons, n'=gain-loss, can be rewritten in a form similar to the transcritical bifurcation. As pump strength increases, the fixed point goes from being a stable fixed point at 0 to an unstable fixed point at 0. This means that the number of photons increases given any slight prodding. A new stable fixed point appears for some positive value of n. The point at which the origin fixed point becomes unstable is called the laser threshold.
CHALLENGES
Some practice drawing bifurcation diagrams would be nice... it seems like you just have to draw a vector field for many values of a parameter, and pretend that they're cross-sections of the bifurcation diagram.
Is there a graphical way to envision Fig. 3.2.1 on the computer?
REFLECTION
3.3 provided a way about thinking about physics that I never thought about. It really is a new dimension of physics when you can state behaviors in terms of their derivatives rather than explicitly...
Also, the emphasizes that first order systems are interesting because of the way we can change parameters... Wouldn't this be similar to simply adding another variable, and making the system second-order?
Thursday, September 18, 2008
Thu Sep 18
Main Points
The book notes that 1-dimensional systems are interesting in the way that they change as parameters change. THese changes (fixed points, stability) are called bifurcations. The saddle-node bifurcation is a typical situation— a saddle shape can overlap the line, creating FP's, it can touch the line (for a half-stable FP), or it can be above the line. Parameters can change this. Several examples are shown where this behavior happens— parameters vary, creating fixed points "out of the blue sky." The book discusses "normal forms," but it doesn't go into much detail about how to use them, it only shows how simple quadratic functions are prototypes for saddle-node bifurcation.
CHALLENGES
The bifurcation diagram looks a little weird to me. Why are they drawn like that? What do they demonstrate? What does the Taylor expansion on page 50 prove?
REFLECTIONS
It's interesting that variability in FP's can be condensed down to thinking about saddle-shapes. Thinking about it, it is inevitable—if a single FP appears when a parameter changes, a second one has to show up, unless it's tangential to a saddle shape. I'm just curious about how this topic is relevant— what it can be used for.
The book notes that 1-dimensional systems are interesting in the way that they change as parameters change. THese changes (fixed points, stability) are called bifurcations. The saddle-node bifurcation is a typical situation— a saddle shape can overlap the line, creating FP's, it can touch the line (for a half-stable FP), or it can be above the line. Parameters can change this. Several examples are shown where this behavior happens— parameters vary, creating fixed points "out of the blue sky." The book discusses "normal forms," but it doesn't go into much detail about how to use them, it only shows how simple quadratic functions are prototypes for saddle-node bifurcation.
CHALLENGES
The bifurcation diagram looks a little weird to me. Why are they drawn like that? What do they demonstrate? What does the Taylor expansion on page 50 prove?
REFLECTIONS
It's interesting that variability in FP's can be condensed down to thinking about saddle-shapes. Thinking about it, it is inevitable—if a single FP appears when a parameter changes, a second one has to show up, unless it's tangential to a saddle shape. I'm just curious about how this topic is relevant— what it can be used for.
Monday, September 15, 2008
Tue Sep 16
MAIN POINTS
Dimensional analysis is based on the premise that physical quantities have dimensions, and physical laws are not changed by a change in units. Dimensions are multiplied along with their associated values, i.e mass, velocity, time, space. Each has an exponent, and when all of them have an exponent of 0, they are called 'dimensionless.' It can be seen immediately if an equation is possible by checking if the dimensions are compatible with each other. This is used to show how the dimensions of a pendulum work out. It is also demonstrated that you can infer what kind of dependence is necessary in an equation if you just know the unit of the answer.
Buckingham's theorem states that an equation is dimensionally homogeneous if it can be stated as a function of dimensionless products. Applying this yields a system of equations that can be used (using linear algebra, for instance), to set up an equation for the function.
CHALLENGES
I don't quite understand the concept of "dimensionally homogeneous," and how it is important that the laws of physics be this way. For instance, I'd like to know, if physics weren't dimensionally homogeneous, what kind of contradictions would we see?
I'd like to see another problem worked out too!
REFLECTION
In science classes we got used to the idea of treating units just like numbers, and multiplying/dividing/adding them so far as they were compatible. That's what dimensional analysis means to me— taking a known equation and seeing that the units cancel out. This approaches that problem from the other end.
Dimensional analysis is based on the premise that physical quantities have dimensions, and physical laws are not changed by a change in units. Dimensions are multiplied along with their associated values, i.e mass, velocity, time, space. Each has an exponent, and when all of them have an exponent of 0, they are called 'dimensionless.' It can be seen immediately if an equation is possible by checking if the dimensions are compatible with each other. This is used to show how the dimensions of a pendulum work out. It is also demonstrated that you can infer what kind of dependence is necessary in an equation if you just know the unit of the answer.
Buckingham's theorem states that an equation is dimensionally homogeneous if it can be stated as a function of dimensionless products. Applying this yields a system of equations that can be used (using linear algebra, for instance), to set up an equation for the function.
CHALLENGES
I don't quite understand the concept of "dimensionally homogeneous," and how it is important that the laws of physics be this way. For instance, I'd like to know, if physics weren't dimensionally homogeneous, what kind of contradictions would we see?
I'd like to see another problem worked out too!
REFLECTION
In science classes we got used to the idea of treating units just like numbers, and multiplying/dividing/adding them so far as they were compatible. That's what dimensional analysis means to me— taking a known equation and seeing that the units cancel out. This approaches that problem from the other end.
Monday, September 8, 2008
Thurs. 9/4
MAIN POINTS
Linear Stability Analysis is a way of examining stability without looking at the system graphically. The derivation uses a Taylor expansion to look at behavior around the point in question, which yields the first derivative term. The other terms are negligable. Thus, f'(x*0 determines te stability at that point.
Section 2.5 gives an overview of when systems may have unique solutions, or when solutions exist at all. A simple example with multiple solutions is dx/dt = x^(1/3). The text explains the Existence and Uniqueness Theorem, which requires that f(x) and f'(x) are continuous in an open interval. This doesn't mandate that solutions are unique over the whole domain, however.
Section 2.6 shows how oscillations are impossible in a first-order system. This is because overshoots can't occur— and obviously, the system is restricted to one dimension.
Section 2.7 I don't understand what the notion of Potential is for.
CHALLENGES
2.7 defines Potential, and it seems like it's just a pedagogical tool for explaining dynamics, but I don't understand how this view contrasts with how they previously explained it.
REFLECTIONS
The Taylor Expansion is used for the derivation of Linear Stability Analysis. Is this necessary? I haven't taken Analysis, but is a Taylor expansion generally regarded as a solid proof, whereas simply taking the derivative isn't?
Linear Stability Analysis is a way of examining stability without looking at the system graphically. The derivation uses a Taylor expansion to look at behavior around the point in question, which yields the first derivative term. The other terms are negligable. Thus, f'(x*0 determines te stability at that point.
Section 2.5 gives an overview of when systems may have unique solutions, or when solutions exist at all. A simple example with multiple solutions is dx/dt = x^(1/3). The text explains the Existence and Uniqueness Theorem, which requires that f(x) and f'(x) are continuous in an open interval. This doesn't mandate that solutions are unique over the whole domain, however.
Section 2.6 shows how oscillations are impossible in a first-order system. This is because overshoots can't occur— and obviously, the system is restricted to one dimension.
Section 2.7 I don't understand what the notion of Potential is for.
CHALLENGES
2.7 defines Potential, and it seems like it's just a pedagogical tool for explaining dynamics, but I don't understand how this view contrasts with how they previously explained it.
REFLECTIONS
The Taylor Expansion is used for the derivation of Linear Stability Analysis. Is this necessary? I haven't taken Analysis, but is a Taylor expansion generally regarded as a solid proof, whereas simply taking the derivative isn't?
Tues. 9/2
MAIN POINTS
The beginning of the first chapter serves as an overview of the history of chaos/dynamics. Then it turns to some examples of differential equations— an oscillator, the heat equation, and then defines the generalized case of a differential equation. The text introduces dummy variables to put equations in standard form, and defines the difference between linear and nonlinear systems (whether the x variables on the right hand side are all in the first power or not). The text defines autonomous vs. nonautonomous systems (systems with the time variable explicit or not), the trajectory, and phase space.
Ch. 2 defines a first-order system as a function of one variable. It gives the example of dx/dt=sinx, which it explains graphically. Stability is defined as whether or not a small peturbation will be rectified, or if it will cause a drastic change in trajectory in the movement of the peturbation. It gives a number of examples of systems, particularly population growth, which uses a logistic equation.
CHALLENGES:
I'm a little unclear on the precise definition of a logistic equation— and how the limitations are incorporated into the system.
REFLECTION:
We did a quick exercise on Chaos in Scientific Computation. I also read the book by Gleick at some point in High School. As for the differential equations, I have little experience working with them apart from using the Runge-Kutte method, etc.
The beginning of the first chapter serves as an overview of the history of chaos/dynamics. Then it turns to some examples of differential equations— an oscillator, the heat equation, and then defines the generalized case of a differential equation. The text introduces dummy variables to put equations in standard form, and defines the difference between linear and nonlinear systems (whether the x variables on the right hand side are all in the first power or not). The text defines autonomous vs. nonautonomous systems (systems with the time variable explicit or not), the trajectory, and phase space.
Ch. 2 defines a first-order system as a function of one variable. It gives the example of dx/dt=sinx, which it explains graphically. Stability is defined as whether or not a small peturbation will be rectified, or if it will cause a drastic change in trajectory in the movement of the peturbation. It gives a number of examples of systems, particularly population growth, which uses a logistic equation.
CHALLENGES:
I'm a little unclear on the precise definition of a logistic equation— and how the limitations are incorporated into the system.
REFLECTION:
We did a quick exercise on Chaos in Scientific Computation. I also read the book by Gleick at some point in High School. As for the differential equations, I have little experience working with them apart from using the Runge-Kutte method, etc.
Thursday, September 4, 2008
First Post
Name: Casey B
Year: Senior
Majors: Math/Computer Science
Math Classes: Discrete Math, Multivariable Calculus, Scientific Computation, Theory of Computation, Abstract Algebra, Graph Theory. Taking: Discrete Applied Math, Linear Algebra, Statistical Modeling
Weakest Part: Probability and Statistics, haven't taken a real course in Linear Algebra
Strongest Part: Discrete + Graph Theory, Calculus (but it's been a while)
Why I'm Taking the Course: Want a more solid background in continuous math, Diff Eqs seem like a lot of fun.
What I want out of it: Becoming very comfortable with the notion of DE's, including terminology and the various methods of solving them.
Interests: Electronic Music / Analog synthesizers, Computer Science, Food, Birds, WMCN
Worst Math Teacher Experience: Too easy, didn't assign enough problem sets to solidify knowledge, didn't energize the class, taught very slowly.
Best Math Teacher Experience: Many problem sets, challenging material, readily available out of class, brought food to class :), had some supplementary handouts but didn't rely on them.
Year: Senior
Majors: Math/Computer Science
Math Classes: Discrete Math, Multivariable Calculus, Scientific Computation, Theory of Computation, Abstract Algebra, Graph Theory. Taking: Discrete Applied Math, Linear Algebra, Statistical Modeling
Weakest Part: Probability and Statistics, haven't taken a real course in Linear Algebra
Strongest Part: Discrete + Graph Theory, Calculus (but it's been a while)
Why I'm Taking the Course: Want a more solid background in continuous math, Diff Eqs seem like a lot of fun.
What I want out of it: Becoming very comfortable with the notion of DE's, including terminology and the various methods of solving them.
Interests: Electronic Music / Analog synthesizers, Computer Science, Food, Birds, WMCN
Worst Math Teacher Experience: Too easy, didn't assign enough problem sets to solidify knowledge, didn't energize the class, taught very slowly.
Best Math Teacher Experience: Many problem sets, challenging material, readily available out of class, brought food to class :), had some supplementary handouts but didn't rely on them.
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