MAIN POINTS
A limit cycle is a closed trajectory surrounded by non-closed trajectories. It can be stable (attracting) or unstable (or even half-stable....?). They cannot exist in linear systems because they cannot be isolated (the trajectory of a linear system can be multiplied inwards and outwards). 7.1 gives examples. The first example shows all trajectories attracted to a unit circle trajectory. The second example shows a trajectory that is not a circle. 7.2 shows that closed orbits are impossible in gradient systems. This is used to show that a given system has no closed orbit. One way this is done is to give an integral for the energy of some closed periodic solution, and show that it is nonzero, reaching a contradiction. Also, Liapunov functions (energy-like function that decreases along trajectoris) can be used to show there is no periodic solution. Otherwise, Dulac's criterion (finding a function that, multiplied by a gradient, has one sign), can be used.
CHALLENGES
The first two chapters were quite short and low on math. Thus, I consider the third:
In example 7.2.2, they use an energy function... when will we not be able to consider a function like this? When it's impossible to show that the integral is nonzero? Is there a typical case when we would use Liapunov functions? Is there a rule to see which method we should use to show that a system has no isolated, closed orbit?
REFLECTIONS
He says that constructing Liapunov functions usually require "divine inspiration" to construct. If we are not prone to divine inspiration, should we still remember their use in this respect?
Monday, November 17, 2008
Monday, November 10, 2008
Tue 11/11
MAIN POINTS
6.5 defines Newton's law, then defines potential energy within the motion equation to show that the total energy is conserved. This makes it a conservative system. A conservative system, by definition, cannot have attracting fixed points. Ex. 6.5.3 shows the energy surface, which seems to be a convenient way of generalizing the potential diagram to 3 dimensions.
Reversible systems work the same when time is increased as when it is decreased... more precisely, thy are invariant as t -> -t and y -> -y. Example 6.6.1 is crazy looking. We are given more terminology with names about trajectories and saddles (could we possibly get a handout that distinguishes all of the different trajectory/saddle terms we've encountered so far?)
CHALLENGES
I don't understand the concept of a the homoclinic orbit... What does the text mean when it says 'centers are more robust when the system is conservative"? (p 163).
On p.165 they say curves "sufficiently close to the origin" are closed. I understand the logic based on symmetry... but is there a closed curve for every reversible system? Do you never see completely divergent behavior?
REFLECTION
Where do reversible systems show up in the real world? Potential energy/velocity are the main example of conservative systems... in this idealized model, does this mean a trajectory will always be moving, because no attracting fixed points are allowed? So.. a marbe would always be rolling around the energy surface...
6.5 defines Newton's law, then defines potential energy within the motion equation to show that the total energy is conserved. This makes it a conservative system. A conservative system, by definition, cannot have attracting fixed points. Ex. 6.5.3 shows the energy surface, which seems to be a convenient way of generalizing the potential diagram to 3 dimensions.
Reversible systems work the same when time is increased as when it is decreased... more precisely, thy are invariant as t -> -t and y -> -y. Example 6.6.1 is crazy looking. We are given more terminology with names about trajectories and saddles (could we possibly get a handout that distinguishes all of the different trajectory/saddle terms we've encountered so far?)
CHALLENGES
I don't understand the concept of a the homoclinic orbit... What does the text mean when it says 'centers are more robust when the system is conservative"? (p 163).
On p.165 they say curves "sufficiently close to the origin" are closed. I understand the logic based on symmetry... but is there a closed curve for every reversible system? Do you never see completely divergent behavior?
REFLECTION
Where do reversible systems show up in the real world? Potential energy/velocity are the main example of conservative systems... in this idealized model, does this mean a trajectory will always be moving, because no attracting fixed points are allowed? So.. a marbe would always be rolling around the energy surface...
Wednesday, November 5, 2008
Thu 11/6
MAIN POINTS
6.0 states that we will begin studying nonlinear systems. (That was fast...) 6.1 defines a vector field on a plane, where each point on the plane has a velocity vector. A phase point traces out a solution on the vector plane. Because analytically finding solutions is so difficult, we focus on qualitative behavior of nonlinear systems, i.e fixed points, closed orbits, stability. The chapter introduces Runge Kutte. 6.2 gives a theorem for the existence of solutions, which are guaranteed it f is continuously differentiable. DIfferent trajectoris do not intersect. There is also a theorem that a trajectory confined to a region without a fixed point will eventually reach a closed orbit. 6.3 looks at the stability near a fixed point by substituting a disturbance into the Taylor expansion to find the Jacobian matrix, which is similar to f'(x) but for many variables. The text defines various sorts of behavior— repellers, attractors, saddles, centers, and fixed points.
CHALLENGES
I would like to practice making and using a Jacobian matrix. Substitution of polar coordinates (used on p. 153) is something I need to brush up on, and going through the Runge Kutte would be nice, too! It's something I didn't really get in Scientific Computation. Also... an explanation of topological equivalence..
REFLECTION
So yeah, we did Runge Kutte in Comp 365. We also used Jacobian matrix then, but I was never sure of the definition of a Jacobian. I see the Taylor expansion show up a lot in proofs, but we have never had to use it in practice— will we ever?
6.0 states that we will begin studying nonlinear systems. (That was fast...) 6.1 defines a vector field on a plane, where each point on the plane has a velocity vector. A phase point traces out a solution on the vector plane. Because analytically finding solutions is so difficult, we focus on qualitative behavior of nonlinear systems, i.e fixed points, closed orbits, stability. The chapter introduces Runge Kutte. 6.2 gives a theorem for the existence of solutions, which are guaranteed it f is continuously differentiable. DIfferent trajectoris do not intersect. There is also a theorem that a trajectory confined to a region without a fixed point will eventually reach a closed orbit. 6.3 looks at the stability near a fixed point by substituting a disturbance into the Taylor expansion to find the Jacobian matrix, which is similar to f'(x) but for many variables. The text defines various sorts of behavior— repellers, attractors, saddles, centers, and fixed points.
CHALLENGES
I would like to practice making and using a Jacobian matrix. Substitution of polar coordinates (used on p. 153) is something I need to brush up on, and going through the Runge Kutte would be nice, too! It's something I didn't really get in Scientific Computation. Also... an explanation of topological equivalence..
REFLECTION
So yeah, we did Runge Kutte in Comp 365. We also used Jacobian matrix then, but I was never sure of the definition of a Jacobian. I see the Taylor expansion show up a lot in proofs, but we have never had to use it in practice— will we ever?
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