A taste of higher mathematics! (2 Viewers)

[Sy123]

What makes a chaotic system chaotic?

ex. Is there a specific reason *why* for example the limiting value of, is chaotic at r > ~3.4?

 

[seanieg89]

It seems to me what you listed all belong to pure mathematics, although part of which have found wide applications in science, engineering and econimics etc. In my opinion, applied and computational mathematics are also interesting branches. Applications can be found in HSC maths such as Newton's method, Trapezoidal formula and Simpson's formula, both of which involve approximating a function by a "simpler" function like linear or quadratic polynomial. Another example is solving linear system of equations. The Gauss elimination method (reduced to row echlon form) theoretically works perfect and is so easy. But do you ever think of solving a million by million linear system of equations (of course letting computer do it)? Due to round off errors, after massive amount of calculations, the row echlon form is merely theoretical, the resulted matrix could even be singular and thus the computer will say no solution!
Back to Newton's method. In year 12 textbook questions, you might see that Newton's method does not always give better approximation of solutions unless the initial guess is "good" enough. Even if the method gives a satisfied approximation, what is the rate of convergence? How can you improve the method?
And Simpson's formula. Some year 12 students have already known that some functions(actually most of the functions) do not a primitive ( in the sense of elementary functions), so in this situation, how can you still compute a definite integral? Simpson's rule gives a good approximation. How can you improve the approximation?
These all belong to computational mathematics.

Yep, I mostly went with things somewhat connected to my interests...as the time I have to devote to such a blog will be a bit limited.

Some aspects of numerical analysis and computational mathematics interest me a bit, but relatively not as much as most of the things I listed.

That said, numerical integration and newton's method in particular both sound like quite good ideas for treatment individually (as well as being worthwhile things for hs students to understand better), so I might well do those.

 

[seanieg89]

What makes a chaotic system chaotic?

ex. Is there a specific reason *why* for example the limiting value of, is chaotic at r > ~3.4?

There is a generally agreed upon definition for chaos in a dynamical system (a system which evolves in a discrete or continuous time variable).

1. It must be sensitive to initial conditions. (The butterfly effect.) This can be made rigorous using Lyapunov exponents, which measure how quickly "nearby" trajectories can diverge.

2. It must be topologically mixing. This roughly means that if you painted something on your phase space, the paint would become spread out equally in the large time limit.

3. It must have dense periodic orbits. (This is a little harder to grasp, but basically means that it can be hard to tell when you are in a periodic orbit, because every point in space is close to infinitely many points that lie in such orbits.)

The importance of all three of these assumptions takes some time to understand, and is somewhat specific to what kind of dynamical system you are talking about. (A sequence of numbers in an interval, a point smoothly moving on a manifold, etc). In some settings, one of the above assumptions will in fact imply one or both of the others.

 

[Sy123]

There is a generally agreed upon definition for chaos in a dynamical system (a system which evolves in a discrete or continuous time variable).

1. It must be sensitive to initial conditions. (The butterfly effect.) This can be made rigorous using Lyapunov exponents, which measure how quickly "nearby" trajectories can diverge.

2. It must be topologically mixing. This roughly means that if you painted something on your phase space, the paint would become spread out equally in the large time limit.

3. It must have dense periodic orbits. (This is a little harder to grasp, but basically means that it can be hard to tell when you are in a periodic orbit, because every point in space is close to infinitely many points that lie in such orbits.)

The importance of all three of these assumptions takes some time to understand, and is somewhat specific to what kind of dynamical system you are talking about. (A sequence of numbers in an interval, a point smoothly moving on a manifold, etc). In some settings, one of the above assumptions will in fact imply one or both of the others.

Thanks for taking the time to answer

I don't really understand much of what you're saying though, at what level do you learn about this?

 

[seanieg89]

Thanks for taking the time to answer

I don't really understand much of what you're saying though, at what level do you learn about this?

Nothing at usyd undergrad I don't think (although Lyapunov exponents were briefly mentioned in a 3rd year biomathematics course). It's pretty niche.

If there are any academics in the department working in chaos (I can't think of any of the top of my head but there are many I haven't met), then you could always do a reading course with one them in a later year. The prerequisites wouldn't be huge, but you would probably want to know some topology and measure theory.

I don't know a huge amount about chaos theory in general, most of what I know has been self-taught during my PhD. From my understanding, a lot of the definitions are pretty vague too.

 

[Axio]

Nothing at usyd undergrad I don't think (although Lyapunov exponents were briefly mentioned in a 3rd year biomathematics course). It's pretty niche.

If there are any academics in the department working in chaos (I can't think of any of the top of my head but there are many I haven't met), then you could always do a reading course with one them in a later year. The prerequisites wouldn't be huge, but you would probably want to know some topology and measure theory.

I don't know a huge amount about chaos theory in general, most of what I know has been self-taught during my PhD. From my understanding, a lot of the definitions are pretty vague too.

What are you doing for your PhD?

 

[seanieg89]

What are you doing for your PhD?

Microlocal analysis, harmonic analysis, quantum ergodicity etc.