📄 math454.htm
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<!-- saved from url=(0042)http://math.arizona.edu/~kglasner/math454/ -->
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-8859-1">
<META content="MSHTML 6.00.2800.1498" name=GENERATOR></HEAD>
<BODY>
<H2>MATH 454: Dynamical Systems </H2><BR><A
href="http://math.arizona.edu/~kglasner/math454/project.pdf">Dynamical Systems
Project (PDF) </A>Here is a full description of the semester-culminating
project. <BR><BR><A
href="http://math.arizona.edu/~kglasner/math454/matlab_primer_3e.pdf">Matlab
Primer (PDF) </A>This will help you get started using MATLAB. <BR><BR><A
href="http://math.arizona.edu/~kglasner/math454/portrait.m">Phase portrait
m-file</A> <BR>This is a script which shows you how to plot 2-d phase portraits.
<HL>
<H2>Computer Projects</H2>
<H3>Project #1 </H3>First, look at this <A
href="http://math.arizona.edu/~kglasner/math454/rk.m">Matlab Runge-Kutta
implementation </A>. Read the code carefully to see what's going on.<BR>Notice
that many things are done for you; most of your job is to correctly implement
the ode to be computed.<BR>By removing the "hold off" command, you can
simultaneously plot solutions for many different initial data,<BR>effectively
producing a slope-field plot. <BR><BR>Do problems 2.8.2 (a-d) in Strogatz. Note
that you may have to scale the plot appropriately using the "axis" command.
<BR><BR><BR>
<H3>Project #2 </H3>Part A. Look at this <A
href="http://math.arizona.edu/~kglasner/math454/linear.m">Matlab code </A>for
plotting phase portraits of linear systems. <BR>This will also tell you the
eigenvalues and plot the eigenvectors (if they are real). <BR>Do problems (5.2)
3,5,7,9 in Strogatz.<BR><BR>Part B. This problem is based on 5.2.14. We want to
know the probability that a randomly selected matrix corresponds to a stable
system. This <A
href="http://math.arizona.edu/~kglasner/math454/random_matrices.m">code
</A>implements what the text suggests. <BR>Find the fraction of stable systems
when the matrix entries are (1) distributed between 0 and 1, (2) distributed
between -1 and 0, (3) normally distributed (see the MATLAB function randn).
Explain why your results make sense. <BR><BR><BR>
<H3>Project #3 </H3>Here is <A
href="http://math.arizona.edu/~kglasner/math454/logistic.m">Matlab code </A>for
plotting that groovy bifurcation diagram associated with the logistic map.
<BR>A1. Change the code to compute the sin map (see text), and also change the
range of the parameter r. Does the result look the same as the logistic map?
Show some plots of your results. <BR>A2. Find the values of the parameter $r$
where the period doubling bifurcations occur. Compute the ratio of their
sucessive differences (see text) to see how close you can get to the Feigenbaum
constant. <BR>A3. Find the point where chaos sets in, and other features, such
as a window where 3-cycles live. <BR><BR><BR>Part B: Here is <A
href="http://math.arizona.edu/~kglasner/math454/lyapunov_exponent.m">Matlab code
</A>for finding the Lyapunov exponent of the attractor.<BR>B1. Set $r=3$ and
determine the Lyapunov exponent. Show that you get the same result independent
of the initial data that you use. Why is the result negative?<BR>B2. Determine
the exponent for some stable 2-cycle, 4-cycle, and a chaotic attractor (indicate
the value of r in each case). Interpret your results. How big can you make the
exponent? <BR>B3. What goes wrong when $r>4$ ? <BR><BR><BR>
<H3>Project #4 </H3>
<SCRIPT>/*#4. Finding dimension of an attractorB. Now let's find the pointwise dimension of a logistic map attractorusing <a href="dimension.m"> this code. </a> The code gives twofigures for output:<br> the attractor, plotted on the x-axis, anda plot of N(epsilon) versus epsilon.<br>B1. The code gives different output each time you run it becausethe initial data is randomized. Find the pointwise dimensionfor several different runs.<br>B2. Approximate the the correlation dimension for the whole attractorby averaging. If you are feeling ambitious, have Matlab averageN(epsilon) for many different initial data.<br><br><br>#5. Here is <a href="phaseplane.m"> Matlab code </a> forplotting particular trajectories in the phase plane. This isespecially useful for seeing limit cycles and other "nonlocal"features. <br><br>A. Problem (8.2.3) is an example of a Hopf bifurcation.By plotting trajectories for several values of mu, decidewhat kind it is.<br><br>B. If you haven't already, do (8.6.8) in Strogatz. This showsquasi-periodic orbits which "fill up" space.</SCRIPT>
</BODY></HTML>
⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -