By all means, learn all you can from outside sources, but then run the programs and gather and report your findings independently, citing these sources and saying how you used them. This is the level of independent work needed for full credit. If you simply report the findings of others (duly acknowledged, of course) this would reduce your grade for the project.
A region with four walls. A
Dynamics in the Stadium. B
Dynamics in the Triangle. B
none
complex roots via Newton-Raphson method B
Using a turtle to draw fractals B
Julia sets. B
The project says that further elaborations will be announced. But I will leave this to the reference cited in the project.
Calculating \alpha and \delta. A
This chapter sketched methods of obtaining \alpha and \delta by approximating g and h as polynomials. The aim of this project is to obtain successive approxi-mations to \alpha and \delta using this analytical scheme or another one. Obtain at least two successive approximations for both \alpha and \delta, to assess how fast the approximations are converging. Explain your procedure. Feel free to use calculational aids such as Mathematica or Matlab. The Introduction in Cvitanovic is a useful starting point. It points to detailed information in the rest of the book, which includes a reprint of Ref. 1. Hilborn, section 5.7 has a heuristic discussion.Diffusion-Limited Aggregation (DLA) A
This project explores the use of random walks to model an interesting growth process of dendritic structures made up of particles that stick together when they come in contact. The particles are assumed to diffuse slowly through space until they come in contact with another particle, at which time they stick. The computer implementation is simple in concept: Put a particle in the middle of an applet window at (i0, j0). Then start another particle from some point (i, j) far away, and allow it to execute a random walk with integer steps. After a time, it will land on one of the points adjacent to the original particle. Stop the particle there, and begin another one from far away etc. The structure that develops has a beautiful, tree-like, fractal shape. Unfortunately, it grows very slowly. You can speed up the growth by starting from a point (i, j) that is closer to the existing cluster. It can be shown that the structure remains unaffected if one chooses a starting point (i, j) at random on a circle around (i0, j0) large enough to contain the cluster which has grown thus far. This is a program that you will want to run as long as possible. Compute the fractal dimension of your clusters, defined by the relation N \proportionalto RD , where N is the number of particles in the cluster, R is the "radius" of the cluster (typically the root-mean-square radius distance of a particle from the starting point), and D is the fractal dimension. Grow several large aggregates to get an idea of the accuracy of your estimate for D. Test whether this D is universal by varying the rules. Instead of stopping a particle when it moves adjacent to the cluster, allow it to continue with some fixed probability p. This can lead to cluster points which are multiply occupied. This is OK. Choose a value of p large enough to make a clear, qualitative difference in the structure, and determine D again. Is it consistent with the previous value with p = 0?A river network model B
Chapter 19 of Gould and Tobochnik (2nd edition) discusses one model of river networks (from R.L. Leheny, Phys. Rev. E 52, 5610 (1995)), in which a rectangular lattice of points describes an eroding terrain with the height of the land, h(x,y), specified at each point. The simulation begins with the landscape as a featureless incline: h(x,y) = l y. Then the following rules are implemented:1) Precipitation lands at a random site on the lattice.
2) Water flows from this site to one of the four nearest neighbors with a probability proportional to eE \Delta h, where \Delta h is the height difference between the site and the neighbor, and E is a parameter of the model. If \Delta h<0, this probability is set equal to zero.
3) Step 2) is repeated until the water reaches the bottom of the lattice, y =0$
4) Each point that has been visited by the flowing water has its height reduced by a constant amount b. This process represents erosion.
5) Any site at which the height difference \Delta h with a neighbor exceeds a threshold M is reduced in height by an amount \Delta h/S, where S is another parameter in the model.
Write a program that implements this model. You can get an idea of suitable parameters to use from Leheny's paper. The resulting river network is defined as follows: every lattice point receives one unit of precipitation which traces a path of steepest descent, without eroding the terrain, until it reaches the lattice edge, y=0; the river network is defined as all points through which at least R units flow. Analyze the network that is generated at different times. Does the river network appear to be fractal? How does evolving the model for longer times affect the network's properties?
Periodic and Non-periodic Orbits in a Central Potential. B
see the chapter pdf file for the formated text. Here is the source text:
By doing the integral in Eq \(6.10b) one can see whether for a given potential and given values of energy $E$ and angular momentum $\ell$, $\Phi$ is a rational number. If it is rational, then the orbit closes; if not, the orbit never repeats itself. One might then ask the question about whether there are any forms of the potential $V(r)$ which permit the orbit to be closed for {\it all} $E$ and $\ell$. However, once $V(r)$ is fixed we know that $\Phi$ is a function of $E$ and $\ell$, which we then write as $\Phi(E,\ell)$. Some applications of calculus theorems indicate that the function is continuous. Hence it can only be rational by being constant, independent of $E$ and $\ell$. It turns out that $\Phi(E,\ell)$ is constant only for very special potentials, those which vary as a power of the distance:
$$
V(r) \proportionalto - \alpha r^{-\alpha} .
\eqno(6.23)$$
One can recognize two familiar special cases, $\alpha =1$, which is the attractive gravitational force, and $\alpha =-2$, which is the harmonic oscillator. Compute the value of $\Phi(E,\ell)$ for these two cases and find out whether the orbit closes. Then do the same for $\alpha =3$. Plot up a few orbits to show what is going on. Notice that the integrals involved must be computed carefully because of their singularities at the two endpoints. To do them accurately, one should first compute $r_{min}$ and $r_{max}$ and then make a transformation like that in Problem \chapno.4 to eliminate the singularities at the endpoints.
Period-doubling bifurcation sequence for motion in a double-well potential: A
Make a bifurcation diagram for the x-value-when-p-equals-zero for the double-well potential using the DoubleWell class introduced in Chapter 6, or your own equivalent. Identify the progression of the first few fixed-points and cycles as the forcing increases from zero. Adapt the methods used in the second Project to test whether there is a counterpart of the scaling exponents α and δ found in the Logistic Map. How does the bifurcation diagram change when the damping strength γ increases from the underdamped limit γ << 1 towards critical damping?
Mapping Regions for a Double Well System: A
In this project, you will investigate the evolution of phase space regions for the damped driven motion of a particle in the double well. The idea is to take a whole bunch of points (at least a thousand) bunched into a fairly small area, propagate them all forward in time using the equations of motion, and see what the resulting region looks like. Can you think of a way of estimating the area of the resulting region? In any case, try to determine if the area seems to decrease exponentially as derived in Chapter 7. The rate of shrinkage of the area of the region after a long time should be independent of the value of F0. However, the shape of the region does depend on F0. Investigate the evolution of the shapes of the regions in the different regimes. This project is quite open-ended, and we expect you to do interesting and imaginative work.
Fractal properties of self-avoiding walks A
The Ising model of a phase transition A