Winter Semester, 2007

Exam project themes

Investigate the phenomena, the problems by any kind of means. You can use any tools and/or development of Mathematica, but do not mechanically copy them; you have to modify them to the given task. Create well-structured notebook (notebooks) and describe briefly each steps, parts of the work. Use elegant and smart methods (like coloring, animations, manipulations; nice, compact generalizable programs), write well parametrizable functions. Develope different version, give examples, if you feel necessary.

During the evaluation, the main points of view will be:

mathematically, and informatically correct solution,

your creativity, smart methods, generalizability,

structure, readability, comments.

In the final grade, your activity during the course will be also taken into account.
At the and of the semester, you will receive a grade, average of the Mathematica and Matlab parts.

Relaxation oscillations can appear in many different fields of Physics and Life sciences. An equation of describing such an oscillation is the van der Pol system

u' = v 
v' = a*(1 - (u^2))*v - u

where a>0. Investigate the properties of the system. Especially, study the solution at different values of the parameter a with small initial values.

Investigate the properties when the parameter a is small, in particular, a->0.

Try to find a small parametric curve which is crossed by the trajectories outwards as well as large crossed inwards (that shows the existence of the limit cycle).

What happens, if there is an inhomogeneity in the system, such as

u' = v 
v' = a*(1 - (u^2))*v - u + c

What happens, if the elastic force is nonlinear, such as

u' = v 
v' = a*(1 - (u^2))*v - g(u)

for example g(u)=(u^n).

Investigate and visualize the problem of resonance for forced linear spring. The  general equation is

x' = v 
v' = F(t) - (Subscript[w, 0]^2) x- a v

where

The driving "force" is of form F(t) = A cos (w t).

Investigate and visualize the resonance phenomenon and  the dependence of the resonance on the value of the parameter a.

Try to use different type of periodic forces such as triangle waves, square waves.

Try to consider the resonance for nonlinear springs with the equation

x' = v
v' = F(t) - g(x)- a v

where g(x) is nonlinear, such as g(x)= (Subscript[k, 1]) x +(Subscript[k, 2]) (x^3) (hard spring) or  g(x)= (Subscript[k, 1]) x -(Subscript[k, 2]) (x^3) (soft spring).

Try to investigate how the resonance frequency depends on the magnitude A of the forcing term.

Write a simple programme for the gradient method to find numerically the local minimum of a function V: (R^k)->R.  One step of the algorithm is the following. Take a point (Subscript[x, 0])∈(R^k). The next pont is taken in the direction of the negative gradient:

(Subscript[x, 1])=(Subscript[x, 0]) - h grad V((Subscript[x, 0]))

where, h is the step size.

Iterate the process until the distance between the consequtive values is less than the given δ. Theoretically (at zero step size), the process should follow the trajectories of the gradient system:

x'=-grad(V)

Compare the path that your algorithm follows and the trajectory of the solution of the gradient system.

Investigate the dependence on the initial values (use a function V with two minima and a saddle point between them, such as (V[x_, y_] := (x^2) + (y^4) + 2 x y).

Try to find similar example in 3D, and present it visually.

Visualize the random motion. Let a point move in 2D (3D) space randomly (Random statement!). Define different moving rules.

For example:

Implement and study the method of linearization for differential equations. Let us consider the equation

x'=f(t,x), x∈(R^n),

and let (Subscript[x, 0])=0 be an equilibrium of it. Then the system can be linearized such a way that the stability properties of the linear system

z'=(∂_x)f(t,0)z

around the origin can be investigated instead of the original system, provided that the eigenvalues of the n×n matrix (∂_x)f(t,0) are nonzeros.

Implement the method to find the linearization, compare graphically and numerically the behavior of the original and the linearized system. Apply the method for the system

x'=y-c (x^3)
y'=-a sin(x)-b (((sin(x)))^2)-k y

find conditions for the parameters such that the both eigenvalues of the linear system are negative, positive, one of them is negative and the other is positive, or one of them is zero.

Compare the original and the linearized system in these cases.

Study the behavior and visualize the motion of the linear double pendulum:

(Subscript[x, 1])' = (Subscript[x, 2])
(Subscript[x, 2])' = -(Subscript[x, 1]) + a*(Subscript[x, 3])
(Subscript[x, 3])' = (Subscript[x, 4])
(Subscript[x, 4])' =(Subscript[x, 1]) - (Subscript[x, 3])

where

The equations above (using suitable time units) model the small-amplitude, undamped oscillations of a double pendulum where the rods are of the same length.  Is the response of the system realistic at large angular velocities?

Visualize the motion (upper -lower pendulum coordinates, the pendulum itself, the coordinates (Subscript[x, 1])-(Subscript[x, 3]).

Does the linear system have periodic orbits in the (Subscript[x, 1])-(Subscript[x, 3]) plane? Can you find one?

What happens, if there is a periodic external force in either (Subscript[x, 2]), or (Subscript[x, 4]), like

(Subscript[x, 2])' = -(Subscript[x, 1]) + a*(Subscript[x, 3])+A cos(w t)

or

(Subscript[x, 4])' = (Subscript[x, 1]) - (Subscript[x, 3])+A cos(w t) .

Are there periodic solutions? What happens at resonance?