Summer Semester, 2013

Version A

(10 pont)

Define a function that is a periodic continuation of the function of the function   (|x|≤1).
Plot the new function on an interval large enough.

(5+5+5+10 points)

Plot the function  h(x)=sin(x) +sin((x^2)) on the interval a [-3,3] , such that the y-range be [-2,2], and the lengths of the x-axis and y-axis are equal. Color the graph according to the second derivative of the function.

Find the extremal points on the given interval.

Mark theses points on the graph.

Draw the tangent lines to the graph at the points of inflection.

(10 points)

Prepare a substitution rule that recognizes a vector with two numeric elements different from zero. The rule has to take the reciprocal of the second element, and keep the first one unchanged. Give an example.

(5+10+5 points)

Consider the following  predator prey system ((Subscript[p, 1]): prey, (Subscript[p, 2]): predator )
 (Subscript[p, 1])' =0.5(Subscript[p, 1])(1-0.5(Subscript[p, 2]))- 0.5 (Subscript[p, 1])(Subscript[p, 2]),
(Subscript[p, 2])'=-0.1(Subscript[p, 2])+0.2 (Subscript[p, 1])(Subscript[p, 2])
where 0≤(Subscript[p, 1]),(Subscript[p, 2]).
Find the equilibria. Plot the vector field in the phase space ((Subscript[p, 1]),(Subscript[p, 2])). (5 pts)
Find the numeric solution if (Subscript[p, 1])(0)=0.5, 1, (Subscript[p, 2])(0)=1,1.5. Plot the solutions (Subscript[p, 1])(t), (Subscript[p, 2])(t) with respect to the time, and the trajectories ((Subscript[p, 1])(t),(Subscript[p, 2])(t)) in the 2D phase space (10 pts). Color the trajectories with respect to the time (5 pts)

(10+10 points)

Consider the following difference equation (finite difference discretization of the differential equation of the damped oscillator (x''+a x'+ sin(x)=0):

{(Subscript[x, n+1]),(Subscript[y, n+1])}:={(Subscript[x, n])+ h (Subscript[y, n]),-h sin((Subscript[x, n]))+(Subscript[y, n])(1-a h)}

where a is the damping coefficient, h is the step-size of the discretization of the derivative (x'(t)≈((x(t+h)-x(t))/h), (Subscript[x, n])=x((Subscript[t, n])), (Subscript[t, n+1])=(Subscript[t, n])+h, (Subscript[t, 0])=0).

Iterate  the from the initial points {1,0} and {0,1} with given 2>a>0 and small enough h>0 (by your choice). Plot the iterated points and the trajectory  (the points joined by a line). (10 pts)
Use coloring of the points by the time.
Change a, h and the initial point interactively in Manipulate. (10 pts)

Version B

(10 pont)

Define a function that is a reciprocal-periodic continuation of the function 1+4(x(1-x)   (0≤x≤1) to the halfline [0,∞), i.e. f(x)=1/f(x-1) for x>1.
Plot the new function on an interval large enough.

(5+5+5+10 points)

Plot the function  h(x)=sin(x) +sin((x^2)) on the interval a [-3,3] , such that the y-range be [-2,2], and the lengths of the x-axis and y-axis are equal. Color the graph according to the second derivative of the function.

Find the zero points on the given interval.

Mark theses points on the graph.

Draw the tangent lines to the graph at these points.

(10 points)

Prepare a substitution rule that recognizes a 3x3 square matrix and it gives the co-Spur, i.e. the sum of the elements of the co-diagonal of the matrix. Give an example.

(5+10+5 points)

Consider the forced nonlinear van der Pol oscillator:

x'=y, 
y'=(1-(x^2))y-sin(x)+cos(t) .

Find the equilibria. Plot the vector field in the phase space (x,y). (5 pts)
Find the numeric solution if x(0)=0.5, 1, y(0)=1,1.5. Plot the solutions x(t), y(t) with respect to the time, and the trajectories (x(t),y(t)) in the 2D phase space (10 pts). Color the trajectories with respect to the time (5 pts)

(10+10 points)

Consider the following difference equation (Henon's map):

{(Subscript[x, n+1]),(Subscript[y, n+1])}:={(Subscript[y, n])+1-1.4 ((Subscript[x, n])^2),0.3(Subscript[x, n])}

Iterate it from the initial points {1,0},{1,1},{0,1},{0,0}. Plot the coordinates with respect to the time, and the trajectory. Use coloring by the time. Create a dynamic application in which the initial point can be changed interactively.