Q1 Consider the system:

with initial condition u = 2 when

1.Determine the closed-form solution for u(t) by integrating numerically.  

2.Based on  a few numerical integration schemes (e.g., Euler, mid-point, Runge-Kutta order 2 and 4 ) and considering a range of integration time steps (from large to small), plot the time evolution of u(t) for 0 ≤ t ≤ 2, using all 4 methods and superimpose with the closed-form solution.  

3.Discuss the agreement between numerically integrated solutions and analytical solution, particularly in relation to the choice of integration time step.  

Q2 Consider the following function:

f(x,y) = (x4+ y4)−(21×2+13y2)+2xy(x + y)−(14x +22y)+170  

where −6 ≤ x,y ≤ 6.

This function admits a number of minima. Use gradient descent to identify them. Your approach must be described and your results presented and discussed, particularly in relation to the suitability of gradient descent. Think on alternative approaches and explain what problems they would address.  

 

Q3 Identify which of the following differential equations:

 

 

 

produces the following direction field.

 

Justify your answer analytically.

 

Q4 Consider the following two-dimensional discrete dynamical system:

xt+1 = xtyt  

yt+1 = yt (xt −1)

1.Find all equilibria.

 

2.Calculate the Jacobian matrix at each of the equilibria.

 

 

3.Calculate eigenvectors and eigenvalues of each of the matrices obtained above.

 

4.Based on the results, discuss the stability of each equilibrium.

 

 

5.Implement the dynamical system and discuss your findings above. In terms of the eigenvectors and eigenvalues found in 3, provide a geometric interpretation of the behaviour of the system.