take-home portion assigned 12/13, due 12/15 at 10:30
Nonlinearities make life interesting
Consider the coupled system:
dx/dt = y - (x^2+y^2-1) x
dy/dt = -x - (x^2 + y^2- 1) y
1) Find the fixed points (the steady-state solutions). Find analytically
using pen and paper or maple.
2) Find the behavior of solutions that start near the fixed point(s). Again,
find analytically.
3) Study the system numerically using fourth-order Runge-Kutta to do
the time-stepping. First, give graphs demonstrating the behavior near
the fixed point(s) (one graph per fixed point). Second, give a
graph demonstrating more global behavior.
Consider the coupled system:
dx/dt = y + (x^2+y^2-1) x
dy/dt = -x + (x^2 + y^2- 1) y
Repeat the above three problems.
Consider the coupled system:
dx/dt = y + a (x^2+y^2-1) x
dy/dt = -x + a (x^2 + y^2- 1) y
where a is some fixed real number.
1) Find the fixed points (the steady-state solutions).
2) Find the behavior of solutions that start near the fixed points.
3) Can you find a geometrical explanation for the behavior of the solutions
for different values of a?
Linear Stability Analysis --- When can it fail?
Consider the single equation:
dx/dt = f(x)
1) Assume that x_0 is a fixed point. I.e., f(x_0) = 0. Give examples
of various plots of f near x_0. (You know that the graph of f will go
through zero at x_0, but what are its various options near x_0?)
2) For each of the above plots of f, explain what that would imply about
the behavior of solutions that start near x_0. ("If a solution starts
to the right
of x_0 then it will...")
3) For each of the plots of f, explain how your answer above is related
to the linearization of f at the fixed point.
4) When can you say that the linearization predicted the behavior of
solutions that started near x_0? When can't you say that the linearization
predicted the behavior? Are there any examples where two ODEs have the
same fixed point, the same linearization, but fundamentally different
behavior for solutions that start near the fixed point?
Integration and orders of convergence
Find a, b, and c so that the integration rule for the integral from
0 to 1
int_0^1 f(x) dx ~ a f(0) + b f(1/2) + c f(1) = I1(f)
is as accurate as possible.
Find A, B, and C so that the integration rule for the integral from
0 to 1
int_0^1 f(x) dx ~ A f(1/4) + B f(1/2) + C f(3/4) = I2(f)
is as accurate as possible.
1) Either use the above calculations or repeat the them to find the
analogous quadrature rules that will
approximate the integral from x0 to x0 + 4 h, int_x0^(x0+4h) f(x) dx.
(I.e., find how a, b, c, A, B,
and C change.)
2) Write up a pair of subroutines, I1.m and I2.m, which implement
the quadrature rules. I1.m will need a subroutine f.m, it will take
inputs x0 and x1 (the left-hand and right-hand endpoints of the interval
of integration) and n (the number of mesh-points in the interval [x0,x1]).
It will return a value which approximates the integral from
x0 to x1, int_x0^x1 f(x) dx.
3) Demonstrate that your two subroutines each have the correct order
of accuracy.
4) Which of the subroutines works better? Does one subroutine need
more mesh-points to get the same accuracy as another?