### 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?