Solutions

Assignment set A2, PHYD38, 2025

This set is a walk through the terrain we have covered up to this point. Such issues can constitue 1/3 of the written part of the final exam, so these are not just the happy memories..

Problem 2.1 [10p] = Strogatz 3.6.2

Problem 2.2 [10p] = Strogatz 4.3.6

Problem 2.3 [10p] Same matrix, same type of fixed point

Consider a 2-d linear system
dx/dt = A x + c,
where x is a variable vector,c is a constant vector, and A a constant-coefficient 2x2 matrix. Denote the fixed point as x*.

Show that in the new coordinate system, with the origin at the fixed point, the new variable, z= x - x*, obeys a simpler equation
dz/dt = A z
with the same matrix A as original system; therefore analysing the type of the fixed point of the original system, we can disregard the non-zero c.

Problem 2.4. [15p] Rotating saddle

Show that any saddle equilibrium point in planar dynamical system is unstable, no matter if and how fast the system in which the coordinates x(t) and y(t) are defined rotates. Assume a quite general form of a saddle below, where the effective potential U(x,y) is due both to a gradient force and a centrifugal force due to frame rotation. Being a saddle, U has a positive curvature in one direction (call it y) and a negative curvature in the perpendicular x direction, such as in U(x,y) = y²/2 - x²/2. This is a general form of saddle point potential, since one can always rotate and rescale axes to yield it, without modifying the problem. Remembering that acceleration is a negative gradient of U(x,y) and that, in a system rotating with angular velocity vector Ω perpendicular to the plane of motion, one needs to also add the Coriolis acceleration:
d²r/dt² = -∇U - 2Ω × dr/dt, where r = (x,y).

Hint: This is a system with dimension 4, but do not panic . Write out the x- and y-components of the equation of motion, substitute trial solutions with time-dependence of the form (x,y) = e^λt(u,v), where u,v do not vary in time.

Problem 2.5 [10p]. Characterize the fixed point (0,0)

dx/dt = -y (1+ ax²)
dy/dt = +x (1+ ay²),
for all real parameters a. Is it linearly stable? Is it nonlinearly stable? Is it time-reversible? Does it have an integral of motion like the system in the 2025 midterm?