Homework Statement
Consider the system x' + x = F(t), where F is smooth, T-periodic function. IS it true that the system necessarily has a stable T-periodic solution x(t)? If so, Prove it; If not, find an F that provides a counterexample.
Homework Equations
x' + x = F(t)
The...
I have the folloring problem:
Given the following flow on the torus (θ_1)' = ω_1 and (θ_2)' = ω_2, where ω_1 /ω_2 is irrational then I am asked to show that each trajectory is DENSE. So I need to prove that Given any point p on the torus, any initial condition q, and any ε > 0, then there...
Homework Statement
For two equal-amplitude waves of nearly equal frequencies w+d/2 and w-d/2 (at a fixed position x), you have shown that the intensity exhibits \beats"
which for d = 2pi Hz can be perceived directly by the ear, even though the
main frequency is perceived only through the...
1. Homework Statement
Write a wave in one space dimension as ARe(ei(kx-wt-d))where A is the
amplitude of the wave. Find a second wave of the same frequency such that
the sum of the two vanishes at x = 0 and x = L. Assuming the wave velocity
c = w/|k| is fixed, for what frequencies ! is...
Homework Statement
Write a wave in one space dimension as ARe(ei(kx-wt-d))where A is the
amplitude of the wave. Find a second wave of the same frequency such that
the sum of the two vanishes at x = 0 and x = L. Assuming the wave velocity
c = w/|k| is fixed, for what frequencies ! is this...