Find all the values of r so that the equation dx/dt=cos(rx) defines a vector field on the circle.
My answer is that ;
By the definition of a vector field on the circle, dx/dt=cos(rx) must be real valued and 2π periodic function. Since cos x has all real numbers in its domain and -1≤ cos x≤ 1, hence cos(rx) is a real valued function. We now need to show that f(x)=cos(rx) is 2π periodic.
now,
f(x)=f(x+2π) cos(rx)=cos(r(x+2π)) = cos(rx+2π)
It implies that 2πr=2πz, z is an integer Hence, r is an integer.
Is my answer correct? The book that we are using is nonlinear dynamics and chaos by steven strogatz.