Assume I have the Duffing oscillator (driven with frequency $\omega$)
$$\frac{dx^2}{dt^2} = x-x^3-\beta v+ \alpha \cos(\omega t)$$ The dynamics in continuous time is given by $$\frac{d\vec{x}}{dt}=f(\vec{x})$$ where in this case $\vec{x}=x,v,\varphi$.
Assuming I know $f(\vec{x})$ for Duffing (and I do), I can calculate the Jacobian matrix using $$J=\frac{\partial{f}}{\partial{x}}$$
Say that $\phi_t(\vec{x})$ is the flow function of the dynamical system, and I define the Poincare section at $\varphi=0$. Two questions:
1) How does one actually show that $P=\phi_{2\pi/\omega}$ is the Poincare map (a.k.a. return map) on this section?
2) How can $detK$ where $K=\frac{\partial{P}}{\partial{x}}$ be calculated from $J$?
Thanks in advance.