The dynamic of the system is showed as below: $$ \begin{cases} \dot x=y \\ \dot y =2\delta y-x+1\ \end{cases}$$ $$[x(t^+),y(t^+)]^T=[-x(t),0]^T\\ for \ x(t)<0 \ and \ y(t)=0, (0<\delta<1)$$ The typical chaotic attractor:
The exact piecewise solution of the above equation: $$x(t)=e^{\delta t}\{\{x(0)-1\}cos(\omega t)+\frac1\omega \{y(0)-\delta x(0)+\delta\}sin(\omega t)\}+1$$ $$y(t)=e^{\delta t}\{y(0)cos(\omega t)+[\omega -\omega x(0)+\frac \delta\omega y(0)-{\delta^2\over\omega}x(0)+{\delta^2\over\omega}]sin(\omega t)\}$$ $$\omega =\sqrt {1-\delta^2}$$
The trajectory rotates divergently around the equilibrium point $(1, 0)$ and reaches the switching threshold at $t=\frac{2\pi}\omega$. A x−coordinate of the reaching point is obtained as $−e^{2πδ\over \omega} + 1$ and as seen in the image above $-A+1$ by substituting $x(0)=0 \ and \ t=\frac{2\pi}\omega$ into the solution. Here I define the Poincare-section as below: $$l\equiv \{(x,y)|y=0\}$$ How I can find the return map: $$f:l\mapsto l,x_{n+1}=f(x_n)$$ for this system? Thanks in advance.