is a diffeomorphism for every point x in U, the positive semi-orbit of x intersects S for the first time at P(x) Example Consider the following system of differential equations in polar coordinates, : The flow of the system can be obtained by integrating the equation: for the component we simply have while for the component we need to separate the variables and integrate: Inverting last expression gives and since we find The flow of the system is therefore The behaviour of the flow is the following: The angle increases monotonically and at constant rate. The radius tends to the equilibrium for every value. Therefore, the solution with initial data draws a spiral that tends towards the radius 1 circle. We can take as Poincaré section for this flow the positive horizontal axis, namely : obviously we can use as coordinate on the section. Every point in returns to the section after a time (this can be understood by looking at the evolution of the angle): we can take as Poincaré map the restriction of to the section computed at the time , . The Poincaré map is therefore : The behaviour of the orbits of the discrete dynamical system is the following: The point is fixed, so for every . Every other point tends monotonically to the equilibrium, for . Poincaré maps and stability analysis Poincaré maps can be interpreted as a discrete dynamical system. The stability of a periodic orbit of the original system is closely related to the stability of the fixed point of the corresponding Poincaré map. Let (R, M, φ) be a differentiable dynamical system with periodic orbit γ through p. Let be the corresponding Poincaré map through p. We define and then (Z, U, P) is a discrete dynamical system with state space U and evolution function Per definition this system has a fixed point at p. The periodic orbit γ of the continuous dynamical system is stable if and only if the fixed point p of the discrete dynamical system is stable. The periodic orbit γ of the continuous dynamical system is asymptotically stable if and only if the fixed point p of the discrete dynamical system is asymptotically stable. See also Poincaré recurrence Stroboscopic map Hénon map Recurrence plot Mironenko reflecting function Invariant measure References External links Shivakumar Jolad, Poincare Map and its application to 'Spinning Magnet' problem, (2005) Dynamical systems Map