Using rotation number to detect sticky orbits in Hamiltonian systems

In Hamiltonian systems, depending on the control parameter, orbits can stay for very long times around islands, the so-called stickiness effect caused by a temporary trapping mechanism. Different methods have been used to identify sticky orbits, such as recurrence analysis, recurrence time statistics, and finite-time Lyapunov exponent. However, these methods require a large number of map iterations and to know the island positions in the phase space. Here, we show how to use the small divergence of bursts in the rotation number calculation as a tool to identify stickiness without knowing the island positions. This new procedure is applied to the standard map, a map that has been used to describe the dynamic behavior of several nonlinear systems. Moreover, our procedure uses a small number of map iterations and is proper to identify the presence of stickiness phenomenon for different values of the control parameter.