Show that the standard map of the previous problem described by

\[x_{n+1}=x_{n}+y_{n} \quad, \quad y_{n+1}=a y_{n}-b \cos \left(x_{n}+y_{n}\right),\]

can be obtained by discretizing time in the Hamiltonian equations of motion of the following physical system: a planar pendulum in the absence of gravity that is periodically kicked in a fixed direction with a fixed force. The phase space would be described by \(\theta(t)\), the pendulum's angle from the vertical, and its canonical momentum \(p_{\theta}(t)-\) which you will need to map to the discrete sequence given by \(x_{n}\) and \(y_{n}\).

Data from previous problem

Consider the recursion relation

\[x_{n+1}=x_{n}+y_{n} \quad, \quad y_{n+1}=a y_{n}-b \cos \left(x_{n}+y_{n}\right)\]

where \(a\) and \(b\) are constants; this system is known as the standard map. Analyze the system as we did for the logistic map in the text. In particular, explore regions of the parameter space where (1) \(a=1\) and (2) \(a=1 / 2\) while varying \(b\), (3) \(b=6\) near point (3,3).