The Last Invariant Circle

Here x is a periodic configuration variable (usually computed modulo 1), y ∈R is the momentum variable, and the parameter k represents the strength of a nonlinear kick. This map was first proposed in 1968 by Bryan Taylor and then independently obtained by Boris Chirikov to describe the dynamics of magnetic field lines. The standard map and Hénon’s area-preserving quadratic map are extensively studied paradigms for chaotic Hamiltonian dynamics. The standard map is an “exact symplectic” map of the cylinder. Because x′(x, y) is a monotone function of y for each x, it is also an example of a monotone twist map (See Aubry–Mather theory). Every twist map has a Lagrangian generating function, and the standard map is generated by F(x, x′)= 1 2 (x′ − x)2 + (k/4π2) cos(2πx), so that y=− ∂F/∂x and y′ = ∂F/∂x′. The map can also be obtained from a discrete Lagrangian variational principle as follows. Define the discrete action for any configuration sequence . . . , xt−1, xt , xt+1, . . . as the formal sum