Lyapunov Graph for Two-Parameters Map: Application to the Circle Map

In a Lyapunov graph the Lyapunov exponent, λ, is represented by a color in the parameter space. The color shade varies from black to white as λ goes from -∞ to 0. Some of the main aspects of the complex dynamics of the circle map (θ_n+1=θ_n+Ω+(1/2π)Ksin(2πθ_n)(mod 1)), can be obtained by analyzing its Lyapunov graph. For K>1 the map develops one maximum and one minimum and may exhibit bistability that corresponds to the intersection of topological structures (stability arms) in the Lyapunov graph. In the bistability region, there is a strong sensitivity to the initial condition. Using the fact that each of the coexisting stable solution is associated to one of the extrema of the map, we construct a function that allows to obtain the boundary separating the set of initial conditions converging to one stable solution, from the set of initial conditions converging to the other coexisting stable solution.