Title: The Circle map, Arnold tongues and Multistability
Abstract:
Circle map helps to study the effect of non-linear coupling of two oscillators. The map contains
two parameters, ω (corresponds to the externally applied frequency) and k (corresponds to the
strength of nonlinearity). It is interesting to look at the behaviour of the map as a function of
these parameters for which we have to look into the rotation number, which gives the average
frequency of motion around the circle. When we look at the map as a function of both the
parameters, we get what is called the Arnold tongues. These are regions in the ω − k plane
where the motion is periodic. Below k = 1, the motion is either periodic (with rational rotation
number) or quasi-periodic (with irrational rotation number). But above k = 1, the tongues
overlap to form regions of multistability where more than one attractor co-exist. The dynamics
of the multistability region is extremely sensitive to the initial conditions. Multistable systems
have important research directions in fields like ecosystems, neuroscience, climate, and so on.