An example of a fully bounded chaotic sea that surrounds an infinite set of nested invariant tori

Precise definitions of elliptic islands and chaotic seas are difficult to give. Models for chaotic seas are rare in theory and practice. A few results on the dynamics in the chaotic region are available [1-2-3-4-6-7-9-15-16]. Some topological characterizations of elliptic islands in a chaotic sea are available in [5-8-11-12-17-18-19]. One of the techniques uses rotation numbers [13-14]. There are many interpretations of the rotation number in real-world models, such as the current in electric oscillators, the frequency of periodically forced pendula, heart rates, the asymptotic firing frequency of pacemaker neurons, and the integrated density of states of certain Schrôdinger operators, and so on. There are some examples of chaotic seas [5-7-8-10-11-12-17-19-20-21], but there is no rigorous proof of their boundedness for these specific cases. Here we will give a rigorous proof that a chaotic sea presented by a particular 2-D map is fully bounded for all values of its bifurcation parameters.