Adiabatic invariants Hamiltonian chaotic systems

B. Adiabatic Invariant for a Simple Hamiltonian Chaotic System... 

The other distribution is the microcanonical equilibrium distribution. More than 15 years ago, Ott-Brown-Grebogi pointed out fractional scaling of deviation from ergodic adiabatic invariants in Hamiltonian chaotic systems [16, 17]. We will reconsider not only ergodic adiabatic invariants but also nonergodic adiabatic invariants, which are important in the mixed phase space. We will show results of our numerical simulation in which a nonergodic adiabatic invariant corresponding to uniform distribution is broken in the mixed phase space. 

Now we consider the adiabatic invariant for a simple Hamiltonian chaotic system. The Hamiltonian is defined as... 

With regard to the microcanonical equilibrium distribution and the extension of the fluctuation-dissipation theorem, we considered a nonergodic adiabatic invariant in a simple Hamiltonian chaotic system. We numerically demonstrated the breaking of the nonergodic adiabatic invariant in the mixed phase space. The variance of the nonergodic adiabatic invariant can be considered as a measure for complexity of the mixed phase space. 

In this section we will consider an nonergodic adiabatic invariant for a chaotic Hamiltonian system. Specifically, we are interested in the mixed phase space, in which tori and chaotic seas coexist. 

---