  The relation between the onset of chaos and critical phenomena, like Quantum
Phase Transitions (QPT) and Excited-State Quantum Phase transitions (ESQPT), is
analyzed for atom-field systems. While it has been speculated that the onset of
hard chaos is associated with ESQPT based in the resonant case, the
off-resonant cases show clearly that both phenomena, ESQPT and chaos, respond
to different mechanisms. The results are supported in a detailed numerical
study of the dynamics of the semiclassical Hamiltonian of the Dicke model. The
appearance of chaos is quantified calculating the largest Lyapunov exponent for
a wide sample of initial conditions in the whole available phase space for a
given energy. The percentage of the available phase space with chaotic
trajectories is evaluated as a function of energy and coupling between the
qubit and bosonic part, allowing to obtain maps in the space of coupling and
energy, where ergodic properties are observed in the model. Different sets of
Hamiltonian parameters are considered, including resonant and off-resonant
cases.
