Quantum chaos in quantum dots coupled to bosons

Chaos transition, as an important topic, has become an active research subject in non-linear science. By considering a Dicke Hamiltonian coupled to a bath of harmonic oscillator, we have been able to introduce a logistic map with quantum corrections. Some basic dynamical properties, such as Lyapunov exponents and bifurcation diagram of the model are studied. we show that in this model, the transition from integrable motion to periodic, chaotic and hyperchaotic as the control parameter $r$ is increased.