Chaotic and regular behaviours of classical and fractional Gross–Pitaevskii equations including two-body, three-body and higher-order interactions

This study investigates the chaotic and regular behaviours of classical and fractional Gross–Pitaevskii equations (GPE) for interacting boson systems under combined harmonic and optical lattice potentials by Poincaré section of phase space, Lyapunov exponents, power spectrum and bifurcation analysis techniques. Also, the effects of system parameters on the system behaviour are discussed. After certain values of the harmonic potential (for β = 0.001 and above), it is seen that the classical GP equation with two-body interaction shows shock wave-like dynamics. In addition, it is found that the harmonic potential is dominant where only binary interaction and three types of interactions exist for β = 0.001 and above. While the boson system exhibits a regular/quasiperiodic behaviour for a small order of fractional derivative operator, it displays a chaotic structure as it approaches the value of 2. (author)