We study nonlinear dynamics of superposition of quantum wavepackets in various systems such as Kerr medium, Morse oscillator, and bosonic Josephson junction. The prime reason behind this study is to find out how the superposition of states influences the dynamics of quantum systems. We consider the superposition states, which are potential candidates for quantum computing and quantum communication, so it is most necessary that we study the dynamics for their proper understanding and usage. Methods in nonlinear time-series analysis such as first return time distribution, recurrence plot, and Lyapunov exponent are used for the qualification and quantification of dynamics. We found a profound change in the dynamics of quantum systems when we consider the superposition of wave packets. These changes are observed in various dynamics such as periodic, quasi-periodic, ergodic, and chaotic dynamics.