This research article mainly focuses on the development of a new six-dimensional dynamical system, principally based on the Lü system. The dynamic characteristics of this introduced system are evaluated and investigated by providing its 2D and 3D plots, time series for each state, discussing bifurcation scenarios and dissipation, the Poincaré map, and calculating its Kaplan–Yorke dimension. This system achieves hyper-chaos with two positive Lyapunov exponents. The generic stagnation points are computed, and their stability analysis is covered. The proposed system has also been studied via circuit implementation on Multisim. Towards the end of the manuscript, hybrid projective combination–combination synchronization among multiple drives and responses is attained through efficient adaptive control. Based on the principles of Lyapunov stability, several nonlinear controllers are created. All the acquired analytical results in the text are validated for their efficacy and viability through simulations on MATLAB.