In this work, we propose a new two-dimensional chaotic map defined by sinusoidal and cubic nonlinear interactions. The system is characterized by the presence of a geometric attractor with particularly complex and symmetrical properties. This structure is validated by detailed phase portraits supported by a maximal positive Lyapunov exponent. A bifurcation diagram also reveals rich dynamical transitions, including the appearance of chaos as system parameters vary, showing that chaos is a transient phase of the system. Taking advantage of these strong chaotic properties, we design a pseudo-random number generator (PRNG) based on map iterates. The proposed PRNG generates binary sequences by extracting bits from the decimal parts of chaotic trajectories and storing them in binary format for evaluation. The quality of the generated sequences is rigorously tested using the NIST SP 800-22 statistical suite and ENT test respectively, and the results confirm high-quality randomness. Furthermore, entropy is close to optimal levels, and autocorrelation analysis demonstrates statistical independence between bits. Given its simplicity, high sensitivity to initial conditions and excellent random characteristics, this PRNG is a promising candidate for applications in cryptography, secure communications and stochastic simulation systems.