Chaos and Complexity in a Novel Three‐Dimensional Nonlinear System: Analysis, Bifurcations, and Applications

ABSTRACT This study introduces a novel three‐dimensional nonlinear system featuring quadratic, cubic, quartic, and quintic nonlinearities, with inherent symmetry about the ‐axis. We analyze its fundamental dynamical properties, including equilibrium points and their stability, dissipative behavior, and high‐periodicity limit cycles. The system's chaotic nature is rigorously examined through Lyapunov exponent spectra, fractal dimension analysis, and Poincaré maps, revealing sensitivity to initial conditions and parameter variations. A detailed bifurcation analysis identifies critical transitions, including Hopf bifurcations, using both numerical simulations and theoretical criteria such as the Routh‐Hurwitz stability conditions. Notably, the system exhibits rich dynamics, from stable equilibria to chaotic attractors, with fractional Lyapunov dimensions confirming its complexity. Practical implications are underscored by parameter ranges yielding chaotic trajectories, validated through circuit‐compatible amplitude constraints. The system's versatility suggests promising applications in cryptography, secure communications, and random number generation, motivating further exploration of its nonlinear phenomena.