Integer–Fractional Order of Identical Eigenvalues Chaotic Oscillator: Analysis and Shadow Economy Application

This study introduces a new three-dimensional chaotic oscillator system characterized by zero eigenvalues, with stability localized in the center manifold, an uncommon feature in chaotic system design. The proposed system is constructed entirely from nonlinear terms and demonstrates complex dynamics validated through bifurcation analysis and Lyapunov exponent computation. The results of this work are the application of the system to model the dynamics of the shadow economy, where the variables represent corruption, enforcement, and hidden economic activity. The model captures the unpredictable feedback interactions inherent in such systems and illustrates how minor changes in parameters lead to vastly different long-term outcomes. Furthermore, a fractional-order extension of the system is investigated using the Caputo derivative to determine the effects of memory in chaotic evolution. Numerical simulations reveal that fractional orders significantly influence attractor behavior, with transitions from chaos to regular dynamics. This paper contributes a structurally novel chaotic model, a fractional-order analysis framework, and an application to economic dynamics—providing valuable insights into both chaos theory and economic system modeling.