Physics-informed symbolic regression and Haar wavelet approaches to study a new fractional-order 3D chaotic system with no equilibrium

Chaotic systems are crucial for security and signal tasks, but many prior systems need higher dimensions or complex nonlinearities, and most give limited validation of fractional-order numerics and security design. This manuscript investigates a new 3D chaotic system containing an absolute-value nonlinearity. The proposed model exhibits no real equilibria and illustrates interesting robust chaotic behaviors affirmed by bifurcation diagrams, Lyapunov exponent, and sensitivity analysis. We generalize the considered new model to the fractional-order system with aid of the Caputo fractional operator. The Haar wavelet method is utilized to derive the numerical results rigorously for the fractional-order system. We portray its dynamical behavior for different fractional orders to show hidden chaotic dynamics. Phase-space portraits affirm the existence of dissipative chaos even at fractional orders ρ<1. A physics-informed symbolic regression framework is implemented to reformulate governing equations from simulated data, attaining high prediction fidelity. On the top of that, the fractional-order system is utilized to gray scale and red-blue-green image encryption. Extensive cryptographic metrics, such as entropy, number of pixels change rate, unified average changing intensity, and correlation coefficients, verify the strength of the algorithm in resisting statistical and differential attacks. The high dimensionality, structural sensitivity, and parameter-tunable complexity of the model make it a powerful tool for uses in secure communication and nonlinear signal processing.