From spiral turbulence to spatiotemporal chaos: Numerical and analytical study of a novel CFGL system

We introduce a novel Complex Fisher–Ginzburg–Landau (CFGL) equation that unifies logistic growth dynamics with complex amplitude evolution, creating a versatile framework for modeling spatiotemporal behaviors in excitable and oscillatory media. This hybrid formulation incorporates nonlinear phase-conjugate feedback and higher-order real-valued saturation, allowing it to capture a wider spectrum of instabilities and patterns than classical Fisher or Ginzburg–Landau models individually. We conduct a thorough linear stability analysis and derive a dispersion relation that delineates the regimes of Turing, Hopf, and mixed-mode instabilities. Through multiple-scale perturbation theory, we construct amplitude equations that govern slow modulations near critical bifurcation thresholds. We also propose an energy-like Lyapunov functional to investigate dissipation mechanisms and boundedness of solutions, establishing conditions for pattern onset and transition to chaos. Numerical simulations based on both exponential time-differencing Runge–Kutta (ETDRK4) and split-step Fourier (SSFM) schemes reveal a wealth of emergent structures, including traveling fronts, defect turbulence, multi-core spirals, and asymmetric pattern drift. Remarkably, ETDRK4 schemes tend to generate spot-like Turing patterns, while SSFM captures robust spiral waves, underscoring the sensitivity of the system to numerical treatment. Our findings provide new insights into real–imaginary coupling effects in pattern formation and demonstrate the CFGL model's applicability across diverse domains such as nonlinear optics, chemical reactions, and biological signal propagation. This work establishes a foundational platform for future studies on symmetry breaking, geometric extensions, and complex bifurcation phenomena in reaction–diffusion systems.