Bifurcation and chaotic dynamics in a fractional five-dimensional Lorenz–Haken laser model

In this paper, a fractional-order five-dimensional Lorenz–Haken laser system is investigated using the Caputo fractional derivative C D t α , where 0 < α < 1 . The mathematical properties of the model are established by proving the existence and uniqueness of solutions together with phase-space volume contraction in the integer-order limit, nonnegativity of the stored-energy variable, and ultimate boundedness under a gain–saturation condition. For pump values satisfying r > 1 + ζ 2 , the system possesses two symmetric nontrivial equilibria whose local stability is analyzed using the Jacobian matrix in conjunction with Matignon’s fractional stability criterion. Hopf bifurcation analysis of the integer-order limit, combined with Matignon’s criterion for α < 1 , reveals the transition from steady states to oscillatory behavior, while Lyapunov spectrum computations at α = 1 confirm the occurrence of chaos ( λ 1 > 0 ) , with a single positive Lyapunov exponent at the sampled parameter sets, indicating chaos rather than hyperchaos. The slow–fast decomposition further reveals the existence of a transversely stable critical manifold and a locally invariant slow manifold whose persistence is established in the integer-order limit. Bifurcation diagrams, local-maxima sections, phase portraits, Lyapunov spectra, and manifold visualizations support the analytical results. Fractional-order trajectories are computed using the Adams–Bashforth–Moulton predictor–corrector method, and integer-order trajectories using RK4. The nonstandard finite difference Grünwald–Letnikov scheme provides a numerical comparison, while the Adomian decomposition method supplies short-time approximations.