Existence of Heteroclinic Orbits in Fractional-Order and Integer-Order Coupled Lorenz Systems

Applying two Lyapunov functions and the concepts of α-/ω-limit sets, this paper reexamines fractional-order and integer-order coupled Lorenz systems and simultaneously proves the existence of twelve heteroclinic orbits, i.e., four ones to S0 and S5,6,7,8, four pairs of ones to S1 and S5,7, S3 and S5,6, S2 and S6,8, S4 and S7,8 when r−1>0, b≥2σ>0 and ac<0. These orbits have not been reported in existing studies on coupled Lorenz-type systems and are verified via numerical simulations. The findings not only uncover new dynamics of the Lorenz system family and expand the application scope of Lyapunov-based methods but also provide insights into heteroclinic orbits of other fractional-order and integer-order Lorenz-like counterparts.