New Insights into a Caputo Fractional-Order Lorenz-like System

The present study is concerned with heteroclinic trajectories of a Caputo fractional-order Lorenz-type model (0<α≤1), a problem that has remained unresolved. Building upon previously known findings, we initially deduce two asymmetric heteroclinic connections that link the globally attracting equilibrium E0, the repelling equilibrium E+ (or E−), and the globally attracting equilibrium E− (or E+), where E0=(0,0,0) and E±=(bd±b2d2+4bc2,bd±b2d2+4bc2,c+d(bd±b2d2+4bc2)). Additionally, a novel heteroclinic trajectory is identified for two separate regimes: (1) joining the repelling E0 with the globally attracting E+ (E−, respectively), alongside a locally attracting E− (E+, respectively); (2) joining the repelling E− (E+, respectively) with the globally attracting E+ (E−, respectively), alongside a locally attracting E0. All analytical claims are supported by numerical experiments.