Accretion onto stars with octupole magnetic fields: Matter flow, hot spots and phase shifts

► Accretion onto stars with complex magnetic fields has been investigated in global three-dimensional simulations. ► The field is taken as a superposition of the dipole and octupole moments. ► Cases of different relative strengths of the octupole and dipole fields were investigated. ► New models of phase shifts based on the complexity of the field were proposed.