The linear scarring theory outlined above has been later extended to include nonlinear effects taking place after the wave packet departs the linear dynamics domain around the periodic orbit. At long times, the nonlinear effect can assist the scarring. This stems from nonlinear recurrences associated with homoclinic orbits. A further insight on scarring was acquired with a real-space approach by E. B. Bogomolny and a phase-space alternative by Michael V. Berry complementing the wave-packet and Hussimi space methods utilized by Heller and L. Kaplan.