The Ashtanga Yoga discussed here does not so much refer to Patanjali´s philosophy but rather to an Indian hatha yoga system (yoga postures/asanas and breath control/pranayama). It is taught today, for example, by Rangaswamy Saraswathi, the daughter of Shri K. Pattabhi Jois, at the Ashtanga Yoga Nilaya in Mysore (India) and by Manju Pattabhi Jois, the son of Shri K. Pattabhi Jois, in Encinitas (USA). This method of yoga involves synchronizing the breath with defined series of asanas. Ashtanga Yoga was developed to keep people physically and mentally fit (or to heal them). The most important elements (the Primary and Intermediate Series) are "Disease Treatment" (roga chikitsa) and "Nerve Cleansing" (nadi shodana). 

Shri K. Pattabhi Jois (the founder of the Ashtanga Yoga Research Institute) who retired from teaching in 2006 at the age of 91 and was still considered the foremost authority in Ashtanga Yoga until his death in 2009 did not invent Ashtanga Yoga but rather learned it from Shri Tirumalai Krishnamacharya, originally in Hassan, then in Mysore. Krishnamacharya also taught famous yogi(ni)s B.K.S Iyengar, T.K.V Desikachar and Indra Devi. The method has (at least partially) perhaps been conveyed orally for many centuries. The oldest known written source for this yoga system is (according to Krishnamacharya) the manuscript Yoga Korunta ("Yoga Groups") by Vamana Rishi who is said to have described the system on palm leaves. This manuscript has not been preserved - "Ants were eating it!" (Shri K. Pattabhi Jois). It is out of respect for a tradition that may have been handed down over many centuries that Ashtanga Yoga teachers do not change the sequence of the series. 

The first Westerner to learn Ashtanga Yoga from Shri K. Pattabhi Jois (the Primary and Intermediate Series of asanas, as well as pranayama) was the Belgian André van Lysebeth in 1964, the first Westerner to learn the complete system of asana and pranayama (i. e., all Ashtanga Yoga asana series and pranayamas) from Shri K. Pattabhi Jois and his sons Manju and Ramesh was the US-American David Williams starting in 1973. David was subsequently the teacher of, e.g., Nancy Gilgoff, David Swenson, and Tim Miller. He also brought Shri K. Pattabhi Jois and Manju to the USA for their first time in 1975.

What follows are statements from the world´s most experienced teacher of Ashtanga Yoga, Manju Pattabhi Jois.  

Manju Pattabhi Jois: Ashtanga Yoga - Understanding the Method
(English)