We consider Lord Mahavishnu as having ten avatars because they descend to this material plane (Earth).
Doesn't that apply to Lord Ayyappa who is the son of Lord Shiva & Vishnu? Shouldn't he be an avatar of Lord Vishnu?
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The difference between a descended avatāra and a foolish embodied being (dehī) is that avatāra does not forget his spiritual identity in the process and maintains a strong connection with the spiritual plane. E.g. in the Kurukšētra battle Krišna was hit by all kinds of divine weapons, including one that Arjuna could not possibly deflect, but Krišna did not suffer. He could just absorb the weapons, himself being unarmed. Likewise, any avatāra is expected to manifest at least some exceptional supernatural prowess.
In Chaitanya-Charitāmrita 2.20.365 Sanātana Goswāmī asks a tongue-in-cheek question "Who is the avatāra in Kali age?". The question is amusing, because he is asking it to that avatāra himself and they both know that. However, the Master answers:
“O Sanātana, give up your intelligent tricks. Listen to the description of the šaktyāveša-avatāras.
To summarize his answer, an avatāra is recognized by the principal (mukhya) and superimposed (gauna) symptoms. In an avatāra, the manifestation of (divine) energy is direct (sākšāt-šaktye), if it is just a semblance (ābhāse), the person is called "vibhūti" instead. The symptoms to be looked for are described in the šāstras by muni Vjāsa (primary symptoms: name, skin color, dress, place of residence; divine energies: knowledge, love, the powers to create and maintain worlds and to protect and destroy villains; superimposed activities: killing of demons, creation, maintenance and destruction of the worlds and other). He also goes on to mention some avatāras and to describe the qualities and pastimes of Krišna in detail.
while we have not discovered a description in šāstras of Lord Ayyappa as an avatar of Lord Mahavishnu, we cannot accept him as such.
Ayyappa isn't an Avatar of Vishnu, rather his son in union with Shiva.
Quoting the official website of Shabarimala
Manikantan, an alias of Ayyappa was born of Hari (Lord Vishnu) and Hara (Lord Shiva), with Hari assuming the form of a female (Mohini). Hence Ayyappan is also named as Hariharasutan (Sutan meaning Son).
Lord Ayyappa  is known as hari-hara suthan, i.e. the son of Hari and Haran. Hari being Vishnu, and Haran being Shiva.
Vishnu took female avatar as Mohini  to kill a demon, Bhasmasura and also to acquire the amrit , or the elixir.
In this avatar, he mated with Shiva, and the child is Ayyappa. He was then adopted by a King.Since he is Vishnu's son, and not an incarnation of Vishnu himself, he is not regarded as an avatar
Where is the proof that Ayyappa is son of Hari- Hara or for that matter Hari or Hara or even whether he is an incarnation of siva or vishnu or dattatreya? These are mere folk lores which have no basis in any Vedas, ithihasa, 18 main puranas or upa puranas.
For your kind information, there is no mention or reference of Ayyappa by any of the Acharyas before 16th century even by any renowned saiva or sakta acharyas forget about vaishnava acharyas.
My intention is not to offend any one, but where is the proof from the valid hindu scriptures. There are none which indicate even remotely the Ayyappa story.
Ayyappa cult or story is more of a modern origin which has absolutely no valid proof in any Vedas, ithihasa, 18 main puranas or upa puranas.
Lord Shiva is depicted to be the yognath who sits on tiger sin, here tiger skin symbolize sexual desires it means that lord Shiva is the one who destroyed Kama as we all know he killed kamadeva or manmada.
Lord Shiva is said to be having the heart of lord Vishnu and vise versa. There is an ancient proverb in Tamil that those who think lord Shiva and lord vishnu as two are fools. Another important thing is that in kerala we are having a lot of temples for Shankaranarayana, the union of lord Shiva and lord vishnu.
Lord ayyappa is the incarnation of lord dattatreya ( is union of the trimurties) born as the son of Anasooya. It is clear that mahishi should only be killed by Hariharasuta and also the child should not to be born out of a womb, and mahishi thought that it was impossible but for that purpose ie to get a child from their union, one have to be purusha and the other have to be prakriti.
So lord Vishnu took the form of mohini and along with lord Shiva the deposited their powers inside a lotus flower and thus from the union lord Ayyappa was born and not from their sexual union.(One thing to remember here is that god can perform anything in any way but lord Shiva and lord Vishnu didn't like to alter the laws of nature so mohini avatar was taken).
Trimurties are the manifestation of absolute Brahman and ayyappa is the incarnation of the trimurties means lord Ayyappa is the manifestation of absolute. Also lord Ayyappa is regarded as the kula devata of kerala, that's the reason lord sasta is present in almost all major and minor temples of Kerala.
Ayyappan has no reference in any scripture.No sexual intercourse happened b/w Lord Shiva and Lord Mohini. We can find it in Srimad Bhagavatam 8th Canto.