I had searched for the same and discussed with many learned people in my vicinity, sent emails to scholars for understanding the concept of incarnation, but in vain.
Finally, I came to a conclusion that the concept of avatar (incarnation) is Puranic.
According to my understanding the order in which the spiritual/religious Literature came into existence in the following manner (correct me if I am wrong).
- Vedas 2. Upanishads/Ramayana 3. Mahabharata 4. Puranas
The concept of incarnation was not mentioned in Ramayana. It first appears in Mahabharata, wherein it was mentioned that Vishnu took incarnation as Varaha/Vamana/Nrisimha. As Mahabharata contains interpolations, mention about Vishnu's incarnations itself may not be genuine.
The incarnations of Vishnu/Shiva are all Puranic concepts.
Further, the division of time into yugas itself, as per my understanding, Puranic, ie., started with the Puranas only, but not even mentioned in Ramayana, which pre-dates Puranas.
In Ramayana there was a mention about earlier period as कृतयुगे. In the entire Ramayana there was no mention about division of time into 4 Yugas.
पूर्वं कृतयुगे तात पर्वताः पक्षिणोऽभवन् |
ते हि जग्मुर्दिशः सर्वा गरुडानिलवेगिनः ||
(Sundara Kanda 1st Sarga 123rd Sloka)
Interestingly, there was a mention about Varaha in Ayodhya Kanda of Ramayana, which was assumed not by Vishnu, but by Brahma.
सर्वम् सलिलम् एव आसीत् पृथिवी यत्र निर्मिता |
ततः समभवद् ब्रह्मा स्वयम्भूर् दैवतैः सह ||
(Ayodhya Kanda 110 Sarga 3rd Sloka)
"All was water only in the beginning from which element the earth was formed. After that, the self-existent Brahma with all the gods came into existence."
स वराहः ततो भूत्वा प्रोज्जहार वसुंधराम् |
असृजच् च जगत् सर्वम् सह पुत्रैः कृत आत्मभिः ||
(Ayodhya Kanda 110 Sarga 4th Sloka)
"Thereafter, that Brahma, assuming the form of boar, caused the earth to rise from water and with his sons of pure soul, created the entire world."
In Ramayana, there was no mention about Matsya avatar!!