The words in पुं. भवान् and भवती masculine and feminine are used in Sanskrit language to show respect. This is a very well-known fact. 'Anyarupa' (अन्यरुप) of the verb is used when the above words are used. So there is no change / difference in meaning when the words भवान् and अत्रभवान् or भवती and अत्रभवती are used, but the idea of respect is expressed (When the person is absent the word तत्र is used in the place अत्र e.g.. तत्रभवान्, तत्रभवती)
Here is this lesson to express respect for Kaikeyi in the sentences विजये, ममागमनं निवेदय अत्रभवत्यै । and दिष्ट्या अनपराद्धा अत्रभवती । spoken by Bharat अत्रभवती is used.