Please tell me why when the ajax is enabled in the admin panel, the names (title) of the categories and subcategories are unchanged during the transition? how to fix this? On your demo site is the same!

No idea what are you talking about.
1) how can you enable ajax in admin panel 
2) what kind of transition are you doing 

Do you change the category name when you move from category to subcategory?

In the admin panel of the site - in the theme settings - enable or disable Ajax - or you do not have it

1) I do not change name when I am moving between categories
2) so it's theme settings not admin panel

Can you send step by step guide how to get issue and what is characteristics of issue?

In the settings of the Veronica theme - the ajax function to enable real-time search for additional fields and to navigate through categories. If one function includes Ajax (Enable direct real-time search without reloading the search page.) Then category titles are not tightened when switching from category to category. How to fix? or is it again I seek from other people for help? or is it your favorite answer - what is the error of the Osclass engine?)