I think you're misunderstanding how single sign-on works.
Lets consider website1 and website2 who want to use single signon.
A login website is created at identityProvider. This is the only place where a logon screen appears.
When the user visits website1 and choose to login website1 sends the user to the logon screen at identityProvider. The user logs onto identityProvider which drops its own login cookie for its domain (and perhaps allows the user to save their authentication information so they're never prompted again). It then redirects the browser back to website1 including a token in the request which website1 cracks open, gets identity information from and performs it's own login bits (dropping it's own authentication cookie which lasts for however it wants).
Then the user visits website2 and selects logon. Website2 bounces the user to identityProvider, who already knows who the user is and, if they user has chosen to save their login information, silently authenticates and then redirects back to website2 with another token which website2 cracks open and then performs its own login bits.
There's a bunch of security around it, limiting tokens to particular websites, only allowing tokens to be sent to whitelisted web sites etc. etc.
So to address your concerns
- User logs on website1 and then moves to website2. How website2 will know user has logged in? It doesn't. website2 must request authentication information from the single signon site first.
- That means I need to marshall all the urls in website1 which takes to website2? No, unless you make website1 the identity provider too. Even then that would be painful, better to have website2 redirect back to the identityprovider if a token is necessary.
- Secondly if user continue to browse website2 for say 1 hour and then move to website1. By that time website1 session has timed out so user will see a login page, isn't it? - It depends how you configure website1, and how long it's authentication cookie lasts for.
- But this behavior is wrong as per single sign on functionality. No it's not. Single signon does not mean you get a floating token that is shared between sites. Each website which uses the single sign-on still creates their own authentication cookie. What might happen is if the user goes back to website1 it detects an expired authentication cookie, then sends the user off to the single signon page again where they're authenticated (silently) and a new token is pushed back to website1 which creates a new authentication cookie for itself.