This is what the SFAuthenticationManager class is for.  You can call [[SFAuthenticationManager sharedManager] logout], which will clean up app state and raise the kSFUserLogoutNotification (@"kSFUserLogoutOccurred") notification.  If you look at the code in AppDelegate.m from either the native sample apps in the SDK workspace, or in the template app that you generate with forceios, you'll see that, in the init method of the class, the app adds an observer for this notification:

[[NSNotificationCenter defaultCenter] addObserver:self selector:@selector(logoutInitiated:) name:kSFUserLogoutNotification object:[SFAuthenticationManager sharedManager]];

 And logoutInitiated:, in this instance, brings up the initial set of view state and prompts the user to login again:

- (void)logoutInitiated:(NSNotification *)notification
{
    [self log:SFLogLevelDebug msg:@"Logout notification received.  Resetting app."];
    [self initializeAppViewState];
    [[SFAuthenticationManager sharedManager] loginWithCompletion:self.initialLoginSuccessBlock failure:self.initialLoginFailureBlock];
}

Your app, of course, can override this behavior in any way you see fit.  But it sounds like this is pretty much the behavior you're hoping to implement in your app, so you should be able to basically copy and paste that code to make it work.