Is it possible to control the strength of the vibration of the phone when ringing? When the phone is ringing, is it possible to set such that it gradually increase the strength of vibration instead a predefined strength of vibration? (When the vibration mode is turn on but the ringer tone is muted)

The concept is something like the 'gentle alarm clock' that will wake me gently, rather than jolting me from my slumber. (It work by having the alarm that gradually grows louder and louder until it becomes impossible to sleep through.)
No. Android doesn't give apps a way to control the strength of the vibration, so you won't find an app that can do this for you.

That said, there's an app that claims to be able to change the vibration strength on Nexus 4 and Nexus 5 phones, using a device-specific method that requires root. Some users report that it doesn't work at all, but if it does work and you have a supported device, and your device is rooted, you may be able to find an app that does what you want.