Looking at 突 carefully, it is composed of upper part - 穴 and lower part 犬. What does it mean here?
Logically two possible options:
a) a hunting dog hiding inside a cave;
b) a hunting dog rush out of a cave.
option a) sounds a bit un-reasonable, if the dog hides inside the cave, we should not be able to see it. Then 犬 would not be part of the character 突.
option b) is a much more natural description of the hunting process: suddenly the hunting dog(s) rush out of the cave where they hide to catch the prey.
Before we continue, it will be also helpful to know the ancient scripts of 突: (image taken from: http://vividict.com)
|ancient scripts of 突|
From the picture of 突, we can perceive the vivid meaning of surprise and sudden. for example 突然.
And the dog will become obvious / visible after it jumps out of the cave. so 突 also has the meaning of prominent. for example: 突出.
That is all for 突, now it is time to enjoy a Chinese calligraphy with 突 inside: