Generally speaking, wolves tend to grow bigger than lynx; a lynx tops at around 30 kg whereas a wolf can weigh 40 or 45 kg. Wolves have bigger heads and jaws, and most importantly, they are pack animals. In other words, wolves should be very dangerous to a lynx. But surprisingly, lynx manage not only to thrive in the same forests as wolves (say, in Belarus, where this picture was taken IIRC), but to actually inspire fear in wolves.

In order to survive among pack-hunting wolves, lynx have to be EXTRA ferocious. Lets analyze what that means. If ferocious means frightening, the lynx, like all cats, has it down to a science. Cats are masters of intimidation because the best fight is the one you dont have to fight; if a wolf gets hurt in a fight, theres a decent chance it will be able to survive because other wolves will continue hunting for the pack, so it will be able to feed and regain strengths as it heals. A lynx, being solitary hunter, doesnt have that luxury. That means if a lynx gets hurt in a fight, it will risk starvation if it cannot hunt. So it is much more pressing for the lynx to avoid fights as much as possible, and that means it has perfected its disuasory apparatus- like other cats, it has a very expressive face, it can arch its back and puff up to appear bigger, as you see in the picture, it can bare its fangs (which are proportionally bigger than any canids fangs), it can make a series of terrifying noises from hissing to growling to howling and screaming, and it can mock charge or strike with its paws at a rival or attacker. All of this makes it very clear to the wolf, or to any enemy, that the lynx IS dangerous. It doesnt even have to touch you; you just dont want to mess with the lynx because it makes it blatantly obvious that it is. The result is that the wolf will likely not want to mess with the lynx. And we know thanks to scientific observations that the scent markings left by lynx on trees and on the ground actually seem to scare wolves, which means they have, after a very long time of coexistence, learned that lynxes are scary.

Now the violent part. If the wolf were to attack anyways, the lynx WOULD fight to the death. This is because, again, it relies exclusively on itself to survive. No one will care for it if it gets hurt. So it uses its formidable weaponry to really fuck the wolf up in a matter of seconds; if you have ever been on the receiving end of a frightened or angry housecat, say, at the vets office, you know how incredibly fast they are and how much damage they can do in the blink of an eye. The wolf, formidable as it is, is no match for a lynxs reflexes. Moreover, the lynx has retractable claws much sharper than the wolfs, as well as much greater arm motion RANGE It can even kick with the sharp claws of its hind feet too. The risk of engaging an enemy in a fight is high for the lynx but when they do fight, they fight desperately, and violently.

Not only that; because of the disadvantage they have being solitary hunters, cats such as lynx, cougar, or leopard, will take any opportunity they have to destroy other predators when they see a chance. Although a lynx will prefer to avoid a face-on confrontation with a wolf, if it finds a wolf den with young pups it will kill them. Sometimes, it will take an adult wolf by surprise. There are records of male lynxes killing adult wolves that were guarding a den, and when this happens, wolves usually abandon the den and find another. Lynxes SCARE wolves, and sometimes kill them, because they need to be that ferocious to survive among wolves. Same happens with cougars/mountain lions; alone, they are at a disadvantage against wolf packs, but they will definitely kill wolves if they find them alone. Leopards will kill any other predator they can overpower, too. Same story. By killing other predators they increase their own odds of survival, get rid of potential dangers and competition, and yes, train them to fear them, lessening the chances of being attacked themselves.

Lions are the big exception because they are social cats; they live in groups called prides. They do however share many of the behaviors of other cats including their tendency to go after other predators to kill them whenever they have a chance. They are also extremely territorial because, being so big, they need enormous ranges with abundant large prey, conditions that are not exactly commonplace, and so the rivalry between lion prides is intense. In order to ensure the survival of the pride, lions have to be ferocious. Same goes for tigers which are solitary but still need huge territories. Big cats often end up dying in territorial fights; they lead violent lives in which every day is a struggle to survive, and no one will come to their aid if they get hurt. Thats why they need to be so fierce.