How does a large-angle pendulum oscillate?

[Peter Jones]

So in high school i studied the small oscillation of a simple pendulum with no air resistance. It reaches harmonic oscillation when the angle is small enough, so it is an approximation right? But what happens if the angle isn't small, will it still oscillate and how? According to energy conservation, it is supposed to. I came across this problem in a Mechanics book from Landau, and the result of cycle is a function of the amplitude of the oscillations. My question is that if this is still considered as a form of oscillation, what type is it?
Here's the picture of the problem from the book.

 

[mpresic3]

It is still considered an oscillaion as Landau states in his Problem statement. It is no longer considered simple harmonic oscillation. It is non-linear in nature. As the problem solution indicates the period takes longer than for simple harmonic motion.

 

[wrobel]

For small oscillations the period does not depend on the amplitude;
for non small oscillations it does
for the cycloidal pendulum the period does not depend on the amplitude even for non small oscillations

 

[Dr.D]

for the cycloidal pendulum the period does not depend on the amplitude even for non small oscillations

What @wrobel says is true regarding the cycloidal pendulum, but it requires a pair of shoulders to constrain the motion.

 

[Chestermiller]

https://www.physicsforums.com/threads/time-average-value-of-pendulum-string-tension.1002681/

 

[wrobel]

There is an interesting problem. The total energy of the pendulum is
$$H=\frac{1}{2}\dot\varphi^2-k\cos\varphi,\quad k>0.$$
Let $\tau(h)$ be a period of the trajectory on the energy level $H=h$.
Find an asymptotic formula
$$\tau(h)\sim \,?\quad\mbox{as}\quad 1)\,h\to k+,\quad 2) \,h\to k-$$

 

[A.T.]

But what happens if the angle isn't small, will it still oscillate and how?

If the deflection angle is π it might get stuck, but otherwise it will oscillate.

 

[vanhees71]

Or it turns around. The solution are of course elliptic functions. There's a plethora of all kinds of approximations for such problems in the literature. Nowadays, I think it's no problem to just use elliptic functions, because they are as easily available via numerics as are the socalled "elementary functions".