Can you have solution around multiple eqiulibrium points for a nonlinear differential equation?

[mint tinsel]

I posted this on the Math Stack Exchange, but have not gotten an answer, so I am reposting it here.

[halcyon pine]

what does the 2 dots above the theta mean

what is g and l?

you have to define what all the variables are - you can't just stick random variables in and not define what they mean.

the biggest question is - are 'g' and 'l' assumed to be constants? or are they also varying?

[mint tinsel]

the dots are a common notation for derivatives, double dots are the double derivative. And they are not random variales, g = constant acceleration due to gravity, and l is the length of the pendulum. It is a common differential equation in mechanical systems and dynamics systems. I am asking a serious question, if you are not familiar with the topic, then please refrain from answering, thank you.

[halcyon pine]

oh yea i remember this in my physics class - barely

a derivative needs 2 variables - what are you taking the derivatie of theta with respect to? I'm going to assume its time. In that case, I'll make x time and y the angle and g and l be constants

that expression at the bottom has no closed-form integral

so that differential equation cannot be solved

what is the taylor series for cosine?

yea, you could probably integrate that

[mint tinsel]

I don't think you understand my question. Usually for a nonlinear differential equation like an inverted pendulum, when talking about control theory and stability of a solution, it is common to use a linearziation of the equation, this of course only an approxmation near that point. For this specific case, using a small angle approximation (same as first order taylor series expansion) a solution near that point becomes doable, and workable. But my question is specifically for a two points of the same system, how one would go around solving this type of question. And as you can see from your solution using integration, it doesn't work, since you're still left with a dy and dx term, which is not what we want.

So my question becomes what method of solution is the best / right way of doing solving a problem like this. In this case, I just made an example of asking for a stable solution around an upright vertical solution, and a horizontal solution. So the question is not if you can solve the inverted pendulum equation, since it doesn't have a closed form solution (https://en.wikipedia.org/wiki/Inverted_pendulum#:~:text=This equation does not have elementary closed-form solutions%2C), but given rather the specific case of given more than one point of operation, how one would go about writting an approximate solution. And I am not interested in a numeric solution either.

[halcyon pine]

so you want to approximate the solution from x=a to x=b where a and b are arbitary numbers?

using like a taylor series expansion?

[mint tinsel]

No. That is not what I am asking for.

[halcyon pine]

if you want me to approximate a differential equation, tell me where to approximate it at