Disclaimer: I am a new learner when it comes to time series.
Typically, before obtaining summary statistics like the mean, or applying models like ARMA, you would want to transform your time series to a stationary process using differencing or log transforms.
But consider that your time series follows a sigmoid shape. In this case, no amount of differencing will convert the time series to a stationary process. I assume this is because the time series is not resultant of a (potentially latent) continuous random variable with a fixed mean and variance. But rather a variable that is a function of time.
Because of this, you are not able to do any kind of ARMA modeling, or determine summary statistics like the mean.
What happens then? Can you just not do any analysis on it?