My problem is as follows. I have to estimate parameters of Lorenz system using given data. Lorenz system is described by following system of ODEs: $$ \frac{dx}{dt} = \sigma(x-y) \\ \frac{dy}{dt} = x(\alpha-z)-y \\ \frac{dz}{dt} = xy-\beta z$$ where $\alpha , \sigma$ and $\beta$ are the parameters of the model. The data are noisy (Gaussian noise $N(0,0.25)$ was added) observations of the simulated system with parameter values set at $(\alpha,\sigma, \beta)$ = $(28,10,2.66)$
I have used ABC approach to avoid a calculation of the likelihood function. Now I would like to employ a non-Bayesian approach - like MLE. However, I find it difficult as I am not able to calculate the likelihood. Therefore, I would like to ask you if there is any elegant solution to this issue.
Thank you