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I would like to compare prediction qualities of three different dynamical models (model A has three free parameters, model B has two free parameters and model C has zero free parameters).

To achieve this I computed correlation coefficients between simulated and empirical time series for each model and tested parameter set.

Now I would like to compute Akaike's information criterion for each model to formally compare the three models.

How do I compute the (maximum of the) likelihood function for each of the three models using the obtained correlations coefficients between simulated and empirical time series?

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To calculate an AIC, you need a likelihood. Correlation coefficients are not a likelihood (crucially, they don't change when you change the data size, while the likelihood is proportional to the number of observations).

Here is an example of how to define a likelihood for a dynamical model https://www.rdocumentation.org/packages/BayesianTools/versions/0.1.3/topics/VSEM

Note that you will have to recalibrate your model parameters, as the parameters might depend on the definition and parameters of your error term, which is part of the likelihood.

For a time-series model, you should probably specify a likelihood that accounts for autocorrelation, e.g. AR1

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