The answer may be, No. recurrent neural networks cannot capture those too-complex patterns even after being trained with more than 100 years of time series data. Perhaps it is not possible with any kind of probability model to exhibit any chaos even with too many constraints and conditions and even with customised activation and design. I am not 100% sure but as far as I know, something more complex like Multiple Time Scales and multiple critical rates of changes or transitions needed to consider modelling any properties of chaos, Recurrent neural networks simply apply the advanced or extended strategies originating from conventional methods like Markov model or hidden Markov models and this kind of model has multiple states and also consider recurrence relation, transitions, ergodicity, periodicity etc. but since the chaos is continuous and it should be model with a continuous system model and also the model should Have the capacity to express all the possible complexities, interactions and influences with the general and all other possible timescales.

It is a very simple fact that a probabilistic model can never ensure the correctness of the predictions and explain the prediction.

I cannot tell how to model or what kind of methods can be used to approach this problem but I have discussed with some prominent and leading mathematicians currently working together to develop the mathematical Foundation To model the complexities and nonlinearities.