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I need an example for pure deterministic time series which is not just dependent on time but on previous values, i.e $Y(t) = βY(t-1) + f(t). It should not have stochastic trends in it. I need this for academic purpose.

Looking for an example of actual data, where I can explain the terms comparing to real world

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Exponential Decay formula would work.

is the remaining amout of material of the decay cycle is complete.
If you start with 100 grams of x this is your
If your decay rate is 2% this is your r.
If you do 1 second increments that would be your t.
The remaining amount becomes your new initial amount and you

Initial Amount         Time          Decay Rate        Remaining Amount
 100 grams              1 sec         2%               98.0198673307 grams
 96.0789438852 grams    1 sec         2%               94.1764532455 grams
 94.1764532455 grams    1 sec         2%               92.3116344834 grams
 92.3116344834 grams    1 sec         2%               90.4837411776 grams

It would take 230.258509299 seconds to only have 1 gram of material left.

Actual examples of academic research, you can find at Exponential decay--Applications and examples

Edit: In my opinion another way of calling what you want would be a non-stochastic iterative time series.

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