Q:

Can you explain this cyclic number sequence

Can you work out the mathematical rule for this cyclic number sequence?  

2, 6, 186, 8, 456, 54, 53, 23, 7, 301, 3, 21, 2, 6,..... 

Hint 1

 this is a mathematical puzzle

From time to time hints will be updated. 
Note: this sequence cannot be found in The On-Line Encyclopedia of Integer Sequences

A:

The answer appears to be:

 The next number in the sequence is derived from the preceding number by the smallest digit cubed, less the smallest digit squared, plus the largest digit. 

 For a single-digit number $x$ this is simply $x^3 - x^2 + x$.  For example $2$:
 $$2 \Rightarrow 2^3 - 2^2 + 2 \Rightarrow 8 - 4 + 2 = 6$$

 As an example for a three digit number, e.g. $456$:
 Smallest digit is $4$, largest is $6$, so:
 $$456 \Rightarrow 4^3 - 4^2 + 6 \Rightarrow 64 - 16 + 6 = 54$$

 As an example of a two digit number, e.g. $54$:
 Smallest digit is $4$, largest is $5$, so:
 $$54 \Rightarrow 4^3 - 4^2 + 5 \Rightarrow 64 - 16 + 5 = 53$$

