What is the difference betwen a time non-homogenous Markov Chain and a non-linear Markov Chain? Example

A time non-homogenous Markov Chain is one in which the transition probabilities are not constant over time. A non-linear Markov Chain is a model that is not linear in parameters and satisfies the Markov property (independence of future and distant past conditional on present/recent past).

(1) What is the relationship between time non-homogeneity and non-linearity of a Markov chain? (2) If time non-homogeneity and non-linearity are identical, please provide an intuitive, real world, example of such a Markov Chain process. If they are different types of Markov chains, please provide an intuitive, real world example of each.

your current state can be expressed probabilistically $$S_i=\{Pr(X=1)=p_1,Pr(X=2)=p_2,...,Pr(X=n)=p_n\}^T$$
all the previous states in theory could affect the current state $$S_i=f_i(S_{i-1},S_{i-2},...,S_{0})$$
time homogeneity means f is always the same $$f_i \equiv f$$
linearity means the function $$f()$$ can be expressed as a linear transform, i.e. multiplying a matrix, which in the order 1 and homogeneous case is $$S_i=Q \cdot S_{i-1}$$ but may also be non-homogenous $$S_i=Q_i \cdot S_{i-1}, \space S_{i-1}=Q_{i-1} \cdot S_{i-2} ...$$