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Suppose you have a linear Markov process, and you can write it as x(t+1) = Ax(t). Here x is the vector of values, and A is the transition matrix. Since this is linear, it can be solved analytically, and simulated numerically. A nonlinear form of the above system is x(t+1) = A[x(t)]x(t). Here A[x(t)] is the transition matrix, whose entries depend on current x(t) values. Due to the nonlinearity, this has not analytical solution. But, how can I run some simulations to see the evolution of such a system. For the simple case, I can use A[x(t)] = A(x(t)^0.5), or any other exponent.
Is there a way to do a simulation like this (e.g., 100 steps)?

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If you make your A matrix a function (but avoid starting with capital letters), like aMatrix[x_] := ... then you can use Nest:

Nest[aMatrix, x0, 100]

where x0 is the starting vector.

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