## Studying a nonlinear system under constraints in linear fashion

Suppose that $a = aBc$ where $a$ and $c$ are vectors and $B$ is some matrix that changes as time "continuously" goes on - making this system dynamical system. But suppose that at any time, if $B$ is linearized into square matrix, it is proven that it is impossible to satisfy $a = aBc$. Does this mean that in non-linear case, there would not be any case that satisfies $a = aBc$? Also, if nonlinear case allows satisfying the equation, what would be the condition? If the latter question is somehow ambiguous, answer to the first question is fine.

Caution: $a$ is transposed form of a vector. About $a$ and $c$ - what happens if we separate the case into two - one that has $a$ and $c$ change as time goes on(which means $B$ changes) and one that has these two fixed?

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 Tags dynamical system, linear algebra