I have the following dynamical system: $$ f\left(x_{t},y_{t},z_{t}\right)=g\left(x_{t+1},y_{t+1},z_{t+1};\alpha\right) $$ where $x_{t},y_{t}$ and $z_{t}$ are dynamic variables indexed by time. The two time periods are linked by this parameter $\alpha.$ To find the steady state, I essentially drop the $t$ subscript, as $x_{t}=x_{t+1}$ for instance, and then solve for the value of $\alpha$ such that: $$ f(x,y,z)=g\left(z,y,z;\alpha\right) $$ By doing this, I have assumed first that the variables are unchanging, and what that implies for the parameter value $\alpha.$ However, does this go the other way? In other words, is it also true that for this value of $\alpha$ , $x,y$ and $z$ is unchanging?
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Are $f,g$ scalar or vector-valued functions? – Lutz Lehmann Oct 13 '22 at 19:49
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Scalar functions. – Kwame Brown Oct 13 '22 at 21:25
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Then all this says is that the next point lies on some hypersurface in 3-space. There is not enough information to get unique values for all 3 variables. – Lutz Lehmann Oct 13 '22 at 21:33