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I'm puzzled by a question of how dimension reduction influences the bifurcation diagram of a dynamical system.

Say we have a 3-D system with variables $x_1,x_2,x_3$ and we reduce it to 2-D by looking at the difference between the two first variables and the third $y : = x_1 - x_2$, $x_3$. If we in this 2-D system find a parameter value for which a saddle node bifurcation appears, what happens in the 3-D system? $\dot{y} = 0$ does not imply $\dot{x}_1 = 0 \wedge\dot{x}_2 = 0$, which would be necessary for a fix point to appear in the 3-D system. Are fixed points in the 2-D system not generally speaking periodic orbits, and so isn't the bifurcation diagram of the 3-D system fundamentally different?

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  • The problem with your approach is that we cannot just define new variables and proceed from there. Namely: why $x_1-x_2$ out of nowhere? You need to reduce the system so that whatever remains is invariant. The idea is that whatever is thrown away is simple enough to study by itself. At the end you need to combine with this last part, sure. – John B May 15 '18 at 12:28

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