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I have a problem that seems similar to the bifurcation problem but that I can't quite seem to mold exactly like that. I have an operator $(I+\lambda A)^{-1}$ acting on a vector $z=x-y$ such that $x$ and $y$ agree for some number of components. For each choice of $\lambda$, there are some number of components of the vector $g=(I+\lambda A)^{-1}z$ such that $g=0$. I want to find the values of $\lambda$ at which the number of $0$ components changes. Since $x$ and $y$ agree in certain places, I believe the number of $0$'s is monotonically decreasing in $\lambda$ with the maximum number of $0$'s in g occuring at $\lambda=0$.

I know very little about bifurcation but this seems sort of like a bifurcation problem (without the derivatives) trying to identify the saddle-node bifurcations. I thought about using power series to write this operator as the derivative of another operator so that I could treat the problem as finding places where the number of equilibria changes.

Does this seems reasonable? If not, can anyone think of a reasonable way to go about this? Thank you so much!

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  • $\begingroup$ using $O$ and $0$ in the same text is never a good idea $\endgroup$ – user354674 Sep 28 '16 at 21:27
  • $\begingroup$ They were both 0 but one of them was within $$ and the other not. It's changed now, thanks for pointing that out. $\endgroup$ – user1801328 Sep 28 '16 at 22:04
  • $\begingroup$ Are you sure that vector $\mathbf {y} =\mathbf {x} −\mathbf {y}$? What means agree, equal maybe? Are there any conditions about A? $\endgroup$ – z100 Sep 29 '16 at 19:32
  • $\begingroup$ whoops. also a typo. Fixed it now. Thanks for pointing that out. $\endgroup$ – user1801328 Sep 30 '16 at 18:39
  • $\begingroup$ Also, there are some conditions on A. I tagged this with splines but didn't say anything about it. A is a matrix of smoothing spline penalties. So component $$A_{i,j} = \int} f^{''}_{i}(x)f^{''}_{j}(x)dx$$ where the f's are the spline basis functions. $\endgroup$ – user1801328 Sep 30 '16 at 18:43

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