Chaos in eigenvalue search methods

Eigenvalue searches for multiplying systems emerge in several applications, encompassing the determination of the so-called alpha eigenvalues associated to the asymptotic reactor period and the adjustment of albedo boundary conditions or buckling in assembly calculations. Such problems are usually formulated by introducing a free parameter into a standard power iteration, and finding the value of the parameter that makes the system exactly critical. The corresponding parameter is supposed to converge to the sought eigenvalue. In this paper we show that the search for the critical value of the parameter might fail to converge for deep sub-critical systems: in this case, the search algorithm may undergo a series of period doubling bifurcations (leading to a multiplicity of solutions) instead of converging to a fixed point, or it may even crash. This anomalous behaviour is explained in terms of the mathematical structure of the search algorithm, which is shown to be closely related to the well-known logistic map for a few relevant applications illustrated in the context of the rod model. The impact of these findings for real-life applications is discussed, and possible remedies are finally suggested.