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Non-Linear Behavior of Iterated Functional Maps

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The universal behavior of certain iterated nonlinear function maps (ie period doubling bifurcation route to chaos): $$x_{i+1}=f(x_i)$$ have been known since Feigenbaum: (see http://theworldismysterious.wordpress.com/2013/10/03/441/)

The usual method of solving the Hartree-Fock equations for interacting fermions is a nonlinear iterated functional mapping: $$f_{i+1}(x)=F[f_{i+1}(x),V_i[f_i(x)]]$$ where $F$ is a non-linear functional of the function $f(x)$. Here $f(x)$ represents the fermion orbitals and $F$ is an integro-differential operator that is non-linear in $f$ because it contains both $f$ and an effective potential function $V$ that also depends upon $f$. The usual method of solution is to guess an effective potential $V_0$ that is used to generate a set of orbitals $f_0(x)$. These orbitals are then used to generate a new effective potential $V_1$ and together they are used to generate a new set or orbitals via the functional iteration above. A solution is obtained when the iteration yields an equivalent set of orbitals on two successive iterations.

Since this mapping is nonlinear it is conceivable that convergence of the iteration may not occur and that something related to the period doubling bifurcations of nonlinear iterated function maps may result. In fact, I found such a situation during research I conducted in 1975.

  1. My question is: has anyone else encountered such situations either in Hartree-Fock calculations or any other physics calculations that employ nonlinear iterated functional mappings?

  2. A secondary question is: are there published mathematical investigations of nonlinear behavior in iterated functional maps?

This post imported from StackExchange Physics at 2015-12-28 22:31 (UTC), posted by SE-user Lewis Miller
asked Dec 28, 2015 in Theoretical Physics by Lewis Miller (10 points) [ no revision ]

1 Answer

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Chaos in nonlinear iterated mappings is very common in higher dimensions - in all sorts of applications, when started far from a fixed point.

But proving its presence rigorously is difficult, as one needs to have full control over a significant part of the space on which the map is defined. See for example my papers with Thomas Rage and papers citing it.

To avoid entering a chaotic regime numerically and enhancing the chance of converging to a fixed point, numerical methods for solving nonlinear systems usually include a damping strategy based on some form of a line search. See, e.g., the last chapter of my book on numerical analysis.

answered Dec 31, 2015 by Arnold Neumaier (12,570 points) [ no revision ]

@ArnoldNeumaier Thanks for answering my post.  What I actually observed was not chaos but truncated period doubling bifurcations (the cascade stopping after a few bifurcations as the map parameter varied).  I have seen experimental papers that report such truncated behavior and they attributed the truncation to noise.  Since I was solving the HF equations on a numerical grid, I wonder if the truncation I observed was due to rounding error and if the true infinite dimensional (functional) mapping I was approximating really does exhibit chaos?

@LewisDudleyMiller: To check whether it is a truncation phenomenon, you need to change the grid and repeat the calculations. If the number of bifurcations changes, it is a numerical artifact. In any case, you are likely close to chaos, and would probably find a chaotic problem by changing the Hamiltonian a bit. If you want to check chaoticity numerically in spite of approximations, you'd have to create maps such as in my work with Rage that show the breakdown of the integrable structure.

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