In this analysis, coordinate variables

take on complex values from

(or

). The Hamiltonian function

has four lines of reflection symmetry in a plane spanned by the real coordinates

and

. Each line

and

intersects one circular point and two hyperbolic points. Reflection symmetry allows us to construct real-valued contours,

.

The surface

is a two-dimensional subset of a four-dimensional space and thus resists geometric intuition. Nevertheless, we color the

cross sections and arrange them along

to suggest the underlying topology. The depiction of this Demonstration raises

to

above

and

to

above

. Curve

meets each curve

and

in four real points of intersection. The transformation model preserves dihedral symmetry up to a permutation of colors (dimensions).

To get a general idea of time dynamics along

, we calculate period integrals along the contour curves. Transforming to action-angle variables,

,

reduces the Hamiltonian from quartic to quadratic degree,

.

Solving

for the root

leads directly to the period-energy function,

,

with the loop integral taken around the curve

. Regardless of toricity, the period function

obviously converges on a region

. Adapting an algorithm from [1], we routinely produce a Picard–Fuchs type differential equation. Application of the left operator,

,

to integrand

makes for a sum over derivatives,

.

Such a sum necessarily leaves a remainder on the right-hand side,

. The certificate function

,

contributes to an exact differential

, which integrates to zero on a complete cycle around

, that is,

. The Picard–Fuchs differential equation

follows immediately.

Definition of the complex period function

around either curve

or

follows from a beautiful symmetry. Inversion of parameters,

,

acts on annihilation operator

as a scale transformation,

.

However, annihilation relations are scale free, so the transformation property implies an identity between real and complex periods,

,

with real period defined as

. Both functions

and

have the same series expansion coefficients, quickly generated by a

-recurrence (the Frobenius solution of the Picard–Fuchs equation with two initial values). Our plot of these functions uses a dynamic sum over 500 powers of

to achieve adequate convergence.

The graph of period functions

and

obviously shows energy inversion symmetry at the elliptic fixed point

. This special configuration corresponds to Edwards's family of elliptic curves [2], with genus

and period function

.

The differential equation

takes a hypergeometric form. In straightforward analysis of

and

, we find that boundary values

also allow solutions in terms of Gauss's hypergeometric function

(see [3–5]). However, the stratum limits do not lead to families of elliptic curves. For nonsingular curves

, toricity parameter

determines the genus,

.

To prove the validity of function

, observe the self-intersection topology of curve

in the limit

(see [6]).

The fixed point

is also an expectation value. We give a dataset for the librational motion of a plane pendulum over a wide range of energies [7]. The data allows construction of a likelihood function with variation of the toricity

. It is then possible to fit function

to period versus energy data at

precision.

[3] N. J. A. Sloane. "The On-Line Encyclopedia of Integer Sequences." (Sep 14, 2018)

oeis.org/A002894.

[4] N. J. A. Sloane. "The On-Line Encyclopedia of Integer Sequences." (Sep 14, 2018)

oeis.org/A113424.

[5] N. J. A. Sloane. "The On-Line Encyclopedia of Integer Sequences." (Sep 14, 2018)

oeis.org/A318417.

[6] S. S. Abhyankar and C. L. Bajaj, "Computations with Algebraic Curves",

*Symbolic and Algebraic Computation*:

* ISSAC 1988* (P. Gianni, ed.), Berlin, Heidelberg: Springer, 1989.

doi:10.1007/3-540-51084-2_26.