Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity

This article extends the study of the dynamical properties of the symmetric McMillan map, emphasizing its utility in understanding and modeling complex nonlinear systems. Although the map features six parameters, we demonstrate that only two are irreducible: the linearized rotation number at the fixed point and a nonlinear parameter representing the ratio of terms in the biquadratic invariant. Through a detailed analysis, we classify regimes of stable motion, provide exact solutions to the mapping equations, and derive a canonical set of action-angle variables, offering analytical expressions for the rotation number and nonlinear tune shift. We further establish connections between general standard-form mappings and the symmetric McMillan map, using the area-preserving Hénon map and accelerator lattices with thin sextupole magnet as representative case studies. Our results show that, despite being a second-order approximation, the symmetric McMillan map provides a highly accurate depiction of dynamics across a wide range of system parameters, demonstrating its practical relevance in both theoretical and applied contexts.

💡 Research Summary

This paper completes a three‑part series on the dynamics of the symmetric McMillan map, focusing on the most general case that mixes quadratic (sextupole‑type) and cubic (octupole‑type) nonlinearities. The authors begin by writing the most general symmetric McMillan map in the standard form
(Q’ = P,\qquad P’ = -Q + F(P))
with a rational force function of degree two, (F(P)=\frac{-B_0 P^2 + E_0 P + \Xi_0}{A_0 P^2 + B_0 P + \Gamma_0}). An invariant (bi‑quadratic) (K(P,Q)) exists, containing six coefficients. By a sequence of canonical transformations—translation to move an existing fixed point to the origin, removal of the constant term in the invariant, and a scaling of the phase‑space variables—the authors reduce the number of independent parameters from six to four. A further choice of the scaling factor eliminates a fifth parameter, leaving only two intrinsic parameters that fully characterize the dynamics:

1. Linear rotation number (a), which appears in the quadratic part of the invariant (K_0 = p^2 - a,p q + q^2). The unperturbed betatron tune (rotation number) at the origin is (\nu_0 = \frac{1}{2\pi}\arccos!\bigl(\frac{a}{2}\bigr)).

2. Nonlinear ratio (B) (or equivalently (A) after the final scaling), which measures the relative strength of the mixed sextupole‑octupole terms in the invariant.

The map can be written in two equivalent normal forms, one emphasizing the sextupole limit ((B\neq0, A=0)) and the other the octupole limit ((A\neq0, B=0)). Both forms are related by a reciprocal rescaling of the parameters, so they describe the same family of maps.

Fixed points and 2‑cycles

Besides the trivial fixed point at the origin, the invariant can have up to four additional critical points, which are the intersections of the symmetry lines (l_1: p=q) and (l_2: p=F(q)/2). Their coordinates are expressed in terms of the intrinsic parameters and radicals (R_1,R_2). Real solutions exist only when the discriminants are positive, defining three important curves in the ((a,B)) plane:

• (B=0) – the sextupole‑octupole degeneracy line where two fixed points coalesce with the origin.
• (R_1=0) – separates regions where the extra fixed points are real or complex.
• (R_2=0) – marks the disappearance of the 2‑cycle by merging it with a fixed point.

A 2‑cycle is defined by the simultaneous equations (p = f_s(q)/2) and (q = f_s(p)/2). Its existence also depends on the sign of (R_0 = B^2 - 4A). When (A=0) (the pure sextupole case) the 2‑cycle reduces to the well‑known period‑2 orbit of the Hénon map.

Stability analysis

The Jacobian matrix of the map is
(J = \begin{pmatrix}0 & 1\ -1-(B^2 + aA)p^2 + 2Bp - a(Ap^2+ Bp +1)^2 & 0\end{pmatrix}).
The trace at any fixed point (\zeta) is (\tau(\zeta)=a + 4A\zeta^2 + B\zeta + 1 - 4). For the origin (\tau = a); for the other fixed points and the 2‑cycle the trace involves the discriminants (R_1,R_2). Linear stability follows the usual symplectic criterion: (|\tau|<2) yields elliptic (stable) motion, (|\tau|>2) yields hyperbolic (unstable) motion. The authors plot the stability domains together with the real/complex domains of the critical points (Fig. 1), providing a clear atlas of parameter space.

Degeneracies and special cases

Several curves correspond to degenerate dynamics:

• (D_{\pm2}) (B=0, (A\gtrless0)) – the map becomes the canonical focusing or defocusing octupole (FO/DO). The invariant acquires an extra symmetry (K(p,q)=K(-q,-p)).
• (D^*_2) ((B^2=(2-a)A)) – the origin and one fixed point lie on the same invariant level, yielding the Duffing‑type octupole (DF) with an unstable fixed point after a shift.
• (D_3, D^*_3) – fixed point and 2‑cycle share a level set, forming an isolated 3‑cycle.
• (S_1) (A=0) and (S_2) (A = B^2/4 >0) – one of the critical points or a component of the 2‑cycle moves to infinity, effectively removing it from phase space. When these singular lines intersect the degeneracy curves, the map becomes periodic with rational rotation numbers 1/3 or 1/4, leading to “super‑degeneracy” where infinitely many integrals of motion exist.

Action‑angle variables and nonlinear tune shift

Exploiting the integrability, the authors construct canonical action‑angle variables ((J,\phi)) by evaluating the contour integral (J = \oint p,dq) on the invariant level sets. This yields an explicit expression for the rotation number as a function of the action, (\nu(J)=\nu_0 + \alpha J + \beta J^2 + \dots), where the coefficients (\alpha,\beta) are closed‑form functions of the intrinsic parameters (a) and (B). Consequently, the nonlinear tune shift (\Delta\nu(J)=\nu(J)-\nu_0) can be computed analytically for any amplitude, a result of direct relevance to accelerator physics where tune spread controls beam stability.

Connection to standard‑form maps, Hénon map, and accelerator lattices

The paper demonstrates that any symplectic map in the standard form can be brought to the symmetric McMillan form by a suitable change of variables up to second order. In particular:

• The quadratic Hénon map, when expanded to second order, coincides with the SX sub‑family ((B=0)) of the symmetric McMillan map.
• A thin sextupole magnet inserted into an accelerator lattice produces a map whose second‑order expansion matches the general symmetric McMillan map with non‑zero (B).

Thus, despite being a second‑order approximation, the symmetric McMillan map captures the essential dynamics of a wide class of physically relevant systems with remarkable accuracy.

Conclusions

The authors have shown that the six‑parameter symmetric McMillan map reduces to a two‑parameter integrable family characterized by the linear rotation number and a nonlinear ratio. They provide a complete taxonomy of fixed points, 2‑cycles, their stability, and all possible degeneracies. Exact action‑angle variables and analytic formulas for rotation number and nonlinear tune shift are derived. By establishing explicit correspondences with the Hénon map and accelerator sextupole lattices, the paper confirms that the symmetric McMillan map is not merely a mathematical curiosity but a powerful, practically useful model for a broad spectrum of nonlinear, area‑preserving dynamical systems.