Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity

Abstract

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.

Figures (14)

• Figure 1: Combined diagram illustrating the stability and real/complex domains for the fixed points $\bm{\zeta}_{2,3}$ (subplot a) and the 2-cycle $\bm{\zeta}^{(2)}$ (subplot b) of the symmetric invariant $\mathcal{K}_\mathrm{s}^+[p,q]$ ($\mathrm{A} > 0$), and for all critical points of $\mathcal{K}_\mathrm{s}^-[p,q]$ ($\mathrm{A} < 0$) (subplot c). Black solid curves represent the boundaries $B_0^\pm$ and $B_{1,2}$, while dashed solid lines denote the singularity $S_2$, corresponding to $r = \pm 2$. Dashed red and orange curves indicate the degeneracies $D_2^\pm$, $D_2^*$, $D_3$, and $D_3^*$. Additionally, vertical lines shown in dotted black and orange represent the asymptotes at $a = -10$ (associated with $B_2$) and $a = -1$ (related to $D_3$ and $D_3^*$), respectively.
• Figure 2: Atlas depicting stable motion regimes around the origin for symmetric invariants $\mathcal{K}_\mathrm{s}^\pm[p,q]$ and $\mathcal{K}_\mathrm{s}^0[p,q]$. The unimodal regime (UM) is shown in gray, the double-well (DW) in magenta, the double lemniscate (DL) in blue, and the simply connected (SC) regimes in varying shades of green. Boundaries of stability ($B$), degeneracy ($D$) and singularities ($S$) are color coded according to Fig. \ref{['fig:StabBIG']}. Additional white curves correspond to the set of parameters with nonlinear tune shift at the origin equal to zero, $\mu_0 = 0$. Auxiliary Figs. \ref{['fig:PhSpaceCN']} and \ref{['fig:PhSpaceSN']} provide typical phase space diagrams for each regime.
• Figure 3: Typical phase space diagrams showing stable trajectories around the origin for the symmetric McMillan map. Isolated fixed points and $n$-cycles, along with their corresponding level sets, are highlighted in color, while other level sets are depicted in black. The plots are schematically arranged in the parameter space $(a,\rho)$ for $|a|<2$ and $\rho > 1/4$. For more details, refer to Fig. \ref{['fig:StabSML']}. The plane is delineated by $B_{1,2}$ representing saddle-node (SN) and period-doubling (PD) bifurcations, degeneracies $D_2^+$, $D_2^*$, and the singularity $S_2$.
• Figure 4: Similar to Fig. \ref{['fig:PhSpaceCN']}, but illustrating simply connected regimes for $|a|<2$ and $\rho < 1/4$. The parameter space is outlined by degeneracies $D_2^-$, $D_3$, $D_3^*$, and the singularity $S_1$.
• Figure 5: Atlas illustrating the space of intrinsic parameters for the invariant $\mathcal{K}_\mathrm{s}^0[p,q]$: the rotation number at the origin $\nu_0$ (linear tune) plotted against the nonlinear parameter $\rho$. The chart is color-coded according to the regimes with stable trajectories around the origin (see Fig. \ref{['fig:StabSML']}). The thick orange curve represents the parameter set $\rho_n(\nu_0)$ corresponding to the second-order approximate invariant $\mathcal{K}^{(2n)}_\text{SX-2}$ for the area-preserving quadratic Hénon map. The white curve indicates the parameter set where the nonlinear tune shift at the origin is zero, $\mu_0 = 0$. The two bottom plots (b and c) provide magnified views of the areas outlined in red.