 Solution

The normal form is invariant under $z\mapsto e^{i\phi }z$. This continuous rotational symmetry makes the axis invariant and reduces the nontrivial geometry to two amplitude variables. The axis connection from $E_{+}$ to $E_{-}$ persists without parameter tuning, while rotating the outer connection produces a family of connections from $E_{-}$ back to $E_{+}$. A two-dimensional autonomous amplitude flow cannot by itself produce the proposed saddle-focus return chaos: symmetry reduction hides the phase-dependent reinjection needed for that mechanism.

Generic higher-order terms need not retain this exact rotational symmetry. A normal form can preserve it to a finite truncation order while the full equations have small symmetry-breaking remainders. These unfold the special connection geometry into phase-dependent returns; nearby parameter curves can contain homoclinic orbits to a saddle-focus, and the Shilnikov bifurcation mechanism becomes available. Their exact placement depends on the omitted terms and is not fixed by the displayed truncation alone.

For example, at the perturbed positive-axis saddle, the real unstable eigenvalue is ${\lambda }_u=2a$ and the stable complex pair has real part ${\mu }_1-2a+a^2$. Near the calculated connection curve ${\mu }_1=-a^2/4+o(a^2)$, the contracting ratio isThis is the positive-saddle-value regime, explaining why the relevant saddle-focus return can be expanding.

In the Shilnikov return map, take a positive return coordinate $y$ and $0<\delta <1$, with nondegenerate $A\ne 0,B\ne 0$. At the connection parameter $\mu =0$, the fixed point condition isThe right side tends to zero, whereas the phase passes through infinitely many oscillations as $y\downarrow 0$. Hence there are infinitely many positive fixed points accumulating at zero, with logarithmically spaced amplitudes. For these points,whose magnitude tends to infinity along the accumulating fixed-point sequence. They represent long-period saddle periodic orbits of the full return, with period growing like ${\lambda }_u^{-1}\log (1/y)$, rather than infinitely many stable cycles.

Successive turning branches of this oscillatory return can stretch and fold across suitable intervals. The full saddle-focus Poincaré return map then admits invariant Smale horseshoes and chaotic itineraries near a generic homoclinic orbit at a Shilnikov bifurcation. Varying $\mu$ unfolds accumulating saddle-node and period-doubling events. Whether any resulting invariant set attracts trajectories depends on the other return direction and the omitted terms; the leading scalar map alone does not prove a chaotic attractor for every perturbation. Its domain also excludes negative iterates, which leave the chosen return branch. The degeneracies $A=0$ or $B=0$ would remove the oscillatory mechanism, so the conclusion uses generic, genuinely saddle-focus reinjection.