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At a saddle-focus homoclinic orbit, the leading Shilnikov return map is $f(x)=A{x}^{\delta }\cos (q\log x+\Phi )$, $x>0$, with $q>0$. If $1/2<\delta <1$, its fixed points solve $\cos (q\log x+\Phi )=x^{1-\delta }/A$. Infinitely many solutions accumulate at zero near the zeros of the cosine, with successive size ratios tending to $e^{-\pi /q}$. The derivative is $A{x}^{\delta -1}[\delta \cos (q\log x+\Phi )-q\sin (q\log x+\Phi )]$, so their expanding multiplier is unbounded. Narrow monotone strips near successive oscillations map across a common interval; inverse branches contract, supplying arbitrarily long symbolic itineraries and infinitely many saddle periodic orbits. Varying the splitting parameter shifts the graph vertically, producing accumulating fold and period-doubling thresholds and stable windows. The condition $\delta >1/2$ makes the local flow volume-contracting, whereas $\delta <1$ makes its saddle value positive.

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.

For repeated small returns, ${\xi }_{n+1}=O(z_n^{\delta })$, so $\rho +{\xi }_n$ is close to the nonzero constant $\rho$. Retain its leading value in the second component. Combining $c\cos \vartheta +d\sin \vartheta$ into a single cosine, and absorbing the section size and fixed phases, gives the Shilnikov return mapChanging the sign of the logarithmic angle changes only the phase after using cosine's evenness. A generic reinjection has $A>0$; $\Phi$ can be represented modulo $2\pi$ by a positive value. The original PDF has exponent $\delta$, which the converted TeX mistakenly replaced by $2$.

This is an asymptotic scalar approximation, not an assertion that the full map has an exact one-dimensional invariant quotient. To see why it captures the thin geometry, differentiate the two components in part (a). Their determinant is of orderwhere $C$ involves $\Omega$ and the determinant of the global linear map. For $\delta >1/2$ this tends to zero, so the return is strongly area-contracting even when its longitudinal derivative is large. Its global quadratic corrections are $O(z^{2\delta })=o(z)$ in this range. For a small fixed point, also $\xi =O(z^{\delta })$, so ignoring $\xi$ contributes that same higher order. Multipassage iterates require all passage coordinates to stay in the small domain; the leading map describes their thin folded return geometry rather than controlling arbitrary points outside it.

Let $\theta (z)=\Omega \log z+\Phi$. A fixed point of the Shilnikov return map is a periodic orbit with one marked global excursion, and satisfiesIts local passage takes logarithmically long. If the global travel time tends to $T_g$, then its physical period isEliminating $z$ and absorbing section sizes and $T_g$ into constants $C_1,C_2>0$ and a phase ${\varphi }_0$ gives the leading period–parameter relationFor the idealized leading map with a constant global travel time, the displayed two terms follow exactly; for the original flow, local and global approximation errors qualify their accuracy.

If $\delta >1$, the oscillatory term is $o(z)$ and $f_S^{\prime }(z)\to 0$. In a sufficiently small interval the map is a contraction. For $\mu <0$ there is a unique small attracting fixed point $z=-\mu +O(|\mu {|}^{\delta })$; it is an attracting periodic orbit also in the transverse direction of the full return. For $\mu \ge 0$, $f_S(z)-z=-\mu -z+O(z^{\delta })<0$ for sufficiently small positive $z$, so no periodic return orbit can remain wholly in this neighborhood. As $\mu \uparrow 0$ from below, the attracting cycle becomes the homoclinic connection andContraction also excludes higher-period cycles wholly inside that small return neighborhood. This does not exclude remote periodic orbits elsewhere in the vector field.

If $1/2<\delta <1$, the oscillatory term dominates $z$. At $\mu =0$ the fixed-point equation isThere are infinitely many positive solutions approaching zero, with phases approaching successive zeros of cosine. Their leading ratio is $z_{j+1}/z_j\to e^{-\pi /\Omega }$, so their periods satisfy $T_{j+1}-T_j\to \pi /\omega$. At these roots the sine tends to $\pm 1$, and the fixed-point multiplier has magnitude asymptotic to $A\Omega z^{\delta -1}\to \infty$. Thus these are longitudinally unstable cycles, with the thin transverse direction contracting: infinitely many saddle periodic orbits accumulate on the homoclinic connection. Expanding oscillatory lobes produce interval-covering walks and horseshoe dynamics in suitable iterates; this is the positive-saddle-value case of the Shilnikov bifurcation.

Away from the connection, the one-passage branches wiggle in the period–parameter plane. To leading order,There is no single monotone logarithmic law: the phase matters, and the exponentially smaller $-C_2{e}^{-{\lambda }_{+}T}$ term matters near cosine zeros. Folds occur when $f_S^{\prime }(z)=1$, asymptotically at $\tan \theta =\delta /\Omega$. Their parameter values alternate in sign and accumulate geometrically toward zero. On a fixed-point branch, stability is $|f_S^{\prime }(z)|<1$; small stable windows occur near stationary points of the oscillatory map. Crossing $f_S^{\prime }=-1$ gives period-doubling bifurcations, while $f_S^{\prime }=1$ gives saddle-node bifurcations of periodic orbits. Subsequent doublings and chaotic invariant sets can occur; neither all periodic orbits being stable nor a particular universal sequence for every coefficient choice is implied.

For completeness, an $m$-cycle of the return map, corresponding to a periodic orbit with $m$ global excursions, obeysand its total period isIts longitudinal multiplier is ${\prod }_j{f}_S^{\prime }(z_j)$. This is the relevant period relation for multipassage cycles; substituting their total $T$ into the one-passage cosine law would generally be incorrect.

A three-dimensional hyperbolic equilibrium point is a saddle-focus when its eigenvalues include a nonreal pair and a real eigenvalue with the opposite sign of real part. In the displayed orientation it has a one-dimensional unstable manifold and a two-dimensional spiralling stable manifold; reversing time interchanges the stable and unstable dimensions. The ratio $-{\lambda }_s/{\lambda }_u$ controls local transverse contraction in a Shilnikov return map.