= Log-periodic accumulation of Shilnikov cycles
At a saddle-focus <homoclinic orbit>, the leading <Shilnikov return map> is $f(x)=Ax^\delta\cos(q\log x+\Phi)$, $x>0$, 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 $Ax^{\delta-1}[\delta\cos(q\log x+\Phi)-q\sin(q\log x+\Phi)]$, 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.
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