Small-signal amplification of period-doubling bifurcations in smooth iterated maps

Various authors have shown that, near the onset of a period-doubling bifurcation, small perturbations in the control parameter may result in much larger disturbances in the response of the dynamical system. Such amplification of small signals can be measured by a gain defined as the magnitude of the disturbance in the response divided by the perturbation amplitude. In this paper, the perturbed response is studied using normal forms based on the most general assumptions of iterated maps. Such an analysis provides a theoretical footing for previous experimental and numerical observations, such as the failure of linear analysis and the saturation of the gain. Qualitative as well as quantitative features of the gain are exhibited using selected models of cardiac dynamics.     

7,8‐(Dichloromethano)homo[70]fullerene,an Unusual,but Unexceptional,Methanoannulene

The crystal structure of 7,8‐(dichloromethano)homo[70]fullerene shows that addition of the dichlorocarbene (:CCl₂) to C₇₀ gives a surprisingly normal methano‐bridged [9]annulene. The structural changes caused by the addition are highly localized; the C(7)? C(8) distance lengthens by 0.705(2) Å but no other distance changes by more than 0.022(3) Å. Bond angles within the [9]annulene increase by as much as 7.8(1)°, but changes outside the C₉ ring are much smaller. The nearly ellipsoidal molecules form hexagonally close‐packed layers that interact in pairs through short Cl???C, and possibly short Cl???Cl and C???C, interactions.     

Synthesis of isomerically pure carboxylate- and sulfonate-substituted xanthene fluorophores

Xanthene-based fluorophores such as fluorescein and rhodamine are typically prepared by acid-catalyzed condensation of the appropriate resorcinol or 3-aminophenol with phthalic anhydride. Condensation of substituted phthalic anhydride species results in functionalized fluorophores that are formed as mixed isomers. Crystallization approaches to isomer separation have been reported elsewhere for symmetric fluorescein carboxylates. We describe crystallization-based separation of protected fluorescein sulfonates and coupling conditions to form sulfonamides, precursors for carboxylate-substituted rhodamines, and precursors for asymmetrically substituted fluoresceins and rhodafluors.