The discretized flow on domains of attraction: a structural stability result

It is well known that, for stepsize sufficiently small, compactattractors of ordinary differential equations persist underdiscretization. The present paper describes the structure ofthe discrete-time dynamical system obtained via discretizationon A(M_h)\M_h where M_h is the approximate attractor and A(M_h)is its domain of attraction. The existence of a smooth embeddinginto a continuous-time parallelizable flow is proved. The constructioncan be used to define sections for discretizations and can beinterpreted as a justification of the method of modified equations.     

Electrochemical reduction of 5,5′-dichlorohydurilic acid. mechanism and analytical applications

The electrochemical reduction of 5,5′-dichlorohydurilic acid has been studied at the dropping mercury electrode (DME) and the pyrolytic graphite electrode (PGE). At the DME the single polarographic reduction wave observed at pH 6–11 involves a direct 4e—2H⁺ reduction of the carbon-halogen bond to give hydurilic acid and chloride. The state of hydration or ionization of the 5,5′-dichlorohydurilic acid has no effect on the electrochemical reaction. At the PGE, 5,5′-dichlorohydurilic acid shows two voltammetric peaks. Peak I_c, observed between pH 5 and 7, arises from an overall 4e—2H⁺ reduction of 5,5′-dichlorohydurilic acid via a mechanism that involves initial electron attack at a carbonyl group alpha to a carbon-halogen bond with simultaneous elimination of chloride ion. The peak II_c process involves an initial 2e—1H⁺ reduction of a partially hydrated form of 5,5′-dichlorohydurilic acid with only one unhydrated halocarbonyl moiety available for reaction. Attack is again via the carbonyl group with simultaneous elimination of chloride and formation of 5-chlorohydurilic acid. A chemical dehydration step then occurs with a rate constant of ca. 0.24 s^?1 at pH 8.2, with formation of a further reducible halocarbonyl group. This is again reduced in an overall 2e—2H⁺ reaction to give hydurilic acid and chloride ion. The peak II_c process hence proceeds via an ECE mechanism. The different mechanisms observed for reduction of 5,5′-dichlorohydurilic acid at mercury and pyrolytic graphite electrodes are unusual. Analytical methods have been developed for the polarographic determination of 5,5′-dichlorohydurilic acid via its reduction wave at the DME, and for the voltammetric determination of hydurilic acid via its first oxidation peak at the PGE.