Routes to Chaos in the Peroxidase−Oxidase Reaction. 2. The Fat Torus Scenario

Abstract

Experimental studies [Hauser, M. J. B.; Olsen, L. F. J. Chem. Soc., Faraday Trans. 1996, 92, 2857−2863] of the peroxidase−oxidase (PO) reaction at pH values in excess of 5.4 suggest the existence of narrow regions of complex dynamics between adjacent mixed-mode oscillations (MMOs) that occur in period-adding sequences. Previously [Hauser, M. J. B.; Olsen, L. F.; Bronnikova, T. V.; Schaffer, W. M. J. Phys. Chem. B 1997, 101, 5075−5083], it was argued that both the period-adding sequences and the transitional regions between neighboring MMOs are predictable by a detailed model of the reaction called BFSO [Bronnikova, T. V.; Fed'kina, V. R.; Schaffer, W. M.; Olsen, L. F. J. Phys. Chem. 1995, 99, 9309−9312]. In the present paper, we study the transitional regions via computer simulation. Our investigations indicate that the motion therein may be periodic, quasiperiodic, or chaotic. In greater detail, we observe a quasiperiodic route to chaos whereby period-doubled cycles give rise to doubled tori that, in turn, undergo homoclinic bifurcations to chaos. Because the latter transitions are a consequence of progressive fattening of the tori, we propose calling this scenario the “fat torus” route to chaos, and the homoclinic bifurcations “fat torus” bifurcations (FTBs). The numerical results are qualitatively consistent with the experimental findings reported to date. FTBs and the resultant period-doubled, fractal tori may provide a criterion for discriminating among alternative models of the PO reaction.

Chaos in the peroxidase–oxidase oscillator

“…Numerical solutions of the original BFSO model show the existence of domains of complex and chaotic dynamics which are both rich and intricate. 32,33,39,[50][51][52][53][54] The numerical simulations reproduce several experimental observations in the PO reaction, e.g. that the chaotic domain can be reached either via a period-doubling route (at conditions mimicking low pH) or a period-adding route (at conditions mimicking higher pH) as the relevant parameter is varied.…”

Nonchaos-Mediated Mixed-Mode Oscillations in an Enzyme Reaction System

“…The latter is accounted for by lumping the effect of pH changes into a single rate constant, namely the rate constant k 9 (Table 1) of the pH-dependent dimerization of NAD • radicals. 24,35,36 The main findings reported here address three issues concerning the global structural organization of MMOs: First, we report the existence of a new type of MMOs where adjacent L S are not mediated by windows of chaos. 24,35,36 The main findings reported here address three issues concerning the global structural organization of MMOs: First, we report the existence of a new type of MMOs where adjacent L S are not mediated by windows of chaos.…”

“…Here, we focus on the dynamics observed in experiments, 23,24 namely in the control parameter space spanned by the NADH influx rate and the pH value of the medium. 24,35,36 This approach was adopted previously in several studies, where selected bifurcation diagrams in the k 9 , k 12 parameter space were calculated to explain experimental observations. 24,35,36 This approach was adopted previously in several studies, where selected bifurcation diagrams in the k 9 , k 12 parameter space were calculated to explain experimental observations.…”

Resolution of Respect William M. Schaffer, 1945–2021

“…About a year later they married. They were often seen walking on the University of Arizona campus together holding hands, and they collaborated on many scientific papers (Bronnikova et al 1995, 1996, 1998, 2001, Hauser et al 1997, Schaffer et al 2001 a , b , Olsen et al 2002, Schaffer and Bronnikova 2007 a , b , 2009, 2011, 2012).…”