Volume 105A, number 4,5 PHYSICS LETTERS 15 October 1984 MULTIPLE PERIODIC REGIMES AND FINAL STATE SENSITMTY IN A BIOCHEMICAL SYSTEM O. DECROLY and A. GOLDBETER Faculte des Sciences, Universitd Libre de Bruxelles, C.P. 231, Campus de la Plaine, B-1050 Brussels, Belgium Received 5 July 1984 We analyze a model biochemical system governed by three nonlinear differential equations, under conditions where multiple stable periodic regimes coexist. We determine the structure of their basins of attraction and show that t-mal state sensitivity is obtained when two stable oscillations are separated by unstable chaos. The effect of fractal basin boundaries on the response of the oscillatory system to perturbations is discussed. Nonlinear dynamical systems often show the coex- istence, for the same set of parameter values, of two or more different attracting states - either stationary or periodic. The question arises then as to what will be the time asymptotic regime as a function of initial conditions. Several authors have investigated this as- pect of nonlinear dynamics, revealing complex struc- tures for the basins of attraction of the different asymptotic regimes [ 1-5]. In addition to simple ba- sin structure, for which the boundary of the basis of attraction can be depicted as a simple surface, one can observe self-similar stripe structure [4], (see also ref. [2], fig. 9, 10), i.e. a basin constituted by an in- finity of "stripes" whose sizes decrease exponentially as they accumulate, and even fractal basin structure [1-3,5]. In the latter case, as a fine structure for the limits of the basins is observed at all scales of precision, it becomes exceedingly difficult to predict the final state as a function of initial conditions, a situation which has been referred to as the final state sensiti- vity [3]. The basin structures have been investigated mainly on discrete-time, one- [5 ] as well as two-dimensional [1-3] return maps. As to differential systems, a basin shape very similar to the stripe structure has been found in a chemical model for bistability and oscilla- tions [6]. There is until now, however, no clear de- monstration of fractal basins in continuous differen- tial systems, although Grebogi et al. [3] have present- 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) ed evidence for such phenomenon in the Lorenz model, as well as in experiments in hydrodynamics. We report here final state sensitivity in a biochemical model governed by three differential equations, and determine the consequences of such phenomenon for the dynamics of the oscillatory system. The model which represents a sequence of two autocatalytic enzyme reactions coupled in series (fig. 1) can be viewed as a simple prototype for the inter- play between two instability-generating mechanisms. In addition to simple and complex periodic oscilla- tions, this system may present chaos and birhythmicity (coexistence of two stable limit cycles under the same conditions) [7], and even a coexistence between three stable limit cycles [8]. The time-evolution of the concentrations of S(a), Pl(/3) and P2(7) is govern- ed by the following ordinary differential equations, where the various parameters, besides o, the constant input, and the sink constant ks, relate to the kinetic ÷ ÷ Fig. 1. Model enzymatic system comprising two autocatalytic reactions coupled in series. A substrate S is transformed into a product P1 by an enzyme El, while the second enzyme E2 converts P1 into product P2. Each enzyme is activated by its reaction product. 259