The Language of Dynamics Rlchard J. Flekl University of Montana, Missoula, MT 59812 Dynamics is the study of change. For a reacting chemical system, there exists a set of polynomial differential equations that describes the dynamics of how the concentrations of all the chemical species (reactants, products, intermediates, and catalysts) change with time. These differential equations may he written down, if the reaction mechanism is known, using the law of mass action. The "path" in time followed by the concentrations of the chemical species throughout concentratinn "space" is called the trajeciory. The polynomial differential equations descrihina a chemir cal reaction may be either linear or nonlinear. ~ i i e a equations have only terms which depend on the concentration of a sinele chemical s~ecies.and each such concentration will a p p e k to the first power:~onlinear polynomial differential eauations will contain terms that d e ~ e n don the oroduct of t d o concentrations or on the conceniration of a single species squared. I t can be shown that, if the dynamic law of a chemical reaction is linear, its dynamic behavior must he simple. That is, the concentrations of reactants must decrease monotonically with time, the concentrations of products must increase monotonically with time, and the concentrations of intermediates may pass through but a single maximum or minimum during the course of the reaction. If the dynamic law for a chemical reaction is nonlinear, however, then much more complicated phenomena may occur, including: in a stirred medium, temporal oscillations in the concentrations of intermediate or catalyst species; in an unstirred medium, traveling spatial waves or zones of chemical reaction; and, in a continuous-flow stirred tank reactor (CSTR), bistability or even tristahility, hysteresis, excitability, and deterministic chaos. These startling phenomena are chemical examples of self-organization. The requisite nonlinearity in the dynamic equations for chemical reactions showing such hehavior is often associated with an autocatalytic component. The concept of stability is central to understanding the appearance of far-from-equilibrium phenomena. We expect chemical reactions running in a closed system to approach an unchanging state of thermodynamic equilihrium. Furthermore we expect that, on the approach to equilibrium, the reaction will follow a line of so-called pseudo-steady states in which the concentrations of intermediates present at low concentrations reach relatively unchanging values calculable from the instantaneous concentrations of reactants and, perhaps, products, present a t much higher concentrations. (Genuine, unchanging steady states can be attained in a CSTR, where fresh reactants are constantly pumped in while products and unused reactants flow out.) Stability, then, relates to the response of a system to an infinitesimal perturbation when the system is in the vicinity of equilibrium or of a steady state. Astable system returns to equilibrium or to the steady state after the perturbation, while an unstable system does not. The stability of a state can be evaluated by means of a linear stabilitv analvsis. This is a standard mathematical technique begun by calculating the pertinent concentrations for thestateof interest.This means. for an eauilibriumstate. finding the concentrations of all leactants; products, and intermediates such that there is no further change in these concentrations. For a steady state, the nonequilihrium con188
Journal of Chemical Education
centrations of reactants and products are held constant (because of in-flow and out-flow or because the steady state is approached much more rapidly than the concentrations of reactants and ~ r o d u c t schance) and the corres~ondineconcentrations of.intermediatesare calculated. he stabiiity of the state in uuestion is then studied bv suhstirutinr into the differential kquations time-dependent perturhzions assumed to he of the form: X = Xo xeht, Y = Yo yekt, and so forth. The quantities Xo, Yo, etc., are the equilibrium or steady-state concentrations; x , y, etc., are the magnitudes of the perturbations of each of the concentrations; t is time; and the Aj are the so-called eigenvalues of the problem. A standard machinery yields the values of the A, which determine the trajectory the system follows in the vicinity of the state in question. The A, are usually complex and so have the form a j i bji. The sign of the real part determines whether the perturbation grows or decays with time. If any a, > 0,then an infinitesimal perturbation will -prow,. and so the state is unstable. If all a j < 0,then infinitesimal perturbations will decay, and the state is stable. If any eigenvalues are complex ( b ; # 0). then the system's trajectorfaround the state will be oscilla: tory; the state is then termed a focus. If all the bj = 0,then "motion"around the state will bemonotonic, and the state is called a node. In any case, the state to which the system ultimatelv settles down is called an attractor. For a ilosed system, equilibrium is always the ultimate attractor. Because of the principle of detailed balance, equilibrium is always a stable node; that is, perturbations to equilibrium will decay monotonicallv hack to equilibrium. k d the final approach t o equilibrium in a reaction also is monotonic. Thus, chemical oscillations and other self-organizing phenomena do not occur in the vicinity of equilihrium. A system must he far from equilibrium before steady states become unstable. What may happen if a steady state is unstable? For a twovariable system, there are theorems that allow determination of where evolution from an unstable steady state will go. However, most chemical reactions of interest involve more than two variables, and for such reactions we can only list possibilities, two of which are that the svstem mav ex~lode. or that i t may evolve to an oscillatory limit cycle. ~ h ~ l a t t e ; is what happens in an oscillating chemical reaction. A limit cycle is a solution to the differential equations for which the concentrations of intermediates follow a cyclic, closed-loop traiectorv determined bv Darameters such as the concentrati&s of the major reactants, the flow rate in ~ C S T R and , the tem~erature.A limit cvcle mav. heeloballv attractive. which means that the same bsci~~atory trajectory is reached from any initial intermediate concentrations. I t is possible for a limit cycle t o exist even if the steady state is stable to infinitesimal perturbations. In this case, both the limit cycle and the steady state are only locally stable or locally attractive. Trajectories starting near the steady state decay to the steady state. Trajectories starting near the limit cycle evolve to the limit cycle. Even a stable, globally attractive steady state may be excitable, in which case the system's response to a large enough finite perturhation will he a very large excursion before the return to the steady state.
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As some parameter of a system is varied, a steady state may change-from stable to unstable when the real part ofnn eipenvalue changes from negative t o positive. This change in stabilitv ia a bilurcalion. The hehavior of a svstem chanees qualitakvely a t a bifurcation. For example, a-system wi& a stable steady state may become oscillatory. The most common sort of bifurcation is a Hopf bifurcation, occurring when the real Dart of a h air of c o m ~ l e xeieenvalues (a; f b;i) passes through zero. The coexistence of a locally'stable steady state and a locally stable limit cycle, as described earlier, occurs in the vicinity of certain Hopf bifurcations. The types of dynamic behavior exhibited by a chemical reaction may increase dramatically when it is run in a CSTR. For example, we normally expect to find only a single steady state for a particular set of reactant concentrations. However, a reaction with an autocatalytic nonlinearity may show three or even five steady states in a CSTR for exactly the same set of experimental parameters. In the case of three steadv states. onlv two mav be even locally stable so as to be exper~mentallyobservable; the third i s always unstable. This situation gives rise to bistabilitv. Alternatively, all three steady states may he unstable, and in this case two of them may he surrounded by stable limit cycles. This situation produces birhythmicity. In the case of five total steady states, only three steady states may be even locally stable. This is tristability. Bistability in a system occurs over a range of parameter values. This means that, for a range of experimental conditions, the system can be in either of two different steady states. BYvarving an experimental Darameter, i t is oossible co move ihesystern thro"gh the bisrability range in htherof the two locally stable steadv staws; if the system is driven out of the range of bistabilky, it will switch to one or the other of these steady states, and, if brought back into the bistahility region, it will remain in that particular steady
state. The system thus "remembers" on which side of the bistahility region i t left and re-entered. This behavior is called hysteresis. Occasionally when an oscillating reaction is run in a CSTR, the observed oscillations are aperiodic. The amplitudes and periods of the oscillations appear random. This could be a result of stochastic environmental perturbations, especially as the autocatalytic nature of many of these reactions codd be expected td amplify perturbations. Such behavior might also be deterministic chaos, which is a form of aperiodicky intrinsic to the governing differential equations themselves. Afeature of chaos is that trajectories which start out arbitrarilv close to each other auicklv diveree. Such aperiodicity will remain no matwr how careful the experimental control. and it is never oossible to oredict the behavior of a chaotic system over long time beriod. There are data treatment methods to help distinguish aperiodicity from experimental noise. The trajectory of a chaotic system is called a strange attractor. Even thoughchaotic behavior appearssuperficially to be random, a strange attractor does not fill concentration space. Detailed examination of a strange attractor will reveal that it is a fractal object, a geometric entity with fractional dimension. One . nroDertv . .of such an obiect is that examination a t increasing levels of resolution shows a neverending succession of very similar detailed structure, hut always with empty space between the occupied points. Thus, although a chaotic system evolves indefinitely without repeating its behavior;there are nonetheless more regions of concentration space which i t does not visit. General References Pippard.A. B . R e a p o ~ eondStobility: Cambridge Univereity.Cambridge. 1985. Field, R. J. Am. Sci. L985,73,112. Field, R. J.; Sehncidsr, F. W. J Chem.Educ. 1989,M.WO.
Volume 66 Number 3 March 1989
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