In the Laboratory
Nonlinear Dynamics of the BZ Reaction: A Simple Experiment That Illustrates Limit Cycles, Chaos, Bifurcations, and Noise Peter Strizhak1 and Michael Menzinger Department of Chemistry, University of Toronto, Toronto, ON M5S.1A1
Chemical reactions not only proceed monotonically as described by classical kinetics, but can also show periodic and chaotic oscillations, multistability, excitability, and the formation of chemical waves and spatial patterns (1–3). These provide some of the most striking examples of the nonlinear phenomena that pervade the physical and biological world. Nonlinear chemical systems are capable of the kind of timing, switching, and signal propagation functions that are the elements of biological and neuronal processing. For instance, Ross and co-workers showed recently that networks of bistable chemical reactions represent chemical realizations of neural networks that are capable of learning and memory (4). Rovinsky and Menzinger showed that spatially extended, excitable systems can act as the kind of analogto-frequency converters that are found in sensory receptors (5). Thus, nonlinear chemical phenomena, apart from their intrinsic interest, are also vehicles for understanding nonlinear phenomena in a larger context. Oscillating chemical reactions have been the subjects of several articles in this Journal (6–20). General dynamic principles were discussed (6, 7) and experiments in flow reactors received particular attention (8, 9). Aperiodic (chaotic) oscillations were discussed in this context. The history, mechanism, and recipes of the Belousov–Zhabotinsky (BZ) reaction were reviewed (9– 11). Chemical waves in reaction-diffusion systems were described (12–15). Recently, a report appeared of transient chaotic oscillations in the Belousov–Zhabotinsky system in a stirred batch reactor (21). This paper revisits the well-known oscillations of the BZ reaction (9–11) in a stirred batch reactor. As this closed system drifts slowly toward thermodynamic equilibrium, it passes through different regimes of steady state, oscillatory, and chaotic dynamics and undergoes the associated bifurcations. While the nonstationarity of batch experiments is often considered a drawback, it provides the careful observer with a rich sequence of dynamic phenomena that illustrate some of the key concepts of nonlinear dynamics. The experiment may be viewed as a self-organized walk through different dynamic domains of parameter space and the unfolding of the associated dynamics. Depending on the initial conditions, the reaction eventually ceases to oscillate either suddenly or gradually—scenarios known as subcritical and supercritical Hopf bifurcations. The experiments are viewed as illustrations of key concepts of the qualitative description of nonlinear dynamic systems, including the notions of separation of time scales, phase space, phase 1
Permanent address: L. V. Pisarzhevskii Institute of Physical Chemistry, National Ukrainian Academy of Sciences, prosp. Nauki, 31, Kiev, Ukraine 252039.
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portraits, periodic and aperiodic (chaotic) oscillations, and the corresponding bifurcations. These concepts are briefly reviewed in the next section. The experimental setup is described in the following section, after which the results are presented and discussed. To conclude, suggestions are made for modifying the experiment for the use of teaching laboratories. Given the rich dynamics, there are several aspects that remain unexplored and that may be profitably addressed by undergraduate students, either singly or in teams. Some Concepts of Nonlinear Dynamics The coupled rate equations that govern the dynamics of a homogeneous and isothermal chemical reaction follow from its reaction mechanism. In compact form they are written as dx/dt = F(x;p) (1) where F is the rate function, x(t) = {x1(t), x 2(t), …, x n(t)} is the vector of dynamic variables in n-dimensional phase space, and p = {p 1, p2, …, pm} is the vector of control parameters. In a chemical context, x(t) represents concentrations and p the constraints (i.e., rate constants, temperature, reactant composition, flow rate, etc.). The concepts of phase space and phase portrait are important tools for visualizing the evolution of system 1. Phase space is the n-dimensional space with coordinates {x1, x2, …, x n}. A phase portrait is a trajectory of the representative point x(t). A dissipative dynamic system (e.g., chemical reaction in which the free energy decreases) is characterized by the contraction of flow in phase space (1) and by the convergence of all initial conditions to socalled attractors. These are low-dimensional subsets, or manifolds, of phase-space to which the system eventually converges after transients have died out. The dynamics is greatly simplified by neglecting the transients and focusing attention on the asymptotic dynamics on these low-dimensional attractors, which organize the flow in phase space. Examples of attractors are the stable steady states or stable fixed-point x oi (zero-dimensional manifold), the stable limit cycle (one-dimensional manifold), and the strange attractor (higher dimensional manifold with noninteger dimension) (22–25). The fixed points x oi are the solutions of the steady-state equation dxo/dt = F(x o;p) = 0. Since the rate function is nonlinear, the steady-state equations may have more than one solution xoi (i=1,2,3 …), and as a result multistability between coexisting attractors is possible. A linear stability analysis (1, 3) of a given fixed point reveals whether it is stable or unstable: this means whether perturbations from the steady state decay or grow. A stable manifold is called attractor;
Journal of Chemical Education • Vol. 73 No. 9 September 1996
In the Laboratory
an unstable manifold, repellor. The type of attractor depends on the chemical kinetic term F(x;p) and on the dimension n of phase space. If the system’s behavior is determined by the rate of consumption and accumulation of a single species, then stable fixed points or steady states are the only kind of attractor. In two-variable systems, two types of attractor are possible: fixed points and limit cycles. A limit cycle is a closed, 1-D curve embedded in phase space, which represents periodic oscillations. For a limit cycle to exist, the chemical mechanism should have at least one autocatalytic stage and a negative feedback (1–3). Autocatalysis provides accelerating growth of one species, and the negative feedback terminates the autocatalytic explosion. In oscillations this sequence repeats periodically. More complex dynamics is possible in systems with three or more variables, including strange attractors and the associated phenomenon of deterministic chaos (22–25). Strange attractors are geometrical objects with dimension that is strangely non-integer. An attractor may lose its stability suddenly when a control parameter is changed. This event is called bifurcation. Figure 1 is a so-called response diagram, which represents the dynamic response (e.g., the value of a variable xi) as a function of control parameter. Solid lines in this figure show stable manifolds; dotted lines represent unstable manifolds. Beyond the bifurcation point, the system diverges from the fixed point that has just become unstable and usually goes to an alternate attractor. In the cases illustrated in Figure 1, this new attractor is a limit cycle. Periodic and aperiodic oscillations may be born through a variety of bifurcations. Periodic oscillations (limit cycles) may be born through a Hopf bifurcation where the stable fixed point loses its stability and the system is forced to a new attractor: the limit cycle. The Hopf bifurcation can appear supercritically or subcritically as shown in Figure 1. In the supercritical case, the limit cycle has vanishing radius and the oscillations are born with zero amplitude. The amplitude grows monotonically with the distance from the bifurcation point and does not depend on the direction of the parameter change (Fig. 1a). In the subcritical case, illustrated by Figure 1b, the oscillations are born suddenly with finite amplitude at one critical parameter value. When the parameter is scanned in the opposite direction, the oscillations disappear at another critical parameter value: hysteresis occurs. The bifurcation may be supercritical or subcritical, depending on the values of the other parameters. This is illustrated by the bifurcation diagram in Figure 2, which represents a map of the dynamic domains in the parameter plane {p1, p2}. It shows three domains, which correspond to steady state, periodic oscillations, and chaotic oscillations. The boundaries of these domains are the sets of bifurcation points. In a closed system the reactants are frequently consumed on a time scale that is long compared to the time scale of the oscillations. While their concentrations are, strictly speaking, slowly changing dynamic variables whose evolution is described by the appropriate component rate equations of eq 1, one often refers to these slow variables as “parameters” and considers them as constant on the time scale of interest (e.g., of oscillations). The quotation marks draw attention to the approximate nature of this terminology. The coordinate axes of Figure 2 are such “parameters”. As the system drifts through “parameter” space along a trajectory that is determined by initial conditions and the stoichiometry, it encounters different dynamic domains. For instance, when the system evolves from the initial,
non-oscillatory conditions (in the BZ system, p1 = [BrO 3–] and p2 = [MA]), it undergoes a Hopf bifurcation and performs almost regular oscillations until it encounters the exit-Hopf bifurcation according to the trajectory 1 in Figure 2. To match the present experiments, assume that the latter is subcritical. For a different initial condition illustrated by trajectory 2, the system begins to oscillate, then encounters a strange attractor where chaotic oscillations exist. As the system leaves the oscillatory domain, the character of the Hopf bifurcation may have changed from subcritical to supercritical, as indicated by the dotted line. The concept of separation of time scales, which allows one to eliminate the slowly varying “parameters” from the set of dynamic variables, is fundamental to all dynamics. For example, when studying a dynamic process, be it the internal motion of a molecule, the expansion of a gas into a vacuum, or a chemical reaction, one is generally only interested in phenomena on a certain time scale t obs. Slow variables that change on a much
Figure 1. Response diagram of (a) supercritical and (b) subcritical Hopf bifurcation.
Figure 2. Schematic bifurcation diagram in the “parameter” space of the batch BZ system, where p1o = [BrO3–] o and p2 o = [MA]o. In the course of its drift through “parameter” space along the trajectories (1), (2) the system encounters Hopf bifurcations and bifurcations to chaos. Subcritical Hopf bifurcations are indicated by the solid line and supercritical bifurcation by the dotted line.
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longer time scale ts>>tobs may be taken as approximately constant and are treated as “parameters”. On the other hand, variables that evolve much faster, on a time scale tf
