Complex systems chaos states

According to Stuart A. Kauffman (1991) there is no generally accepted definition for the term complexity . However, there is consensus on certain properties of complex systems. One of these is deterministic chaos, which we have already mentioned. An ordered, non-linear dynamic system can undergo conversion to a chaotic state when slight, hardly noticeable perturbations act on it. Even very small differences in the initial conditions of complex systems can lead to great differences in the development of the system. Thus, the theory of complex systems no longer uses the well-known cause and effect principle. 

Structures come in two kinds within the agent, there are schemas in the environment, there are sociomaterial resources. (The predicated sociomaterial is used to approach social and material phenomena symmetrically.) The two are again dialectically related, for the schemas allow us to recognize environmental structures for what they are but the structures in the environment have led to the formation of the schemas in the first place. This may sound like a chicken-and-egg situation, which would be difficult to explain in traditional logic. But such systems are as easy to explain in dialectical logic, or even in chaos- and catastrophe-theoretic approaches, where new, multi-state variants emerge as complex systems move through bifurcation points (e.g., Roth Duit, 2003). 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

The key property in this complex, unpredictable, random-like behavior is nonlinearity. When a system (process, or model, or both) consists only of linear components, the response is proportional to its stimulus and the cumulative effect of two stimuli is equal to the summation of the individual effects of each stimulus. This is the superposition principle, which states that every linear system can be studied by breaking it down into its components (thus reducing complexity). In contrast, for nonlinear systems, the superposition principle does not hold the overall behavior of the system is not at all the same as the summation of the individual behaviors of its components, making complex, unpredictable behavior a possibility. Nevertheless, not every nonlinear system is chaotic, which means that nonlinearity is a necessary but not a sufficient condition for chaos. 

It has been very fashionable to blame the kinetic behavior on chaos, which for our introductory purpose we can think of as the sensitivity to the precise initial conditions. If the initial conditions are fully and tightly specified (i.e., when one excites an eigenstate) then indeed the isolated system cannot forget. But in reality one can seldom excite a molecule to a fully specified initial state and if one did, then there would only be the spectroscopic mode of behavior. Otherwise, the point is that even quite simple mechanical systems but of realistic complexity (e.g., allowing for anharmonicity) exhibit sensitivity to initial conditions by which one means that two rather similar initial states follow quite diverging trajectories. If the initial state is not sharply defined then its subsequent evo-... 

Fig. 4.30. Diagram showing the various modes of dynamic behaviour of the multiply regulated biochemical system in the parameter space v-k. The indications stable and unstable relate to the stability properties of the unique steady state admitted by eqns (4.1). The domain of coexistence of such a stable state with a stable lirnit cycle is represented by the dotted area. Two domains of birhythmicity are observed their overlap gives rise to trirhythmicity. Three regions of chaos are represented in black, while the domains of simple or complex periodic oscillations occupy the rest of the space. The diagram is established for the parameter values of fig. 4.2 (Decroly Goldbeter, 1982 Goldbeter, Decroly Martiel, 1984).

While the coexistence between two limit cycles or between a limit cycle and a stable steady state is also shared by the two-variable models of fig. 12.1b and c, new modes of complex dynamic behaviour arise because of the presence of a third variable in the multiply regulated system. The coexistence between three simultaneously stable limit cycles, i.e. trirhythmicity, is the first of these. Moreover, the interaction between two instability-generating mechanisms allows the appearance of complex periodic oscillations, of the bursting type, as well as chaos. The system also displays the property of final state sensitivity (Grebogi et ai, 1983a) when two stable limit cycles are separated by a regime of unstable chaos. 

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