Steady state attractor

The phase space representation of trajectories computed numerically, as described above, has been introduced in another chapter of this volume. TTie systems considered there are Hamiltonian systems which arise in chemistry in the context of molecular dynamics problems, for example. The difference between Hamiltonian systems and the dissipative ones we are considering in this chapter is that, in the former, a constant of the motion (namely the energy) characterizes the system. A dissipative system, in contrast, is characterized by processes that dissipate rather than conserve energy, pulling the trajectory in toward an attractor (where in refers to the direction in phase space toward the center of the attractor). We have already seen two examples of attractors, the steady state attractor and the limit cycle attractor. These attractors, as well as the strange attractors that arise in the study of chaotic systems, are most easily defined in the context of the phase space in which they exist. 

In dissipative systems, many trajectories with different choices ot initial conditions will be attracted to the same region in phase space and end up, asymptotically, on the same attractor. The phase space portrait, then, will usually consist only of the asymptotic state, that is, a trajectory that represents the final state for many different initial conditions. This asymptotic trajectory traces over the attractor and reveals its shape. Figure 22(a) shows a steady state attractor, whereas Figure 22(b) shows a limit cycle attractor. 

Figure 22 Examples of attractors (a) shows a steady state attractor (here, a node), whereas (b) shows a limit cycle attractor.

Chaotic behavior in nonlinear dissipative systems is characterized by the existence of a new type of attractor, the strange attractor. The name comes from the unusual dimensionality assigned to it. A steady state attractor is a point in phase space, whereas a limit cycle attractor is a closed curve. The steady state attractor, thus, has a dimension of zero in phase space, whereas the limit cycle has a dimension of one. A torus is an example of a two-dimensional attractor because trajectories attracted to it wind around over its two-dimensional surface. A strange attractor is not easily characterized in terms of an integer dimension but is, perhaps surprisingly, best described in terms of a fractional dimension. The strange attractor is, in fart, a fractal object in phase space. The science of fractal objects is, as we will see, intimately connected to that of nonlinear dynamics and chaos. 

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

Naturally, the state of an RBN will change with time. If there are no external influences, then it has been shown that the state of any RBN will settle into either a point attractor (i.e., steady state) or a cycle. 

In the present chapter, steady state, self-oscillating and chaotic behavior of an exothermic CSTR without control and with PI control is considered. The mathematical models have been explained in part one, so it is possible to use a simplified model and a more complex model taking into account the presence of inert. When the reactor works without any control system, and with a simple first order irreversible reaction, it will be shown that there are intervals of the inlet flow temperature and concentration from which a small region or lobe can appears. This lobe is not a basin of attraction or a strange attractor. It represents a zone in the parameters-plane inlet stream flow temperature-concentration where the reactor has self-oscillating behavior, without any periodic external disturbance. 

Yet another type of complex oscillatory behavior involves the coexistence of multiple attractors. Hard excitation refers to the coexistence of a stable steady state and a stable limit cycle—a situation that might occur in the case of circadian rhythm suppression discussed in Section VI. Two stable limit... 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

Sincic and Bailey (1977) relaxed the assumption of only one stable attractor for a given set of operating conditions and showed examples of some possible exotic responses in a CSTR with periodically forced coolant temperature. They also probed the way in which multiple steady states or sustained oscillations in the dynamics of the unforced system affect its response to periodic forcing. Several theoretical and experimental papers have since extended these ideas (Hamer and Cormack, 1978 Cutlip, 1979 Stephanopoulos et al., 1979 Hegedus et al., 1980 Abdul-Kareem et al., 1980 Bennett, 1982 Goodman et al., 1981, 1982 Cutlip et al., 1983 Taylor and Geiseler, 1986 Mankin and Hudson, 1984 Kevrekidis et al., 1984). 

A Jacobi or Gauss-Seidel iteration on (6) will provide us with the coordinates of the steady state, or it will cycle indefinitely, depending on the slope of the functions. On the other hand, determining the trajectory by numerical integration of equation (5) will lead to a stable steady state or to a limit cycle depending on the slope of the functions. There is thus an obvious formal similarity between the two situations. However, the steepness corresponding to the transition from a punctual to a cyclic attractor is much smaller in the first case (in which the cyclic attractor is an iteration artifact) as in the second case (in which the cyclic attractor is close to the real trajectory). 

Figures 4.44 and 4.45, best viewed in color, show a benign complication of the problem caused by the Lewis numbers. If, however, we reduce the Lewis number LeA further to 0.07, the system trajectories indicate periodic explosions of the underlying system throughout all time, and the trajectories do not converge to the steady state at all, even with what we thought to be proper feedback. The trajectory that these curves settle at is called a periodic attractor of the system in contradistinction to the earlier encountered point attractor of Figures 4.43 or 4.44, for example. A point attractor, or more accurately a fixed-point attractor, is a more commonly encountered steady state in chemical and biological engineering systems. It could be called a stationary nonequilibrium state to distinguish it from the stationary equilibrium states associated with closed or isolated batch processes.

At Dpu the periodicity of the system changes form period one (PI) to period two (P2). Figure 7.31(A) shows that as D decreases further, the periodic attractor P2 grows in size till it touches the middle unstable saddle-type steady state and the oscillations disappear homoclinically at D bt = 0.041105 hr-1 without completing the Feigenbaum25 period-... 

There are many other interesting and complex dynamic phenomena besides oscillation and chaos which have been observed but not followed in depth both theoretically and experimentally. One example is the wrong directional behavior of catalytic fixed-bed reactors, for which the dynamic response to input disturbances is opposite of that suggested by the steady-state response [99, 100], This behavior is most probably connected to the instability problems in these catalytic reactors as shown crudely by Elnashaie and Cresswell [99]. Recently Elnashaie and co-workers [102-105] have also shown rich bifurcation and chaotic behavior of an anaerobic fermentor for producing ethanol. They have shown that the periodic and chaotic attractors may give higher ethanol yield and productivity than the optimal steady states. These results have been confirmed experimentally [105],... 

From the viewpoint of experimental workers, slow relaxations are abnormally (i.e. unexpected) slow transition processes. The time of a transition process is determined as that of the transition from the initial state to the limit (t -> oo) regime. The limit regime itself can be a steady state, a limit cycle (a self-oscillation process), a strange attractor (stochastic self-oscillation), etc. 

In the case of steady state bifurcations, certain eigenvalues of the linear-approximation matrix reduce to zero. If we consider relaxations towards a steady state, then near the bifurcation point their rates are slower. This holds for the linear approximation in the near neighbourhood of the steady state. Similar considerations are also valid for limit cycles. But is it correct to consider the relaxation of non-linear systems in terms of the linear approximations To be more precise, it is necessary to ask a question as to whether this consideration is sufficient to get to the point. Unfortunately, it is not since local problems (and it is these problems that can be solved in terms of the linear approximations) are more simple than global problems and, in real systems, the trajectories of interest are not always localized in the close neighbourhood of their attractors. 

The co-limit set is a very natural object from the qualitative viewpoint. co(x, k) is the set towards which the x-motion tends for t - oo (for a given k). Therefore it seems a natural formalization of the concept of a "limit regime . The co-limit set can consist of one fixed point (steady-state) and points belonging to one limit cycle. But its structure can also be more complex, i.e. it can include several fixed points and cycles, various surfaces, strange attractors, etc. 

Under the assumption C = 1 at each time r, the system evolves toward steady-state conditions that can be located graphically on the Semenov diagram of Fig. 4.4 as the intersections of the curves < r and q, this condition implies, indeed, that d%/dx = 0 in (4.16). For the sake of simplicity, let us first assume that %o = 7j = 1. When qE is given by line 1 with slope i, the steady-state condition is given by point A, characterized by a low operating temperature. Point A is an attractor since its temperature is spontaneously restored after any small perturbation of the system and, consequently, in these conditions thermal explosion does not occur. 

From Fig. 18b it is clear that under galvanostatic conditions the limit cycle coexists with a stationary state at high overpotentials. The latter is the only attractor at large current densities. Hence, when the current density is increased above the value of the saddle-loop bifurcation, the potential jumps to a steady state far in the anodic region. Once the system has acquired the anodic steady state, it will stay on this branch as the current density is lowered until the stationary state disappears in a saddle-node bifurcation. 

---