Attractors different types

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

In this section the application of the optimal path approach to the problem of escape from a nonhyperbolic and from a quasihyperbolic attractor is examined. We discuss these two different types of chaotic attractor because it is known [160] that noise does not change very much the structure and properties of quasi-hyperbolic attractors, but that the structure of non-hyperbolic attractors is abruptly changed in the presence of noise, with a strong dependence on noise intensity. Note that for optical systems both types of chaotic attractor [161-163] (nonhyperbolic and quasihyperbolic) are observed, but a nonhyperbolic attractor is much more typical. 

Oscillations of physicochemical elements such as temperature, concentrations of chemical elements, etc. correspond to oscillatory solutions of the dynamic equations (differential equations) of the system under consideration. As discussed above, various reactions possess different types of these oscillatory mathematical solutions. The well known limit cycles were first observed and named as such by Poincare exactly a century ago. Although formulated by Poincare in general terms, the other mathematical solutions have recently been explored and named by mathematicians, e.g. attractors and exploded points consequently their application to chemical systems are more recent than the limit cycles. 

This approach needs modification as soon as multiple attracting periodic trajectories exist for a particular set of operating parameters. A conceptually different modification will be necessary to account for attractors which are not simply periodic. Quasi-periodic solutions, characterized by multiple frequencies, are the first type one should expect these are by no means exotic but occur generally in several periodically forced systems. Deterministic chaotic situations, arising from the system nonlinearities (and not the stochastic responses due to random noise) need not be discarded as intractable (Wolf et al., 1986 Shaw, 1981). 

It is necessary to emphasize one principal peculiarity of the copolymerization dynamics which arises under the transition from the three-component to the four-component systems. While the attractors of the former systems are only SPs and limit cycles (see Fig. 5), for the latter ones we can also expect the realization of other more complex attractors [202]. Two-dimensional surfaces of torus on which the system accomplishes the complex oscillations (which are superpositions of the two simple oscillations with different periods) ate regarded to be trivial examples of such attractors. Other similar attractors are fitted by the superpositions of few simple oscillations, the number of which is arbitrary. And, finally, the most complicated type of dynamic behavior of the system when m 4 is fitted by chaotic oscillations [16], for which a so-called strange attractor is believed to be a mathematical image [206]. 

In the non-linear systems (5.2), a second type of attractor — a closed curve (limit cycle) is also possible. For example, the system of van der Pol equations (representing oscillations of current in electrical circuits and oscillations of concentrations, or more precisely the differences between the concentrations and their stationary values, in chemical systems)... 

Fig. 6.19. Evolution in the phase space at different values of the relative proportions of (initially) chaotic and periodic cell populations in a mixed suspension containing various amounts of the two types of cells, (a) Oscillations of the limit cycle type obtained for Vj = 4.5 x 10 min" when the suspension contains only cells of periodic population 2 (fj = 0, fj = l)l arrows show the direction of movement and the trajectory has been broken to indicate the part that comes behind (the portion of the curve in front corresponds to a decrease in all three variables after a peak in cAMP). (b) Period-2 oscillations obtained upon adding to periodic population 2 cells from the chaotic population 1, for which Vi = 4.396875 x 10" min" the value of the fraction of the (initially), chaotic population is = 0.5. (c) Period-4 oscillations obtained when is increased up to 0.86 notice that two of the loops of the trajectory over a period are very close to each other, which is also apparent in the bifurcation diagram of fig. 6.20. (d) Chaotic behaviour corresponding to a strange attractor when the suspension contains only cells of population 1 (Fj = 1). The curves are obtained by numerical integration of eqns (6.9) for the above-indicated values of Vj and Vj other parameter values, which hold for the two populations, are as in fig. 6.2. Variables pr and a relate to population 2 in (a)-(c), and to the homogeneous population 1 in (d) variable y is shared by the two populations. Ranges of variation for pr, a and y are 0-1,0.65-0.68 and 0-2.2, respectively. Initial conditions were a = 0.6729 and pr = 0.2446 for both populations, while 7=1.7033. The curves were obtained after a transient of 500-1000 min. The period of the oscillations shown in (a)-(c) is of the order of 8-10 min thus for F. = 0.3 and Fj = 0.7 the period is equal to 8.7 min (Halloy et al. 1990).

Two types of attractors were known since the times of Poincare points or closed curves (limit cycles). The third type was discovered in 1971 [60]. It is so-called strange attractor , which can exist in three- and more-dimensional systems. In accordance with these three known types of attractors, three different kind of system s behaviour are possible after bifurcation (1) transition into a new stable steady state (2) undamped self-oscillations, and (3) chaotic regime (turbulence). 

A family of phase trajectories starting from different initial states is called a phase diagram or phase portrait. A phase portrait graphically shows how the system moves from the initial states and reveals important aspects of the d)mamics of the system. Certain phase portraits display one or more attractors, which are long-term stable sets of states toward which a system tends to evolve (is attracted) dynamically, from a wide variety of initial conditions, the basin of attraction. The simplest type of attractor is a so-called rest point. This is a point for which the derivatives of all the state variables are zero ... 

Finally, interesting question arises what is the driving force behind the N-O bond length variation. The answer is not straightforward as the studies on FONO and HOONO show that the topology of ELF depends on the quality of wave function used for analysis. Furthermore, different conformers can exhibit different number and type of attractors in the N-O bond as has been shown for example for HOONO, (CF3)2N0-N0 and (CH3)2NO-NO. Nevertheless some generalisations are possible. 

Fixed points are not the only possible states of dissipative, far-from-equilibrium systems more complex macroscopic attractors, like limit cycles or even strange attractors, are commonly observed [19]. Bistabilities between the different attractor types may occur and give rise to interesting transition rate processes when these systems are subjected to external noise. We examine some of the new features that enter the calculation of the transition rate by examining some specific examples of systems displaying bistability between a fixed point and a limit cycle, but the discussion can be generalized to other situations. 

The spiral-like shape of this attractor follows from the shape of homoclinic loops to a saddle-focus (2, 1) which appear to form its skeleton. Its wildness is due to the simultaneous existence of saddle periodic orbits of different topological type and both rough and non-rough Poincare homoclinic orbits. 

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