Attraction basin

Even the states of systems with infinite dimension, like systems described by partial differential equations, may lie on attractors of low dimension. The phase space of a system may also have more than one attractor. In this case the asymptotic behavior, i.e., the attractor at which a trajectory ends up, depends on the initial conditions. Thus, each attractor is surrounded by an attraction basin, which is the part of the phase space in which the trajectories from all initial conditions end up. 

Phase plane analysis indicates that the two limit cycles possess different sensitivities toward perturbations. It is much easier to pass from the small limit cycle to the large one than to achieve the reverse transition. This differential sensitivity results from the relative sizes of the attraction basins of the two cycles. Moreover, to pass from the large cycle to the small one, the quantity of substrate must be sufficient to cross the border defined by the unstable trajectory, but not so large so as to avoid bringing the system across the basin of the small cycle, into the other side of the attraction basin of the large cycle in such a case, the perturbation would only cause a phase shift of the large oscillations. In... 

When the system admits several, simultaneously stable periodic solutions, the boundary that separates their attraction basins is not always as sharp as in the case of figs. 4.4 and 4.7. There, an abrupt threshold characterizes the evolution towards either one of the stable limit cycles. In certain cases, the structure of the attraction basins is more complex. 

Another complex structure of the attraction basins is observed in certain cases of birhythmicity. Thus, the system can evolve towards either one of limit cycles LCl and LC2, starting from proximate initial... 

The behaviour illustrated by figs. 4.13 and 4.14 provides an example of final state sensitivity (Grebogi et al., 1983a). The evolution towards one or other final state is unpredictable when the unstable trajectory that defines the boundary of their attraction basins is a strange attractor rather than a simple limit cycle. 

Fig. 4.14. Self-similar structure of the attraction basins of the two limit cycles in the case of final state sensitivity described in fig. 4.13. The initial values of p and y being fixed, the initial value of a is varied in a continuous manner. A vertical line is traced when the system evolves towards limit cycle LC2. The white zones correspond to values for which the system evolves towards limit cycle LCl. Successive enlargements of the domains of variation of a illustrate the self-similarity of the random alternation between the two limit cycles, as a function of the initial substrate concentration (Decroly Goldbeter, 1984b).

Fig. 4.15. Influence of the fractal structure of the attraction basins on the sensitivity to perturbations in the course of oscillations. The system is perturbed at two different phases of limit cycle LC2 by an instantaneous increase in a. A short horizontal line is traced at that value of a when the system returns to limit cycle LC2 an empty space corresponds to a transition to limit cycle LCl (Decroly Goldbeter, 1984b).

Attraction basin fractal structure, 131-5 of limit cycle, 101,125,131, 251,252 see also Final state sensitivity Attractor, see Limit cycle Steady state ... 

Basin of attraction, see Attraction basin Bell-shaped dependence, of Ca release on Ca, 358,359,379,499 Belousov-Zhabotinsky reaction chaos, 12,283,511 chemical waves, 169,513 excitability, 102,213 oscillations, 1,508 temporal and spatiotemporal organization, 7,169 Bifurcation... 

The set of all points of the phase space whose trajectories converge to L as t —> +00 (—cx)) is called the stable (unstable) manifold of the periodic orbit. They are denoted by W and W , respectively. In the case where m = n, the attraction basin of L is Wf. In the saddle case, W is (m + l)-dimensional if m is the number of multipliers inside the unit circle, and is (p + 1)-dimensional where p is the number of multipliers outside of the unit circle, p = n — m — 1. In the three-dimensional Cctse, Wl and are homeomorphic either to two-dimensional cylinders if the multipliers are positive, or to the Mobius bands if the multipliers are negative, as illustrated in Fig. 7.5.1. In the general case, they are either multi-dimensional cylinders diffeomorphic to X S, or multi-dimensional Mobius manifolds. 

Recall that a fixed point 0 x = xq) is called structurally stable if none of its characteristic multipliers, i.e. the roots of the characteristic equation (7.5.2), lies on the unit circle, A topological type (m,p) is assigned to it, where m is the number of roots inside the unit circle and p is that outside of the unit circle. If m = n (m = 0), the fixed point is stable (completely unstable). The fixed point is of saddle type when m 0,n. The set of all points whose trajectories converge to xq when iterated positively (negatively) is called the stable (unstable) manifold of the fixed point and denoted by Wq Wq). In the case where m = n, the attraction basin of O is Wq. If the fixed point is a saddle, the manifolds Wq and Wq are C -smooth embeddings of and MP in respectively. 

If li > 0, then for /x < 0, there exists a stable fixed point O at the origin as well as an unstable orbit of period two bounding the attraction basin of O at /i = 0, the period two orbit merges into O the latter becomes unstable and all trajectories, except O, leave a neighborhood of the origin for /x > 0, see Fig. 11.4.2. 

Fig. 11.4.2. Transformations of the Lamerey spiral for the case l > 0. The unstable period-two cycle bounds the attraction basin of the origin.

If Zi > 0, then for all sufficiently small negative /i, there exists a period-two orbit (Oi, O3) of saddle (-h, -h) type. Its invariant stable and unstable manifold separate the attraction basin of the fixed point O2. As /i tends to zero the orbit of period two approaches O and collapses into it at /i = 0. When /i > 0, the point O becomes a saddle (—, —) (see Fig. 11.4.4). 

If the first Lyapunov value L is positive, then for small fJL>0, the equilibrium state O is unstable and any other trajectory leaves a small neighborhood U of the origin. When fx <0, the equilibrium state is stable. Its attraction basin is bounded by an unstable periodic orbit of diameter /i which contracts... 

If Li > 0, the phase portraits are depicted in Fig. 11.5.5. Here, when // < 0, there exists a stable equilibrium state O (a focus) and a saddle periodic orbit whose m-dimensional stable manifold is the boundary of the attraction basin of O. As /i increases, the cycle shrinks towards to O and collapses into it at /i = 0. The equilibrium state O becomes a saddle-focus as soon as p increases through zero. 

Fig 11.5.5. A subcritical Andronov-Hopf bifurcation, (a) An attraction basin of a stable focus is bounded by a stable manifold of a saddle periodic orbit, (b) The periodic orbit narrows to the stable focus at /x = 0, and the latter becomes a saddle-focus (1,2). 

If the first Lyapunov value Li > 0, then the fixed point of the map (11.6.6) is unstable for sufficiently small /x > 0. When /x < 0 the fixed point is stable its attraction basin is the inner domain of an unstable smooth invariant curve of the form (11.6.7). As p —0, the curve collapses into the fixed point see Fig. 11.6.2). 

If Li > 0, then when /i > 0, the fixed point is a saddle-focus of the above type, but its unstable manifold is the whole plane y = 0. Upon entering the region M < 0, the fixed point becomes stable. Meanwhile a saddle invariant curve C bifurcates from the fixed point its unstable manifold is (m -h 1)-dimensional and consists of the layers x — constant, restored at the points of the invariant curve. The stable manifold separates the attraction basin of the point O all trajectories from the inner region tend to O, and all those from outside of Wq leave a neighborhood of the origin. 

Here, the following scenarios is played when approaching the stability boundary the attraction basin of Oe Le) is getting smaller and smaller, and in the limit at e = It degenerates into a stable set (Oeo)(Leo))-This set is not empty because, by definition, Oeo Q " (- eo))-... 

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