Basin of attraction

F after transients have decayed. This final set of phase-space points is tire attractor, and tire set of all initial conditions tliat eventually reaches tire attractor is called its basin of attraction. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

Since we will be dealing with finite graphs, we can analyze the behavior of random Boolean nets in the familiar fashion of looking at their attractor (or cycle) state structure. Specifically, we choose to look at (1) the number of attractor state cycles, (2) the average cyclic state length, (3) the sizes of the basins of attraction, (4) the stability of attractors with respect to minimal perturbations, and (4) the changes in the attractor states and basins of attraction induced by mutations in the lattice structure and/or the set of Boolean rules. 

Figure 10.4 shows a schematic representation of how Hopfield s net effectively partitions the phase space into disjoint basins of attraction, the attractor states of which represent some desired set of stored patterns. 

Fig. 10.4 Basins of attraction in the partitioned phcise space of a Hopfield neural net.

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

From the computational point of view the Fourier space approach requires less variables to minimize for, but the speed of calculations is significantly decreased by the evaluation of trigonometric function, which is computationally expensive. However, the minimization in the Fourier space does not lead to the structures shown in Figs. 10-12. They have been obtained only in the real-space minimization. Most probably the landscape of the local minima of F as a function of the Fourier amplitudes A,- is completely different from the landscape of F as a function of the field real space. In other words, the basin of attraction of the local minima representing surfaces of complex topology is much larger in the latter case. As far as the minima corresponding to the simple surfaces are concerned (P, D, G etc.), both methods lead to the same results [21-23,119]. 

Leaving the details, the equation describing the motion of one particle in two electrostatic waves allows perturbation methods to be applied in its study. There are three main types of behavior in the phase space - a limit cycle, formation of a non-trivial bounded attracting set and escape to infinity of the solutions. One of the goals is to determine the basins of attraction and to present a relevant bifurcation diagram for the transitions between different types of motion. 

From a thermodynamic perspective, Stillinger and Weber demonstrated that the total entropy of the liquid can similarly be divided into two additive terms, a configurational and a vibrational contribution.5,6 The configurational part Sc measures the number of structurally distinct basins of attraction on the PEL that the configuration point accesses at a given temperature, whereas the vibrational contribution Svib characterizes the number of states associated with intra-basin fluctuations. Thus, the AG relationship, when viewed from the PEL perspective, suggests that it is the thermodynamic availability of basins on the landscape that dominates the rate of liquid-state diffusive processes. 

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. 

The map (p determines discrete dynamical system on a set of components V — A,. We call it the auxiliary discrete dynamical system for a given network of monomolecular reactions. Let us decompose this system and find the cycles Cy and their basins of attraction, Att(Cy). 

If the ergodicity boundary reaction starts in the ergodic component Gi and ends at B which does not belong to the "opposite" basin of attraction II2, then T >S> This is the first possible obstacle. 

Systems that display strange kinetics no longer fall into the basin of attraction of the central limit theorem, as can be anticipated from the anomalous form (1) of the mean squared displacement. Instead, they are connected with the Levy-Gnedenko generalized central limit theorem, and consequently with Levy distributions [43], The latter feature asymptotic power-law behaviors, and thus the asymptotic power-law form of the waiting time pdf, w(r) AaT /r1+a, may belong to the family of completely asymmetric or one-sided Levy distributions L+, that is,... 

We then report and discuss the results of recent investigations of fluctuational escape from the basins of attraction of chaotic attractors (CAs). The question of noise-induced escape from a basin of attraction of a CA has remained a major scientific challenge ever since the first attempts to generalize the classical escape problem to cover this case [92-94]. The difficulty in solving these problems stems from the complexity of the system s dynamics near a CA and is... 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

Figure 14. The basins of attraction of the SC (shaded) and CA (white) for a Poincare cross section with

In the presence of weak noise there is a finite probability of noise-induced transitions between the chaotic attractor and the stable limit cycle. In Fig. 14 the filled circles show the intersections of one of the real escape trajectories with the given Poincare section. The following intuitive escape scenario can be expected in the Hamiltonian formalism. Let us consider first the escape of the system from the basin of attraction of a stable limit cycle that is bounded by an saddle cycle. In general, escape occurs along a single optimal trajectory qovt(t) connecting the two limit cycles. 

Since the basin of attraction of the CA is bounded by the saddle cycle SI, the situation near SI remains qualitatively the same and the escape trajectory remains unique in this region. However, the situation is different near the chaotic attractor. In this region it is virtually impossible to analyze the Hamiltonian flux of the auxiliary system (37), and no predictions have been made about the character of the distribution of the optimal trajectories near the CA. The simplest scenario is that an optimal trajectory approaching (in reversed time) the boundary of a chaotic attractor is smeared into a cometary tail and is lost, merging with the boundary of the attractor. 

Figure 15. Escape trajectories found [173] in the analog simulations for the parameters h — 0.19, tof 1.045, coo 0.597,D ss 0.0005 are shown in comparison with the Poincare cross section of a quasiattractor and its basins of attraction for (sift = 0.

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