Attractor basins

Quantum chemical topology (QCT) is a branch of theoretical chemistry that uses the language of dynamical systems (e.g. attractor, basin, gradient path, critical point and separatrix) to partition chemical systems and characterise them via associated... 

Hanson and Crutchfield linked their analysis of intrinsic computation in CAs to nonlinear dynamics by analyzing the attractor-basin portrait of ECA 18 (Hanson and Crutchfield, 1992), which gave additional insight into the dynamics of the CA s particles, and showed that small perturbations in the configurations do not appreciably change the CA s behavior or intrinsic computational complexity. 

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. 

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). 

At the heart of the AIM theory is the definition of an atom as it exists in a molecule. An atom is defined as the union of a nucleus and the atomic basin that the nucleus dominates as an attractor of gradient paths. An atom in a molecule is thus a portion of space bounded by its interatomic surfaces but extending to infinity on its open side. As we have seen, it is convenient to take the 0.001 au envelope of constant density as a practical representation of the surface of the atom on its open or nonbonded side because this surface corresponds approximately to the surface defined by the van der Waals radius of a gas phase molecule. Figure 6.15 shows the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and the p = 0.001 au envelope. It is clear that atoms in molecules are not spherical. The well-known space-filling models are an approximation to the shape of an atom as defined by AIM. Unlike the space-filling models, however, the interatomic surfaces are generally not flat and the outer surface is not necessarily a part of a spherical surface. 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

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. 

Typically, gradient paths are directed to a point in space called an attractor. It is obvious that gradient paths should be characterized by an endpoint and a starting point, which can be infinity or a special point in the molecule. All nuclei represent attractors, and the set of gradient paths is called an atomic basin, This is one of the cornerstones of AIM theory becanse the atomic basin corresponds to the portion of space allocated to an atom, where properties can be integrated to give atomic properties. For example, integration of the p function yields the atom s population. 

Though rare, there are cases in which the total density shows minor maxima at non-nuclear positions. As all (3, — 3) critical points are attractors of the gradient field, basins occur which do not contain an atomic nucleus. These non-nuclear basins (which have been found in Si—Si bonds1 in Li metal, and some other cases, distinguish the zero-flux partitioning from other space partitioning methods. 

Notice that for the graph that represents a discrete dynamic system, attractors are ergodic components, whereas basins are connected components. 

Atom Nucleus + basin 2 Basin filled by trajectories that terminate at the nucleus as the attractor of these trajectories... 

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. 

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. 

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. 

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