Attractor selection

Figure 10.5 Attractor selection (left, low Activity, right, high Activity).

Fukuyori, I., Nakamura, Y., Matsumoto, Y., Ishiguro, H., 2009. Control method for a robot based on adaptive attractor selection model. In The 4th International Conference on Autonomous Robots and Agents, pp. 618—623. 

Kashiwagi, A., Urabe, I., Kaneko, K., Yomo, T., 2006. Adaptive response of a gene network to environmental changes by fitness-induced attractor selection. PLoS One 1 (1), 1—9. 

Nurzaman, S.G., Yu, X., Kim, Y., Dda, F., 2014. Guided self-organization in a dynamic embodied system based on attractor selection mechanism. Entropy 16, 2592—2610. 

The table below gives a few selected values of for the logistic and Henon map attractors ... 

Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

The limit cycle is an attractor. A slightly different kind occurs in the theory of the laser Consider the electric field in the laser cavity interacting with the atoms, and select a single mode near resonance, having a complex amplitude E. One then derives from a macroscopic description laced with approximations the evolution equation... 

With this model, we need only apply the method already used to derive the selection rules for electrocyclic reactions (p. 53). From the Coulson equations, we can deduce that in the in conrotatory cyclization of pentadiene, the MO generates a destabilizing C5-C4 secondary interaction, a stabilizing and Fg a destabilizing interaction. The absolute values of these contributions rise steadily because the terminal coefficients increase from Fg to Fg. Therefore, the sign of their sum is given by the HOMO contribution. If R is an attractor, the HOMO is Fg and rotation inwards is favored. If R is a donor, the HOMO is 4T and rotation inwards is disfavored. As the Coulson equations are valid only for polyenes, these conclusions are correct insofar as R can be modeled by a carbon 2p orbital. It follows that the Rondan-Houk theory works better for conjugative than for saturated substituents. 

A recent theoretical study has suggested that persistent activity in the PFG is considered to be an attractor state, in that relatively small amounts of variation in this state lead it back to the same state. This idea has been examined in detail theoretically, especially by Amit, who described persistent activity in terms of dynamical attractors (Amit and Brunei, 1997 Rolls et al., 2008). The spontaneous state and stimulus-selective memory states are assumed to represent multiple attractors, such that a memory state can be switched on or off by transient inputs. This formulation is plausible, insomuch as stimulus-selective persistent firing patterns are dynamically stable in time. These properties of attractors result from interactions in neuronal circuits. Neural synchrony is a general mechanism for dynamically linking together cells coding task-relevant information (Salmas and Sejnowski, 2001). The dynamics of neuronal activities and the representations they reflect are two sides of a coin. 

An algorithm due to Wolf et al. 2 jg the most widely used for calculating Lyapunov exponents. It can be applied to a reconstructed phase portrait or one found by measuring more than one dynamical variable. Two nearby points, A and B, that are not part of the same orbit around the attractor are located the latter criterion can be ensured by not considering the first m points in the time series after the selection of one of the two points. The evolution of the two points can be followed since the time series is available, and it is known which points on the attractor follow in time. In a chaotic system, the initial distance... 

Rossler argued that the very long-term qualitative dynamic behaviour of an evolving chemical system cannot be analysed. The attractor can be different from the present known attractors dynamic systems describing evolution internally select a sequence of subsystems in which each generates a viable successor even if the environment changes during the process. 

This perturbation will send the system toward x,. Since the parameter value is no longer exactly p, we will not be exactly at the steady state, and we will need to make another correction using eq. (8.21) at the next return, that is, at a time T later, where T is the time interval used to discretize the time series. In practice, the corrections soon become small and the fixed point is stabilized. Higher period orbits can also be stabilized using the same algorithm. Figure 8.22 shows phase portraits and time series of period-1 and period-2 orbits selected from a chaotic attractor in the BZ reaction. 

Table 15.1 A-H bonds of selected molecules and cations (A designates heavy atom) r— the distance between A and H attractors R —the distance between A and H nuclei AR% is equal to [(R-r)/R] 100 % e— electronegativity of the A heavy atom in the Pauling scale distances in A...

The first step in the Wolf and Swift method is to construct an attractor from the time series, as described in Section 2.1. Then a starting point at time t is selected on an arbitrarily chosen fiducial trajectory, as illustrated in Fig. 3. A nearby point is then found the distance between the two points is called L(t ). The separation L(t) between the two trajectories is monitored until it becomes large compared to L(t ) yet still small compared to the size of the folds in the attractor. A new point is then found near the fiducial trajectory the new point is chosen to be (to the extent possible) in the same direction from the fiducial... 

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