Attractor unique

Figure 4.11 shows sample trajectories starting from various initial conditions for four different values of q. (Keep in mind that these are not plots of attractors each trajectory is unique to a particular starting position on the unit torus). For q = 0,... 

Universality in Unimodal Maps A seminal work on the 2-symbol dynamics of one-dimensional unimodal mappings due to Metropolis, Stein Stein [metro73]. Specifically, they studied the iterates of various mappings within periodic windows, labeling the attractor sequences by strings of the form RLLRL , where R and L indicate whether f xo) falls to the right or left of xq, respectively. Each periodic sequence therefore corresponds to a unique finite length word made up of R s and L s. 

The spatial and temporal dimensions provide a convenient quantitative characterization of the various classes of large time behavior. The homogeneous final states of class cl CA, for example, are characterized by d l = dll = dmeas = dmeas = 0 such states are obviously analogous to limit point attractors in continuous systems. Similarly, the periodic final states of class c2 CA are analogous to limit cycles, although there does not typically exist a unique invariant probability measure on... 

Points on the zero-flux surfaces that are saddle points in the density are passes or pales. Should the critical point be located on a path between bonded atoms along which the density is a maximum with respect to lateral displacement, it is known as a pass. Nuclei behave topologically as peaks and all of the gradient paths of the density in the neighborhood of a particular peak terminate at that peak. Thus, the peaks act as attractors in the gradient vector field of the density. Passes are located between neighboring attractors which are linked by a unique pair of trajectories associated with the passes. Cao et al. [11] pointed out that it is through the attractor behavior of nuclei that distinct atomic forms are created in the density. In the theory of molecular structure, therefore, peaks and passes play a crucial role. 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

It can be seen that the solution of the problem of the energy-optimal guiding of the system from a chaotic attractor to another coexisting attractor requires the solution of the boundary-value problem (33)-(34) for the Hamiltonian dynamics. The difficulty in solving these problems stems from the complexity of the system dynamics near a CA and is related, in particular, to the delicate problems of the uniqueness of the solution, its behaviour near a CA, and the boundary conditions at a CA. 

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. 

In Figure 4.47, best viewed in color, we observe a parallel time-shifted behavior of the three profiles that start from our three different initial values after the first 50 time units. This seems to indicate that for infinite time, the solutions that start at any initial value will eventually travel with the same period along the same unique loop that is depicted in three dimensions in Figure 4.46, but time-shifted one from the other like trains on the same track. More specifically, the profiles plot of Figure 4.47 indicates that in 2,000 time units, this periodic attractor loop is traversed about five and a half times, or approximately once every 380 time units. The system reaches an explosive state eventually (after an early possible higher-level explosion, depending on the choice of the initial value) every 380 time units. These periodic explosions occur with temperature... 

The very narrow region 4 of Figure 7.30(A) with Dslp < D < DHB has a unique periodic attractor with period one which emanates from the HB point at which the stable static branch loses its stability and becomes unstable as D decreases, see Figure 7.30(A). Region 5 with D > Dhb has a unique stable static attractor. 

Due to their characteristic course, there is always one unique intersection point of both curves for any given temperature and voltage. At this intersection point, both reaction rates become zero. Once the gas has reached it, both reactions stop and the gas composition does not change anymore along the spatial coordinate, unless the temperature or the cell voltage is changed. Thus, this is a stationary point. Because the reactions always run towards their equilibrium, this stationary point is an attractor. 

In many applications, and especially in population dynamics, one is not so much interested in the transient behaviour of (1.2.1), but rather in its asymptotic behaviour for n oo. Of special interest is the question whether the population described by (1.2.1) will settle down to some constant value Xoo(r) for n- oo, and here especially whether Xoo is finite or zero. In order for the as3onptotic value of Xn to be more significant, the result Xoo should be independent of the starting value xq of the population. This is possible if Xoo is an attractor of initial values 0 < xq < 1. For r < r = 3.5699... this is indeed the case and Xoo(r) is defined independently of Xq. But, as we saw above, the function Xoo(r) is not always unique. In order to capture even cyclic asymptotic behaviour, the following procedure 3delds an excellent representation of Xqo (r) in a single... 

Since the [ 0jt(q) basis does not depend on the PCB geometry, the class-1 functions generate Uk i) potentials that are confining, i.e., stationary with respect to -displacements around the vector of a unique minimum, dU , 4>k)/d = 0 at 1=1. In other words, there is a single global attractor in the 91 -space of -configurations that is totally determined by each 0jt(q)-function. This result also applies to the case of asymptotic attractors. 

For the trajectories defined in Equation (51) it is possible to prove that there is a unique limit cycle attractor. If p, q are two points on the boundary between regions I and II, and the distance between the two points d[p, ] = /, it is possible to show that the distance between the images of these points on the boundary between regions II and III is smaller than 4 II 20 + 5). If f ip) is the map that takes point p on the border between regions I and II back to this border after one complete circuit through the four regions, we compute... 

A Psuedo Maximum of Periodic Attractor Average on Periodic Attractor H Unstable Saddle Middle Steady Slate Unique Stable ARractor T Psuedo Minimum of Periodic ARractor... 

Unique Stable Attractor X Unstable Steady State o Average on Periodic Attractor Average on Chaotic Attractor A Psuedo Maximum for Qiaotic Attractor... 

HANUSSE - In fact the morphology analysis is performed on the complete trajectory given by the simulation of the full model. One interesting result is that the results do not seem to depend on the space in which you work (concentrations, reaction rates, combinations of them). You obtain the same essential dynamical features. This stability of the solution is to be compared to that of the reconstruction procedure of three dimensional attractors from a unique time series, in studies of chaotic behaviour. 

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