Invariant set

The long term behavior of any system (3) is described by so-called invariant measures a probability measure /r is invariant, iff fi f B)) = ft(B) for all measurable subsets B C F. The associated invariant sets are defined by the property that B = f B). Throughout the paper we will restrict our attention to so-called SBR-measures (cf [16]), which are robust with respect to stochastic perturbations. Such measures are the only ones of physical interest. In view of the above considerations about modelling in terms of probabilities, the following interpretation will be crucial given an invariant measure n and a measurable set B C F, the value /r(B) may be understood as the probability of finding the system within B. 

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

In the unperturbed case, say 7 = 70, let the system have two disjoint invariant sets Bi and B2 associated with two eigenmodes and invariant inea-... 

After these preliminaries we are now ready for a mathematically precise definition of an almost invariant set. Let p M he any probability measure. Wc say that the set B is 5-almost invariant with respect to p if... 

For more than two almost invariant sets one has to consider all eigenmea-sures corresponding to eigenmodes for eigenvalues close to one. In this case, the following lemma will be helpful. 

Let us introduce a suitably simple example in order to illustrate the notion of almost invariant sets and the performance of our algorithm for Hamiltonian systems. For p = pi,P2),q = (91,92) consider the potential... 

Almost Invariant Sets Recall that the relevant almost invariant sets correspond to eigenvalues X 1 with A < 1 of the associated Frobenius-Perron operator. 

Based on observations concerning the dynamical behavior we already conjectured that there exist seven almost invariant sets - a conjecture that we now want to check numerically. We employ the subdivision algorithm for subtrajectories of length mr = 0.1. The final box-collection corresponding to the total energy E = 4.5 after 18 subdivision steps consists of 18963 boxes. 

The invariant measure corresponding to Aj = 1 has already been shown in Fig. 6. Next, we discuss the information provided by the eigenmeasure U2 corresponding to A2. The box coverings in the two parts of Fig. 7 approximate two sets Bi and B2, where the discrete density of 1 2 is positive resp. negative. We observe, that for 7 > 4.5 in (15) the energy E = 4.5 of the system would not be sufficient to move from Bi to B2 or vice versa. That is, in this case Bi and B2 would be invariant sets. Thus, we are exactly in the situation illustrated in our Gedankenexperiment in Section 3.1. 

The third eigenmeasure 1 3 corresponding to A3 provides information about three additional almost invariant sets on the left hand side in Fig. 8 we have the set corresponding to the oscillation C D, whereas on the fight hand side the two almost invariant sets around the equilibria A and B are identified. Again the boxes shown in the two parts of Fig. 8 approximate two sets where the diserete density of 1/3 is positive resp. negative. In this case we can use Proposition 2 and the fact that A and B are symmetrically lelated to conclude that for all these almost invariant sets 5 > A3 = 0.9891. 

Fig. 9. Illustration of four almost invariant sets with respect to the probability measure i/4. The coloring is done according to the magnitude of the discrete density.

We examined the reaction of triose reductone with both periodate and iodate (55,56), and found that, whereas iodine was invariably set free from both sodium periodate and sodium iodate if the concentration of the reductone were greater than 10 3M, no iodine was liberated at lower concentrations (e.g. 6 x 10 4M) of substrate, even in the presence of relatively large amounts of the oxidants. 

J. Aguirre, J. C. Vallejo, and M. A. F. Sanjuan, Wada basins and chaotic invariant sets in the Henon-Heiles system, Phys. Rev. E 64, 066208 (2001). 

A mathematical formulation based on uneven discretization of the time horizon for the reduction of freshwater utilization and wastewater production in batch processes has been developed. The formulation, which is founded on the exploitation of water reuse and recycle opportunities within one or more processes with a common single contaminant, is applicable to both multipurpose and multiproduct batch facilities. The main advantages of the formulation are its ability to capture the essence of time with relative exactness, adaptability to various performance indices (objective functions) and its structure that renders it solvable within a reasonable CPU time. Capturing the essence of time sets this formulation apart from most published methods in the field of batch process integration. The latter are based on the assumption that scheduling of the entire process is known a priori, thereby specifying the start and/or end times for the operations of interest. This assumption is not necessary in the model presented in this chapter, since water reuse/recycle opportunities can be explored within a broader scheduling framework. In this instance, only duration rather start/end time is necessary. Moreover, the removal of this assumption allows problem analysis to be performed over an unlimited time horizon. The specification of start and end times invariably sets limitations on the time horizon over which water reuse/recycle opportunities can be explored. In the four scenarios explored in... 

This is a holomorphic map from M to 0l(P). (More detailed explanation of pc will be given later.) Then is GL(P)-invariant set. Hence by Theorem 3.12, we have... 

Fig. 3.11. Six gauge invariant sets of diagrams for corrections of order a Za) m...

This correction is induced by the gauge invariant set of diagrams in Fig. 3.11 (d) with the polarization operator insertions in the radiative photon. The respective radiatively corrected electron factor is given by the expression [50]... 

Relying on the experience with the corrections of order a ZaY we expect that the numerically dominant part of the corrections of order Y ZaY will be generated by the gauge invariant set of diagrams with insertions of three... 

The next correction of order a Za.)EF is generated by the gauge invariant set of diagrams in Fig. 9.8(c). The respective analytic expression is obtained from the skeleton integral by simultaneous insertion in the integrand of the one-loop polarization function Ii k) and of the electron factor F k). 

By far the most difficult task in calculations of corrections of order a Za)EF to HFS is connected with the last gauge invariant set of diagrams in Fig. 9.8(f), which consists of nineteen topologically different diagrams [17] presented in Fig. 9.9 (compare a similar set of diagrams in Fig. 3.12 in the case of the... 

This independency is related to the fact that the semiclassical calculation of the scattering amplitudes involves classical orbits belonging to an invariant set that is complementary to the set of trapped orbits in phase space [56]. The trapped orbits form the so-called repeller in systems where all the orbits are unstable of saddle type. The scattering orbits, by contrast, stay for a finite time in the scattering region. Even though the scattering orbits are controlled... 

We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [19]. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like Hgl2, CO2, and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. 

Although unstable, this periodic orbit is an example of classical motion which leaves the molecule bounded. Other periodic and nonperiodic trajectories of this kind may exist at higher energies. The set of all the trajectories of a given energy shell that do not lead to dissociation under either forwarder backward-time propagation is invariant under the classical flow. When all trajectories belonging to this invariant set are unstable, the set is called the repeller [19, 33, 35, 48]. There also exist trajectories that approach the repeller in the future but dissociate in the past, which form the stable manifolds of the repeller Reciprocally, the trajectories that approach the... 

The invariant set undergoes bifurcation sequences in the course of which its topology and its stability are modified. These bifurcations are responsible... 

The fact that the antipitchfork bifurcation is supercritical implies that all these new trapped orbits of the invariant set are bom above the bifurcation... 

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