Wandering trajectory

Equilibrium states and periodic orbits are non-wandering trajectories. In the former case, any neighborhood of an equilibrium state will contain it forever in the case of a periodic orbit, any of its points returns infinitely many times to an initial neighborhood simply because of periodicity. 

In fact, it can be shown that periodic orbits and equilibrium states are the only non-wandering trajectories of Morse-Smale systems. Axiom 1 excludes the existence of unclosed self-limit (P-stable) trajectories in view of BirkhofF s Theorem 7.2. The existence of homoclinic orbits is prohibited by Theorems 7.9 and 7.11 below. Next, it is not hard to extract from Theorem 7.12 that an u)-limit (a-limit) set of any trajectory of a Morse-Smale system is an equilibrium state or a periodic orbit. 

The fact that there are a finite number of non-wandering trajectories in Morse-Smale systems implies that any chain has a finite length which does not exceed the total number of non-wandering trajectories. Moreover, a maximal chain can end only at a stable equilibrium state or a periodic orbit. 

There is no doubt that some subtle aspects of the behavior of homoclinic and heteroclinic trajectories might not be important for nonlinear dynamics since they refiect only fine nuances of the transient process. On the other hand, when we deal with non-wandering trajectories, such as near a homoclinic loop to a saddle-focus with i/ < 1, the associated fi-moduli (i.e. the topological invariants on the non-wandering set) will be of primary importance because they may be employed as parameters governing the bifurcations see [62, 63]. 

The numerical trajectory will wander off this energy surface. If the trajectory is stable it will wander on an energy shell... 

Although the n-tori (conventionally called KAM-tori or KAM surfaces) may be distorted versions of the curves appearing for the integrable Hamiltonian Ho alone, the qualitative nature of the motion remains the same. Initial conditions starting from the exceptional set result in trajectories that wander freely over the energy surface (defined by H = constant). 

One increasingly popular method which lead to non-dynamical trajectories is replica-exchange MC or MD [11-13], which employs parallel simulations at a ladder of temperatures. The "trajectory" at any given temperature includes repeated visits from a number of (physically continuous) trajectories wandering in temperature space. Because the continuous trajectories are correlated in the usual sequential way, their intermittent — that is, non-sequential — visits to the various specific temperatures produce non-sequential correlations when one of those temperatures is considered as a separate ensemble or "trajectory" [14]. Less prominent examples of non-dynamical simulations occur in a broad class of polymer-growth algorithms (e.g., refs. 15-17). 

Such a trajectory has an important property. It could converge towards a limiting scheme, it could repeat itself, or it could wander around randomly. Discount the last possibility. If it repeats itself after a fixed number of steps, then we can take that cycle of steps together as a single scheme of high arity and discover that it is essentially a stationary scheme. 

To visualize some of the effects described in the previous section, Poincare showed that the behavior of two degree-of-freedom nonlinear systems can be profitably studied by mapping the dynamics onto a well-chosen plane. This is because the conservation of energy requires all trajectories to wander on a three-dimensional hypersurface. In his honor, these maps are often referred to as Poincare maps. The plane chosen to map the dynamics onto is referred to as a surface of section. 

The condition of self-avoidance of a random walk trajectory on //-dimensional lattice demands the step not to fall twice into the same cell. From the point of view of chain link distribution over cells it means that every cell cannot contain more than one chain link. Chain links are inseparable. They cannot be tom off one from another and placed to cells in random order. Consequently, the numbering of chain links corresponding to wandering steps is their significant distinction. That is why the quantity of different variants of iV distinctive chain links placement in Z identical cells under the condition that one cell cannot contain more than one chain link is equal to Z I Z-N) ... 

We will skip further details of this adventure story. We just need to emphasize one more thing before we get back to polymers. Since a Brownian particle moves due to collisions with molecules, its path breaks into a sequence of many very short flights and turns. In this sense, a Brownian trajectory is pretty similar to the shape of the pol Tner chains which we saw in Section 2.4 (Figure 2.6). Another obvious example of this sort is of a man who is lost in a forest, with no compass, and has no choice but to wander at random. 

For an isolated polymer chain, the problem is purely geometrical. Indeed, the spatial shape of an ideal chain resembles the path of a randomly wandering Brownian particle (see Chapter 6). What new features will the shape of the chain acquire, if we allow for the excluded volume Clearly, since the private space of each monomer is not available to the rest, the chain cannot possibly cross itself at any stage. This sort of behavior can be described as self-avoiding. For example, if there were an equivalent Brownian particle, it would not be allowed to cross its own track. A two-dimensional version of such a trajectory is sketched in Figure 8.2. Thus, we have made it a purely geometrical problem of self-avoiding random walks. 

The proof of this theorem relies on showing that if trajectories are started from a point sufficiently close to z they cannot wander away to infinity. Although the result holds in greater generality, it is easy to show under assumptions of local smoothness of U (which we are normally happy to make in molecular dynamics). For more discussion see the text [216]. If the potential is C, the linearized version of this system has kinetic plus potential form with Hamiltonian... 

One can see that the change of sign at s, does not change the value coin). Thus, fflfn) represents the probability that the trajectory of random wandering after n,- steps in i directions will end in one of 2 cells Mp s), coordinates of which are determined by vectors j= (s ), (=, d having a distinction in signs of their components s, only. These cells or states of chain end are equiprobable. 

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