Stable integral trajectory

FVom the results of Anosov, Klingenberg, and Takens, it follows that in the set of all geodesic flows on smooth Riemannian manifolds there exists an open everywhere dense subset of flows without closed stable integral trajectories [170], 17l. This means that the property of a geodesic flow to have no stable trajectories is the property of general position. Recall once again that we mean strong stability (see Definition 2.1.2). 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

Indeed, as is seen in Fig. 3, under small pertubation of initial data near the major and minor axeSy the integral trajectory of the Euler equations remains a small closed curve (circle) and, therefore, the stationary solution K t) is stable. In the case of the mean axis, the small perturbation of the stationary solution makes the endpoint of the angular momentum vector K(t) move along a closed trajectory of "large diameter, as a result of which the vector K(t) quickly moves away from the mean ellipsoid axis (Fig. 3). 

In recent years many new results have been obtained concerning integration of Hamiltonian systems on symplectic manifolds. See, for instance, the survey in [143], [188]. Of particular interest in this connection is the discovery of stable closed integral trajectories of integrable systems. Such trajectories correspond to periodic motions of the system. 

Definition 2.1.2 Let 7 be a closed integral trajectory of a system t on a surface Q (that b, a periodic solution). We say that the trajectory 7 b stable if its certain tubular neighbourhood b fully fibred into two-dimensional tori which are invariant with respect to the system v, that b, all integral trajectories close to 7 "are located on invariant two-dimensional tori whose common axb b the circle 7 (Fig. 23). 

In both versions of the surgery, it is obvious that a deformed circle (or a doubled circle r ) describe a two-dimensional disk with two holes. Moreover, after one revolution along the circles 5, this disk returns to its initial place, but the two holes of this disk exchange places. We have obtained none other than the manifold fibred over the base with fibre JV. Let be a critical circle of index 2 or 0. Consider its tubular neighbourhood U(S ), Since / is a Bott function, it follows that U S ) fibres into tori which contract onto the circle and are level surfaces of the function /. Since these are Liouville tori, the integral trajectories of the system v lie on them. Then the circle 5 is obviously stable. 

Along stable separatrix surfaces, integral trajectories tend to the critical point (with increasing time), while along unstable separatrix surfaces they move away from the critical point (with increasing time). Condition 2 consists in the fact that A-i- = A., that is, the stable separatrix surface A+ of the point (x+, +) coincides with the unstable separatrix surface A- of the point (x, ). FVom this it follows, in particular, that ffo( +> +) = o( - -) ... 

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

Thus these integrators are measure preserving and give trajectories that satisfy the Liouville theorem. [12] This is an important property of symplectic integrators, and, as mentioned before, it is this property that makes these integrators more stable than non-symplectic integrators. [30, 33]... 

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

A Jacobi or Gauss-Seidel iteration on (6) will provide us with the coordinates of the steady state, or it will cycle indefinitely, depending on the slope of the functions. On the other hand, determining the trajectory by numerical integration of equation (5) will lead to a stable steady state or to a limit cycle depending on the slope of the functions. There is thus an obvious formal similarity between the two situations. However, the steepness corresponding to the transition from a punctual to a cyclic attractor is much smaller in the first case (in which the cyclic attractor is an iteration artifact) as in the second case (in which the cyclic attractor is close to the real trajectory). 

On the other hand, the trajectory sensitivity equations method requires simultaneous integration of a greater number of equations than the adjoint system approach. However, it is more stable than the adjoint system approach due to the requirement of forward integration only. It is usually preferred in the area of parameter estimation and sensitivity (Kalogerakis and Luus, 1983 Caracotsios and... 

The trajectory of an ion will be stable if the values of x and y never reach r0, thus if it never hits the rods. To obtain the values of either x or y during the time, these equations need to be integrated. The following equation was established in 1866 by the physicist Mathieu in order to describe the propagation of waves in membranes ... 

We will not attempt to integrate these equations [6], We only need to recognize that they establish a relationship between the coordinates of an ion and time. As long as x and y, which determine the position of an ion from the centre of the rods, both remain less than r0, the ion will be able to pass the quadrupole without touching the rods. Otherwise, the ion discharges itself against a rod and is not detected. Figure 2.6 shows stable and unstable trajectories in a quadrupole [7]. 

To have a stable trajectory, the movement of the ions must be such that during this time the coordinates never reach or exceed r0 (r-stable) and z0 (z-stable). The complete integration of the Mathieu equation by the method of Floquet and Fourier requires the use of a function e( +I. Real solutions correspond to a continuously increasing, and thus unstable, trajectory. Only purely imaginary solutions correspond to stable trajectories. This requires both a = 0 and 0 ftu < 1 (Figure 2.14). 

Integrate the trajectory along this direction one finds the stable/unstable manifold itself. 

Figure 28. Numerically integrated dynamics around the main RE at = 0.001 atomic units. R = Pr = 0 is the RE. The dynamics is projected onto the R pr plane. Thick lines represent the stable and unstable manifolds of the RE thin lines represent several trajectories. Ti and 72, nonreactive and reactive trajectories, respectively.

Further efficiency is obtained by making the path length variable [87]. As only shots from the barrier itself are useful in stochastic path sampling, it is natural to stop with the integration of the equation of motion once one reaches a stable state, provided that the trajectory is then really committed to a stable state [87,88]. 

The Langevin equation [Eq. (Ill)] was integrated numerically following the procedure developed in Ref. 90. Whence, we obtained the trajectories of the particle shown in Fig. 17. In the Brownian limit, we reproduce qualitatively the behavior found in Ref. 89. Accordingly, the fluctuations around the positions of the minima are localized in the sense that their width is clearly smaller than the distance between the minima and barrier. In contrast, for progressively smaller stable index a, characteristic spikes become visible, and the individual sojourn times in one of the potential wells decrease. In particular, we note that single spikes can be of the order of or larger than the distance between the two potential minima. 

Figure 17. Typical trajectories for different stable indexes a obtained from numerical integration of the Langevin equation [Eq. (111)]. The dashed lines represent the potential minima at 1. In the Brownian case a = 2, previously reported behavior is recovered [89]. In the Levy stable case, occasional long jumps of the order of or larger than the separation of the minima can be observed. Note the different scales.

In support of the philosophical appeal is the fact that Eq. (4) is an integral while in Eqs. (2) we use derivatives. Numerical estimates of integrals are, in general, more accurate and more stable compared to estimates of derivatives. On the other hand, computations of the whole path are more expensive than the calculation of one temporal slice of the trajectory at a time. The computational effort is larger in the boundary value formulation by at least a factor of N, where N is the number of time shces, compared to the calculation of a step in the initial value approach. To make the global approach computationally attractive (assuming that it does work), the gain in step size must be substantial. 

---