Asymptotic solutions convergence

We remark that if (4.1) is autonomous and is either competitive or cooperative then we are free to choose w and a corresponding Poincare map Theorem 4.2 implies that every bounded solution of (4.1) is asymptotic to an w-periodic solution. Since w is arbitrary, it follows that every bounded solution converges to a rest point. 

This equation is valid for all times, but the convergence of the series is very slow for short times. For short times, a better solution can be obtained by taking Laplace transform of the governing equation and then find the asymptotic solution when s ->... 

What about the sequence for A, = 3.7 It does not seem to repeat. In fact, no matter how far we extend the sequence [assuming that our calculator computes eq. (8.13) to infinite precision], the numbers will never repeat the sequence is chaotic. One can demonstrate, either numerically or analytically, that the set of bifurcation points X/ converges to the value X = 3.569946, and that at higher values of X, the sequence is chaotic. In Figure 8.12, chaotic solutions appear as continuous, filled regions the asymptotic solution eventually comes arbitrarily close to all values of x within a certain range. You may have noticed that within the chaotic solutions are embedded narrow windows of periodic solutions. We shall not discuss these windows in any detail here, but we point out that not only the existence but also the order of the periods of these regions is predicted by theory (Metropolis et al., 1973) and found in experiments on chemical systems (Coffman et al., 1987). Because the order in which the windows occur is predicted... 

Figure 12.3b shows the widths of the Bragg layers as a function of the layer number of the CBNL depicted in Fig. 12.3a. There are two notable properties of the Bragg layers (1) the width of the high-index layers is smaller than the width of the low-index layers, and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional DBRs as well and stems from the dependence of the spatial oscillation period, or the wavelength, on the index of refraction. The second property is unique to the cylindrical geometry and arises from the nonperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

In the preceding paragraphs of this section we have summed the terms arising from the partial expansion of the exponentials occurring in the coefficients of the powers of particle concentrations to obtain a series of multiple infinite sums, the terms of which are convergent. The terms in S(R) are of the same form as those in the Mayer solution theory, apart from replacement of integration by summation and the fact that mu differs from the solution value because of the discreteness of the lattice. The evaluation of wi - is outlined in the next section. It is found that the asymptotic form is... 

Perturbation method as asymptotic expansion. In the preceding section we have shown that the solution power series of k, which is absolutely convergent for every k and for every initial state bounded operator. But this is the only case in which we have succeeded in proving the convergence of the series (16. 9). 

Such a full investigation is virtually impossible except in some very simple cases, and is even then usually very difficult. In particular for quadratically convergent methods, the convergence region is usually bounded by a fractal instead of a regular curve. Out of necessity, the convergence properties studied are usually some necessary criterion on f and c near the desired solution, and the influence of that criterion on the asymptotic error. 

Chaotic solutions are those which are neither periodic nor asymptotic to a periodic solution but are characterized by extreme sensitivity to initial conditions. A solution that is asymptotic to a stable periodic solution is not sensitive to starting point, for, if we start from two nearby values, the trajectories will both converge on the same periodic solution and get closer and closer together. With a chaotic solution, the trajectories starting at two nearby points ultimately diverge no matter how close they may have been at the beginning. If /( )( ) denotes the nth iterate,... 

Realizing that Eq. (13) gives an explicit solution of (1) with an appropriate V, in terms of logarithmic derivatives, it is possible to identify u with the well-known Jost solution denoted as/(r, 2), see more below and Ref. [44], which here must be proportional to the Weyl s solution x(f, )- With this identification, we obtain the generalized Titchmarsh formula (generalized since it applies to all asymptotically convergent exponential-type solutions commensurate with Weyl s limit point classification)... 

Any two solutions lying in the same balance polyhedron converge in the metric (161) and p(z(1)(t), z(2)(t)) -> 0 at t -> co. It results, in particular, in the existence, uniqueness and asymptotic stability (in the large) of the steady state in the balance polyhedron. This was confirmed by Vol pert et al. [45] and partly and simultaneously by Bykov et al. [46-48], (Note that all the considerations given also hold for the n A -> LmiBi- type reaction systems.)... 

P 7] The topic has only been treated theoretically so far [28], A mathematical model was set up slip boundary conditions were used and the Navier-Stokes equation was solved to obtain two-dimensional electroosmotic flows for various distributions of the C, potential. The flow field was determined analytically using a Fourier series to allow one tracking of passive tracer particles for flow visualization. It was chosen to study the asymptotic behavior of the series components to overcome the limits of Fourier series with regard to slow convergence. In this way, with only a few terms highly accurate solutions are yielded. Then, alternation between two flow fields is used to induce chaotic advection. This is achieved by periodic alteration of the electrodes potentials. 

In contrast, the asymptotic approach puts minimal strain on the computer but demands more of the modeller. The convergence of the computed solutions is usually easy to test with respect to spatial and temporal resolution, but situations exist where reducing the timestep can make an asymptotic treatment of a "stiff" phenomenon less accurate rather than more accurate. This follows because the disparity of time scales between fast and slow phenomena is often exploited in the asymptotic approach rather than tolerated. Furthermore, the non-convergence of any particular solution is often easier to spot in timestep splitting with asymptotics because the manner of degradation is usually catastrophic. In kinetics calculations, lack of conservation of mass or atoms signals inaccuracy rather clearly. 

Intuitively, one can imagine that the field generated by an arbitrary source, located within a bounded domain of space, can be approximated accurately enough by a spherical wave at a large distance from the source. We can expect, therefore, that the asymptotic behavior of this field at infinity in the case of divergent and convergent waves can be characterized by formulae (13.161) and (13.162) as well. These heuristic considerations provide a basis for choosing only those solutions of the problem defined by (13.155) and (13.156) which satisfy additionally condition (13.161). 

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