Stability eigenvalues

Gutzwiller trace formula should present a peak. At the leading order, the amplitude is predicted to have a divergence because some stability eigenvalues pass through = 1 so that the denominator vanishes in Eq. (2.13). Consequently, uniform semiclassical approximations are required in the vicinity of bifurcations, which show that the amplitude is strongly peaked but still remains finite. 

The resonances are then obtained by searching for the complex zeros of the zeta functions (4.12) in the complex surface of the energy. Assuming that the action is approximately linear, S(E,J) = T(E - Ei), while the stability eigenvalues are approximately constant near the saddle energy E, the quantization condition (4.12) gives the resonances [10]... 

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal iepeller. As we mentioned in Section II, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodic-orbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. 

In the three-branch horseshoe, the periodic oibit 0 is hyperbolic with reflection and has a Maslov index equal to no = 3 while the off-diagonal orbits 1 and 2 are hyperbolic without reflection with the Maslov index n = 2 [10]. Fitting of numerical actions, stability eigenvalues, and rotation numbers to polynomial functions in E can then be used to reproduce the analytical dependence on E. The resonance spectrum is obtained in terms of the zeros of (4.16) in the complex energy surface. 

To completely specify all the resonance energies as a function of 1 and y, we have fit our stabilization eigenvalues to the following multinomial... 

The type of steady state illustrated by the Brusselator example is called a focus because it is the pivot point for the spiraling trajectories that move toward or away from it. As we will see, the existence of a focus is often the prerequisite for the existence of oscillatory solutions to the full equations of motion. In particular, we look for an unstable focus (one for which the real part of the stability eigenvalues is positive) because the trajectories that spiral away from the focus may eventually reach a stable cyclic path surrounding that focus called a limit cycle. 

A Lyapunov exponent is a generalized measure trf the growth or decay of small perturbations away from a particular dynamical state. For perturbations around a fixed point or steady state, the Lyapunov exponents are identical to the stability eigenvalues of the Jacobian matrix discussed in an earlier section. For a limit cycle, the Lyapunov exponents are called Floquet exponents and are determined by carrying out a stability analysis in which perturbations are applied to the asymptotic, periodic state that characterizes the limit cycle. For chaotic states, at least one of the Lyapunov exponents will mm out to be positive. Algorithms for the calculation of Lyapunov exponents are discussed in a later section in conjunction with the analysis of experimental data. These algorithms can be used for simulations that yield possibly chaotic results as well as for the analysis of experimental data. 

As described in a previous section, the Lyapunov exponents are a generalized measure of the growth or decay of perturbations that might be applied to a given dynamical state they are identical to the stability eigenvalues for a steady state and the Floquet exponents for a limit cycle. For aperiodic motion at least one of the Lyapunov exponents will be positive, so it is generally sufficient to calculate just the largest Lyapunov exponent. 

Systems without delay that obey mass action kinetics give rise to polynomial equations of order m for the stability eigenvalues cOj]j= m- As eq. (10.20) shows, whenever a derivative is taken with respect to a delayed variable in a DDE, the resulting term in the Jacobian J,. must be multiplied by a factor exp (-cur). The inclusion of delay leads to a transcendental equation which, in general, has an infinite number of roots a>j. 

We wish to relate the eigenvalue problem (2.9.13)—(2.9.16) to the eigenvalue problem for uniqueness. Now, while the uniqueness eigenvalue problem refers to a time independent situation and can be described by the extents, the stability eigenvalue problem (2.9.13)- (2.9.16)... 

There are a number of routes to self organization in reaction transport systems. The types of linear instability resulting in the monotonic growth of perturbations has been discussed [ 3 ] and are summarized in Fig. 1. Shown are four cases of the dependence of stability eigenvalues for perturbations, of wave vector k, from the uniform state. The first two cases were distinguished in [4] where the extrinsic type of instability was introduced. In the intrinsic case patterns arise at a well defined wave vector... 

For stability reasons, the micro-step-size 5t has to be chosen smaller than the inverse of the largest eigenvalue of the (scaled) truncated quantum operator % This can imply a very small value of 5t compared to... 

This set of ordinaiy differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler s method is used for integration, the time step is hmited by... 

The stability of the (lAe)-family is lost at a Hopf bifurcation point denoted by the open circle (o) on Fig. 7, where the real parts of a complex conjugate pair of eigenvalues change sign. No stable time-periodic solutions were found near this point, indicating that the time-periodic states evolve sub-critically in P and are unstable. Haug (1986) predicted Hopf bifurcations for codimension two bifurcations of the form shown in Fig. 7. but did not compute the stability of the time-periodic states. 

Expanding the sample size to 2Xc admits the other shape families shown on Fig. 6 into the analysis and leads to additional codimension-two interactions between the shapes is the (1A<.)- family and shapes with other numbers of cells in the sample. The bifurcation diagram computed for this sample size with System I and k = 0.865 is shown as Fig. 11. The (lAc)- and (Ac/2)-families are exactly as computed in the smaller sample size, but the stability of the cell shapes is altered by perturbations that are admissible is the larger sample. The secondary bifurcation between the (lAc)- and (2Ae/3)-families is also a result of a codimension two interaction of these families at a slightly different wavelength. Two other secondary bifurcation points are located along the (lAc)-family and may be intersections with the (4Ac and (4A<./7) families, as is expected because of the nearly multiple eigenvalues for these families. 

Practical experience has shown that (i) if we have a relatively large number of data points, the prior has an insignificant effect on the parameter estimates (ii) if the parameter estimation problem is ill-posed, use of "prior" information has a stabilizing effect. As seen from Equation 8.48, all the eigenvalues of matrix A are increased by the addition of positive terms in its diagonal. It acts almost like Mar-quadt s modification as far as convergence characteristics are concerned. 

Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems and stability problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundary conditions are prescribed. 

The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations... 

The question of stiffness then depends on the solution at the current time. Consequently nonlinear problems can be stiff during one time period and not stiff during another. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stability and accuracy of the numerical scheme and the step size used. 

Note that the stability of the spectra with respect to k m and um must be carefully checked. Within this enlarged base, the eigenvalues equation of the total Hamiltonian (104) may be written... 

The exact approach to the problem of dynamic (linear) stability is based on the solution of the equations for small perturbations, and finding eigenvalues and eigenfunctions of these equations. In a conservative system a variational principle may be derived, which determines the exact value of eigenfrequency... 

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

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