### Table IV. Non-linear dynamics of frame: CPU time for grade zero Jaumann stress rate hypoelasticity with updated Lagrangian formulation.

in SUMMARY

### Table V. Non-linear dynamics of frame: CPU time for St. Venant-Kirchhoff hyper- elasto-plasticity with total Lagrangian formulation.

in SUMMARY

### Table VI. Non-linear dynamics of frame: CPU time for grade zero Jaumann stress rate hypoelasto-plasticity with updated Lagrangian formulation.

in SUMMARY

### Table 1. Analogy between the Chirikov criterion in 1-D nonlinear dynamics and the Reynolds apos; condition for an incompressible #0Duid. As suggested by Teng, it is interesting to draw an analogy between chaos in 1-D nonlinear dynamics and turbulence in #0Duid dynamics.4 The idea is based on the observation #28Liouville theorem in a Hamiltonian system#29 that the beam distribution in phase space behaves as a viscous incompressible #0Duid in the real space. By writing down the Hamilton equation on the one hand and the #0Duid equation on the other, it is possible to establish the analogy as given in Table 1. Chirikov criterion is then equivalent to the Reynolds

### Table 6: Herd dynamics

2006

"... In PAGE 21: ... The sample splitting generated by the regression trees method thus reinforces the finding of a unique equilibrium for lower ability herders and multiple equilibria for the rest. Our estimates of the herd growth models associated with each terminal node appear in Table6 and are graphed in figure 12.29 Expected herd dynamics appear highly nonlinear in each regime.... ..."

### Table 1 Number of iterations and CPU time (in seconds), with np = 60, for single shooting (SS), multiple shooting (MS), and modi ed multiple shooting (MMS) In general, the single shooting technique requires more iterations as compared to the other techniques. During the process of obtaining optimal trajectories, we con rmed the lack of robustness of single shooting with respect to the initial guess and bounds on optimizing variables. For instance, the method failed to converge unless the control was constrained to be non-negative. For multiple shooting, the total time interval was divided into 10 equal shooting intervals. The computation times were signi cant and increased rapidly as the mesh became ner. The multiple shooting technique provided optimal trajectories for a varying degree of nonlinearity in the dynamical system as compared to single shooting.

### Table 6: Comparison of linear and non-linear estimates of the RMS energy data where the energy values are estimated from either static, dynamic or combined static and dynamic visual data representations. Figures show the correlation coe cients (and their standard deviations) measured between the estimates and the true data.

### Table 2: Key Properties of Surface Systems Most models are usually constructed by combining basic principles (e.g. conservation laws, constitutive laws) with insight about what the important aspects of the dynamics are likely to be. In our view, the following are the critical issues that are generic to sur- face-dynamics models. Most of these are associated with nonlinearity in one way or an- other:

2003

"... In PAGE 13: ...Table2... ..."

### Table 3. Comparison between di erent non-linear approxima- tions to the for the unsmoothed elds up to the rst corrective term beyond tree-level (the one-loop term). The asterisk denotes the results obtained within the diagrammatic approach for the relevant dynamics.

1998

"... In PAGE 12: ... Only a monotonic enhancement of the scaling properties is observed. In Table3 we compare the results from di erent approx- imations to the exact dynamics in the perturbative regime (exact PT), available in the literature and we give as well the SC predictions in the same regime. We recall that the FFA is based on a linearization of the peculiar velocity eld.... In PAGE 12: ... (1994). As summa- rized in Table3 , the SC model yields the best estimates for the loop corrections in PT within the available non-linear approximations. 5.... ..."

### Table 4: Percentage of SIR steps: nonlinear time series This model requires simulation of more samples than the preceding one. In fact, the variance of the dynamic noise is more important and more trajectories are necessary to explore the space. The most interesting algorithm is the SIS with a suboptimal importance function which greatly limits the number of resampling steps over the prior importance function while avoiding a MC integration step needed to evaluate the optimal importance

1998

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