Finite-parameter family

Therefore, Andronov s approach (Sec. 8.4) for studying dynamical models has to be corrected in cases where a complete bifurcation analysis may not be possible without moduli. We note, however, that if some fine delicate phenomena may be ignored, or if the problem is restricted to the analysis of non-wandering orbits like equilibrium states, periodic and quasiperiodic motions, a study of the main bifurcations in systems with simple dynamics still remains realistic within the framework of finite-parameter families under certain reasonable requirements (Sec. 8.4). 

Bifurcations in finite-parameter families of systems. Andronov s setup... 

Consider some finite-parameter family of smooth systems Xg, where e = ( 1,..., 6p) assumes its values from some region V e W. If is non-rough, then q is said to be a bifurcation parameter value. The set of all such values in V is called a bifurcation set. It is obvious that once we know the bifurcation set, we can identify all regions of structural stability in the parameter space. Hence, the first step in the study of a model is identifying its bifurcation set. This emphasizes a special role of the theory of bifurcations among all tools of nonlinear dynamics. 

In principle, to solve a bifurcation problem we need to consider all systems close to XsQ, This means that we must consider the Banach space of all small perturbations." On the other hand, when it is possible to reduce the analysis to some appropriate finite-parameter family of systems, the study is simplified significantly. 

This idea was proposed by Andronov and Leontovich in their first work [9] which deals with primary bifurcations of limit cycles on the plane. Further developments of the theory of bifurcations, internal to the Morse-Smale class, has also confirmed the sufficiency of using finite-parameter families for a rather large number of problems. 

Note that she did not use finite-parameter families. Nevertheless, it is understood that an appropriate transverse family can be found such that the governing parameters are /i, e,, crj,..., (Tn-i in the first case of Theorem 13.5, or /i, e, si, 0 1,..., in the second case. 

Using the Bessel approximation as a start-up artifice always gives us a two-parameter family of solutions (or a one-parameter family for each initial abscissa xq) as is always the case with a second-order differential equation. The parameter x+ is directly related to the curvature yo or y" " of a given profile. However, to singularize a profile that passes through a point (x°, y°) three parameters are necessary (x°, y°, q>°), although solutions may not exist for some combination of parametric values (for example, if x° = 0, no profiles with finite nonzero slope at x° exist). In all cases, once x" " and y" " have been determined, we may proceed... 

It was solved numerically using the alternating-direction implicit (ADI) finite difference method (5). The steady-state results were obtained as a long time limit and presented in the form of two-parameter families of working curves (5). These represent steady-state tip current or collection efficiency as functions of K = akc/D and L. 

An explicit mathematical formulation to the finite-parameter approach to the local bifurcations was given by Arnold [19], based on the notion of versal families. Roughly speaking, versality is a kind of structural stability of the family in the space of families of dynamical systems. Different versions of such stability are discussed in detail in [97]. 

It should be noted that constructing the versal families is realistic only in these simple cases, and in a few special cases. For example, a finite-parameter versal family cannot be constructed for the bifurcation of a periodic orbit with one pair of complex multipliers Nevertheless, this problem does... 

Families of finite elements and their corresponding shape functions, schemes for derivation of the elemental stiffness equations (i.e. the working equations) and updating of non-linear physical parameters in polymer processing flow simulations have been discussed in previous chapters. However, except for a brief explanation in the worked examples in Chapter 2, any detailed discussion of the numerical solution of the global set of algebraic equations has, so far, been avoided. We now turn our attention to this important topic. 

For each zero-one law specific sharp transitions exist if two systems in a one-parametric family have different zero-one steady states or relaxation modes, then somewhere between a point of jump exists. Of course, for given finite values of parameters this will be not a point of discontinuity, but rather a thin zone of fast change. At such a point the dominant system changes. We can call this change a... 

Two of them occur when (T, Tp)>(T, Tp). Then both families of chains order simultaneously at a temperature between max(T, Tp) and min(T, Tp). Depending on the parameters, the value of q at this temperature is either finite or equal to... 

The parameters of Eq. (7) can be varied to delineate spin-charge separation or other properties with e-e correlations. Such variations are not possible experimentally, however. The sp carbons of conjugated polymers suggest similar e-e interactions in these chemically related systems. We have systems with almost constant V R) and variable [41] 8 due to bond lengths, topology, or heteroatoms. As shown by the open circles in Fig. 6.16, the observed [135] Eja/Eib ratios indicate both polymers with 2A above and below B. When 2A is higher, we have Si = B, and fluorescence is in fact typical in these families [41]. Experimental data on polyenes [136] and a PDA oligomer [137] are shown as open circles at finite VN in Fig. 6.16. 

Another approach to TG/SC experiments does not rely on the mediator feedback [56]. The reactant galvanostatically electrogenerated at the tip diffuses to the substrate and undergoes the reaction of interest at its surface. The substrate current is recorded as a function of either time or the tip/ substrate separation distance (approach cnrves). The theory for transient responses, steady-state TG/SC approach curves, and polarization cnrves (i.e., 4 vs. E ) was generated solving the diffnsion problem numerically (an explicit finite difference method was used). The substrate process was treated as a first-order irreversible reaction, and the effects of its rate constant and the experimental parameters were illnstrated by families of the dimensionless working curves (Figure 5.11). 

The finite element description of the nonlinear viscoelastic behavior of technical fabric was presented by Klosowski et al. [65]. The technical fabric called Panama used in this model was made of two polyester thread families woven perpendicularly to each other with the 2/2 weave. The long term uniaxial creep laboratory tests in directions were conducted at five different constant stress levels. The dense net model [66] together with the Schapery one-integral viscoelastic constitutive model [67] was assumed for the fabric behavior characterization and the least square method in the Levenberg-Marquardt variant was used for the parameters identification. 

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