Leading characteristic exponents

To distinguish between these two cases, we will call the equilibrium state a saddle in the first case, and a saddle-focus in the second case, for the sake of brevity. Note that this terminology differs from what we have used throughout the first part of this book. Namely, in this section we do not take into account whether the leading characteristic exponent A is real or complex. Thus, in this particular section, we call O a saddle if (13.5.1) and (13.5.2) are satisfied, even if Ai is complex. 

In the case of a saddle (the leading characteristic exponent Ai is real), the bifurcation diagram depends on the signs of the separatrix values Ai and A2, as well as on the way the homoclinic loops F1 and F2 enter the saddle at t = -f 00. Let us consider first the case where F1 and F2 enter the saddle tangentially to each other, i.e. bifurcations of the stable homoclinic butterfly. 

Note also that these results can be extended immediately to the case of a saddle with a multi-dimensional unstable manifold. Namely, if O has several characteristic exponents with positive real parts but the leading characteristic exponent 71 is real, i.e. if... 

Here p is the saddle index at the saddle-focus Oi, and z/ is the saddle index at the saddle O2 and P = Re A2/7I, where 7 denotes the positive characteristic exponents of O2, and A2 is the non-leading characteristic exponent of O2 nearest to the imaginary axis (Ai is the leading exponent sou = Xi/ and I < u < u recall also that p > 1 by assumption — the saddle values are negative). 

In particular, let the dimension of the unstable manifold of Oi be equal to the dimension of the imstable manifold of O2. Besides, let both the stable and unstable leading characteristic exponents at both 0 and O2 be real. Assume also that both heteroclinic orbits F 1 2 enter and leave the saddles along the leading directions. We also assume that the extended unstable manifold of one saddle is transverse to the stable manifold of the other saddle along every orbit Fi 2, and that the extended stable manifold of one saddle is transverse to the unstable manifold of the other saddle along F 1,2 as well. Under... 

When studying homoclinic bifurcations, an important characteristic of saddle equilibria is the sign of the saddle value a defined as the sum of the real parts of the two leading characteristic exponents nearest to the imaginary axis from the left and from the right. 

Introduce the second saddle value <72 as the sum of the three leading characteristic exponents at the saddle-focus. In the three-dimensional case, it is the divergence of the vector field at the origin. Here, the curve 0 2 = 0, given by the equation a = 6, intersects HS at (a = 6,6 = 7.19137). Above this point, <72 > 0. 

L. Mandelstam and N. Papalexi were first to establish the theory of the subharmonic resonance based directly on the theory of Poinear6 (Section 6.18). The derivation of this theory, together with the details of the electronic circuits, is given in25, or in an abridged version in6 (pages 464-473). The difficulty of the problem is due to the fact that this case is nonautonomous so that conditions of stability are determined in terms of the characteristic exponents, which always leads to rather long calculations. 

As mentioned above, for conservative scattering, where = 1, the leading eigenvalue turns to infinity, i.e., the characteristic exponent A turns to zero. On expanding the eigenfunction vector for small true absorption in powers of k, we find, by means of Eqs. (2), (3), (4), and (6), the following asymptotic formulae for (1 - a) 1 ... 

Note that an asymmetric density of jump lengths leads to the Riesz-Feller space-fractional derivative of order a and skewness 9 with the characteristic exponent... 

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

As we have seen above, the dynamics near the homoclinic loop to a saddle with real leading eigenvalues is essentially two-dimensional. New phenomena appear when we consider the case of a saddle-focus. Namely, we take a C -smooth (r > 2) system with an equilibrium state O of the saddle-focus saddle-focus (2,1) type (in the notation we introduced in Sec. 2.7). In other words, we assume that the equilibrium state has only one positive characteristic exponent 7 > 0, whereas the other characteristic exponents Ai, A2,..., are with negative real parts. Besides, we also assume that the leading (nearest to the imaginary axis) stable exponents consist of a complex conjugate pair Ai and A2 ... 

Consider an (n + l)-dimensional C -smooth (r > 4) system with a saddle equilibrium state O. Let O have only one positive characteristic exponent 7 > 0 the other characteristic exponents Ai, A2,..., A are assumed to have negative real parts. Moreover, we want the leading stable exponent Ai to be real ... 

The phase separation process at late times t is usually governed by a law of the type R t) oc f, where R t) is the characteristic domain size at time t, and n an exponent which depends on the universality class of the model and on the conservation laws in the dynamics. At the presence of amphiphiles, however, the situation is somewhat complicated by the fact that the amphiphiles aggregate at the interfaces and reduce the interfacial tension during the coarsening process, i.e., the interfacial tension depends on the time. This leads to a pronounced slowing down at late times. In order to quantify this effect, Laradji et al. [217,222] have proposed the scaling ansatz... 

The exponent a depends on the type of experiment performed, and it has been introduced to allow for a consistent description of the different measurements. a=l for PCS, a=0 for short exposure TDFRS, which guarantees strictly concentration proportional amplitudes, and a=b for long exposure TDFRS. a = 1 leads directly to the z-average diffusion coefficient (D)cM characteristic for... 

According to these models, the turbulent field for chemical reaction is determined once Ls and e are known at any position in the reactor. This leads to interesting scale-up criteria in stirred reactors GO the mixing characteristics are the same if g/l is kept constant. As e v P/pV and V L, the power input P should be proportional to L Lg. This may be as high as if one assumes that Ls L. The empirical exponent actually lies in between 3 and 5. 

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

Pearson and Helfand [12] used a somewhat similar approach to determine the characteristic relaxation time for a branch to disentangle their calculation leads to a similar form, with a different exponent for the front factor ... 

The studies of Wiesenfeld [28] and Lai et al. [43] on the classical dynamics of a one-electron atom in a sinusoidal external field provide a physically realistic example in which the presence of KAM tori surrounding stable periodic orbits leads to deviations from the generic behaviour characteristic of a hyperbolic scattering system as discussed in Sect. 2. Although this system (10) seems simple, further studies illuminating the mathematical structures behind the scattering process, e.g. calculation of the Liapunov exponents of the unstable trapped orbits and the fractal dimension of the trapped set, have yet to be performed. 

The general characteristics of these groups of powder are summarized in Tables 118-120, according to Roth and Capener [ J ]. Potassium salts arc added (Tables IIS and 119) as flash reducing agents. Lead salts (Table 119) are decomposition moderators which play a role in producing low temperature coefficients of burning propellants and a low exponent n in the expression... 

Physically, the reason for the dramatic difference between performances of cathode and anode active layers is the exchange current density ia at the anode the latter is 10 orders of magnitude higher than at the cathode [6]. Due to the large ia, the electrode potential r]a is small. The anode of PEFC, hence, operates in the linear regime, when both exponential terms in the Butler-Volmer equation can be expanded [178]. This leads to exponential variation of rja across the catalyst layer with the characteristic length (in the exponent)... 

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