Two-parameter family

Figure Al.6.30. (a) Two pulse sequence used in the Tannor-Rice pump-dump scheme, (b) The Husuni time-frequency distribution corresponding to the two pump sequence in (a), constmcted by taking the overlap of the pulse sequence with a two-parameter family of Gaussians, characterized by different centres in time and carrier frequency, and plotting the overlap as a fiinction of these two parameters. Note that the Husimi distribution allows one to visualize both the time delay and the frequency offset of pump and dump simultaneously (after [52a]).

On the other hand, a two-parameter family of difference operators... 

Aronson, D. G., Chory, M. A., Hall, G. R. and McGehee, R. P., 1982, Bifurcations from an invariant circle for two-parameter families of maps of the plane a computer assisted study. Comm. Math. Phys. 83, 303-354. 

Hall, G. R. 1984 Resonance zones in two-parameter families of circle homeomorphisms. SIAM Jl Math. Anal. 15(6), 1075-1081. 

Gouesbet, G., Berlemont, A. and Picart, A. (1984), Dispersion of discrete particles by continuous turbulent motions. Extensive discussion of the Tchen s theory, using a two-parameter family of lagrangian correlation functions, Phys. Fluids, 27(4), 827. 

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. 

However, the effect of a small perturbation in action-action-angle type flows is quite different. The two-parameter family of invariant cycles coalesce into invariant tori that are connected by resonant sheets defined by the u(h,l2) = 0 condition. The consequence of this is that contrary to action-angle-angle flows in this case a trajectory can cover the whole phase space and no transport barriers exist. Thus, in this type of flows global uniform mixing can be achieved for arbitrarily small perturbations. This type of resonance induced dispersion has been demonstrated numerically in a low-Reynolds number Couette flow between two rotating spheres by Cartwright et al. 

Bifurcations om an Invariant Cirde for Two-Parameter Families of Maps of the Plane A Computer-Assisted Study. 

On M-y there are locally stable and unstable manifolds that are of equal dimensions and are close to the impertm-bed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold intersects each of the 5D level energy sm-faces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem. The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. 

KAM TheoremP The KAM theorem determines whether the recurrent motions occm- on the pertm-bed normally hyperbolic locally invariant manifold M. and whether any of the two parameter families of 2D nonresonant invariant tori survive the Hamiltonian pertm-bation. The unpertm-bed Floquet Hamiltonian (P)(7 = 0) = Po(p,PljPj) + He(q,P,Pi,P2) satisfies the following nondegeneracy (or noiu eso-nance) condition ... 

It is now clear that (6.2.6) represents a two-parameter family of solutions, each representing a junction of two periodic waves. It is easy to confirm that... 

We have seen that bifurcations in one-parameter families transverse to the stability boundary OT may develop in completely different ways depending on the sign of the first Lyapunov value. If the value L vanishes at e = 0, at the very least we have to consider two-parameter families. To explore such a situation let us reduce the system on the center manifold to the normal form up to the terms of fifth order... 

The behavior of trajectories of system (13.2.13) near 0(0,0) is shown in Fig. 13.2.8. To investigate the bifurcations near this point let us consider a two-parameter family which can be written in the following form ... 

Consider a smooth two-parameter family which is transverse to the... 

Byragov, V. S. [1987] Bifurcations in a two-parameter family of conservative mappings that are close to the Henon mapping , in Methods of the Qualitative Theory of Differential Equations (Gorky Gorky State Univ. Press), 10-24. 

---