Homoclinic tangency

Subsidiary elliptic islands of very small area continue to exist until a last homoclinic tangency occurs at Eht, above which all the trapped orbits of the invariant set are unstable of saddle type. The system is then fully chaotic. According to this scenario, the invariant set may contain quasiperi-odic motion for energies Ea < E < Eht, while the main elliptic island exists only for Ea < E < Ed < Eh,- The interval /, - Ea turns out to be small as compared with the energy interval above Eht, where full chaos has set in and the invariant set is a repeller. 

There are also two local bifurcations. The first one takes place for r 13.926..., when a homoclinic tangency of separatrixes of the origin O occurs (it is not shown in Fig. 20) and a hyperbolic set appears, which consists of a infinite number of saddle cycles. Beside the hyperbolic set, there are two saddle cycles, L and L2, around the stable states, Pi and P2. The separatrices of the origin O reach the saddle cycles Li and L2, and the attractors of the system are the states Pi and P2. The second local bifurcation is observed for r 24.06. The separatrices do not any longer reach to the saddle cycles L and L2. As a result, in the phase space of the system a stable quasihyperbolic state appears— the Lorenz attractor. The chaotic Lorenz attractor includes separatrices, the saddle point O and a hyperbolic set, which appears as a result of homoclinic tangency of the separatrices. The presence of the saddle point in the chaotic... 

Similar to unstable periodic orbits, an NHIM has stable and unstable manifolds that are of dimension 2 — 2 and are also structurally stable. Note that a union of the segments of the stable and unstable manifolds is also of dimension 2n — 2, which is only of dimension one less than the energy surface. Hence, as far as dimensionality is concerned, it is possible for a combination of the stable and unstable manifolds of an NHIM to divide the many-dimensional energy surface so that reaction flux can be dehned. However, unlike the fewdimensional case in which a union of the stable and unstable manifolds necessarily encloses a phase space region, a combination of the stable and unstable manifolds of an NHIM may not do so in a many-dimensional system. This phenomenon is called homoclinic tangency, and it is extensively discussed in a recent review article by Toda [17]. 

Fig. 4.21. Inverse parabolic bursting obtained in system (4.1a-c) for v = 0.25 s and Its =1.86 s, near the point of homoclinic tangency in the rapid, two-variable subsystem (/3, y). Other parameter values are those of fig. 4.3 (Decroly Goldbeter, 1987).

Tracqui, P. 1993. Homoclinic tangencies in an autocatalytic model of interfacial processes at the bone surface. Physica 62D 275-89. 

Homoclinic (or heteroclinic) tangency would be a common phenomenon for systems of more than two degrees of freedom. Therefore, its role in reaction dynamics has to be taken seriously. Toda [4] noticed that the homoclinic tangency would lead to crisis where a transition between chaos... 

In Figure 3.11, a typical case of homoclinic tangency is indicated by showing those sections of the stable and unstable manifolds. Here, we can... 

Figure 3.11. An example of homoclinic tangency in a model of Helj that includes the freedom of internal rotation. [Reprinted with permission from R. E. Gillilan and G. S. Ezra, J. Ghent. Phys., Vol. 94 (1991), p. 2648. Copyright 1991, American Institute of Physics.]...

The above reasoning is not limited to homoclinic tangency, but also can be applied to heteroclinic tangency such as that shown in Figure 3.12(b). 

Figure 3.12. (a) An example of homoclinic tangency. (b) An example of heteroclinic tangency. Here, H, Hj, and H2 indicate saddle points. 

We note parenthetically that the situation becomes drastically different for the systems with complex dynamics. In the majority of cases (at least in those cases where homoclinic tangencies appear) the introduction of the moduli is inexorable because they serve as the essential parameters governing the bifurcations (see [63]). 

For the former type of boundary system, we consider an example of a homoclinic tangency. Let a C -smooth family of diffeomorphisms T(/i) have, at... 

Nevertheless, in their later works (see also the books [11, 12]) when investigating similar bifurcations they use the Banach space of all small perturbations. Note that the Banach space approach to the bifurcations becomes, in essence, necessary in the case of systems with complex dynamics, due to the persistence of homoclinic tangencies (see [60, 61, 62]). 

Gonchenko, S. V. and Gonchenko, V. S. [2000] On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies , Preprint No. 556, WIAS, Berlin. 

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