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Nonlinear Dynamics New Directions pp 57–78Cite as

Homoclinic Ω-Explosion: Hyperbolicity Intervals and Their Bifurcation Boundaries

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Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 12))

Abstract

It has been established by Gavrilov and Shilnikov (Math USSR Sb 17:467–485, 1972) that at the bifurcation boundary, separating Morse–Smale systems from systems with complicated dynamics, there are systems with homoclinic tangencies. Moreover, when crossing this boundary, infinitely many periodic orbits appear immediately, just by “explosion.” Newhouse and Palis (Asterisque 31:44–140, 1976) have shown that in this case, there are infinitely many intervals of values of the splitting parameter corresponding to hyperbolic systems. In the present chapter, we show that such hyperbolicity intervals have natural bifurcation boundaries, so that the phenomenon of homoclinic Ω-explosion gains, in a sense, complete description in the case of 2D diffeomorphisms.

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Notes

  1. 1.

    Evidently, the case \(\sigma> 1\) is reduced to the case under consideration for f -1. Thus, only the case \(\sigma = 1\) is not in our competence; however, it is very specific and requires a special consideration (see, for example, [15, 16]).

  2. 2.

    Moreover, they possess Ω-moduli, i.e., continuous invariants of topological conjugacy on the set of nonwandering orbits. It means that any change of value of an Ω-modulus leads to a bifurcation of an orbit (periodic, homoclinic, etc) from the set N(f). As it was established in [19], the Gavrilov–Shilnikov invariant, \(\theta = - \ln|\lambda|/\ln|\gamma|\) introduced in [5], is the principal Ω-modulus here.

  3. 3.

    The existence of such coordinates was proved in [17]. Note that analogous \(C^{r-1}\)-coordinates was found in [19]. The form (1) of a saddle map is called the main normal form or normal form of the first order: such a form exists for any multidimensional saddle map, [17].

  4. 4.

    The number \(\bar k\) is chosen, in principle, depending on sizes of \(\Pi^+\) and Πi and it equals the minimal index “i” of the strips \(\sigma^{0}_{i}\) and \(\sigma^{1}_{i}\) from these neighborhoods. That is, the strips with index \(\bar k\) are “border”: they form the boundary, the so-called “special neighborhood” from U, [8,24] (so that, for example, \(\Pi^+\) does not contain points which reach Π− for a number of iterations (by T 0) less than \(\bar k\)), see Fig. 4.

  5. 5.

    Note that the minimal number \(k(\mu)\) of the strips is chosen here to be depending on μ (in particular, \(k(\mu)\to +\infty\) as \(\mu\to 0\)). It follows from the fact that if \(i<k(\mu)\), then the condition \(T_{1\mu}\sigma^{1}_{i}\cap\sigma^{0}_{j}\neq\emptyset\) means that j > i. Therefore, \(N(f_\mu)\) does not contain those orbits which intersect the strips \(\sigma^{0,1}_{i}\) with numbers \(i<k(\mu)\); it means that all such strips can be “eliminated from” the initial neighborhoods \(\Pi^+\) and Π− and, thus, one can consider smaller neighborhoods of \(O\cup\Gamma\), the so-called “special neighbourhoods,” see [14] for more detail.

  6. 6.

    It can easily be seen from Fig. 7d that if a planar diffeomorphism has a homoclinic tangency with \(c<0\), then other homoclinic orbits necessarily exist. Therefore, here takes place the so-called local Ω-explosion, i.e., the sharp change in the structure of nonwandering orbits from some subdomain of the phase space. Although this case can be described in the same way as the global Ω-explosion, it is not so interesting and, therefore, we do not consider it especially.

  7. 7.

    Here we assume that c is positive only for the sake of definiteness: it can be always realized for the appropriate choosing of a pair of homoclinic points. Indeed, take a pair of points \(M^{+\prime} = T_0(M^+)\) and \(M^-(0,y^-)\) instead \(M^+(x^+,0)\) and \(M^-(0,y^-)\). Then it is easy to check that \(x^{+\prime}= -\lambda x^+, b^\prime = b\lambda, c^\prime = c\gamma\) for the new global map \(T_1^\prime = T_0T_1:\Pi^-\to T_0(\Pi^+)\). Making the coordinate change \(x\mapsto-x, y\mapsto y\), we obtain that \(x^{+\prime}= |\lambda| x^+, b^\prime = -b\lambda, c^\prime = -c\gamma\), that is, the “new c” will have the opposite sign than the “old” one (making the change \(x\mapsto-x\), we arrive to our agreement that the coordinates \(x^+\) and y - of choosing homoclinic points must be positive).

  8. 8.

    An analog is the well-known “last bifurcation” in the Hénon family after which the nonwandering set becomes hyperbolic.

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Acknowledgments

The authors thank D. Turaev for fruitful discussions. This research was carried out within the framework of the Russian Federation Government grant, contract No.11.G34.31.0039. The paper was supported also in part by grants of RFBR No.10-01-00429, No.11-01-00001, and No.11-01-97017-povoljie.

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  1. Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University, Nizhny Novgorod, Russia

    Sergey Gonchenko & Oleg Stenkin

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  1. Sergey Gonchenko
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Correspondence to Sergey Gonchenko .

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  1. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Hernán González-Aguilar

  2. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Edgardo Ugalde

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Gonchenko, S., Stenkin, O. (2015). Homoclinic Ω-Explosion: Hyperbolicity Intervals and Their Bifurcation Boundaries. In: González-Aguilar, H., Ugalde, E. (eds) Nonlinear Dynamics New Directions. Nonlinear Systems and Complexity, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-09864-7_3

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