Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits

Lamb, Jeroen S. W.; Sten'kin, O V

doi:10.1088/0951-7715/17/4/005

Cited by 50 publications

(69 citation statements)

References 24 publications

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“…In particular, such local coordinates are given by Lemma 2 in which the normal form of the first order for a saddle map is derived. ‡ Moreover, as we show, symmetry breaking bifurcations have also another nature, in comparison with [23]. We find a two-step "fold⇒pitch-fork" scenario of bifurcations in the first-return maps leading to the appearance of non-conservative fixed points which can be either attracting and repelling or saddle with Jacobian greater and less than 1 (see Theorem 1 and Figure 4).…”

Section: Introductionmentioning

confidence: 55%

“…Thus, one realizes that the phenomenon of mixed dynamics in the case of two-dimensional reversible maps should be connected with the coexistence of infinitely many attracting, repelling, saddle and elliptic periodic orbits. The existence of Newhouse regions (intervals) in which this property is generic was already established in [23] for the case of reversible two-dimensional maps close to a map having a heteroclinic cycle of the type depicted in Figure 1(a).…”

Section: Introductionmentioning

confidence: 66%

“…Precisely, it will be shown that in some regions of the space of parameters (c,M ) the map T km possesses chaotic dynamics and has four saddle fixed points, two of them symmetric conservative and a symmetric couple of fixed points (that is, symmetric one to each other and with Jacobian greater and less than 1, respectively). According to [12,23,20], the following result holds:…”

Section: A Short Description Of the Main Resultsmentioning

confidence: 99%

“…We find a two-step "fold⇒pitch-fork" scenario of bifurcations in the first-return maps leading to the appearance of non-conservative fixed points which can be either attracting and repelling or saddle with Jacobian greater and less than 1 (see Theorem 1 and Figure 4). Notice that in the case of heteroclinic cycles of type 1), such non-conservative points appear just under fold bifurcations [23] in a symmetric couple of first-return maps, whereas elliptic points appear in symmetric first-return maps.…”

Section: Introductionmentioning

confidence: 99%

“…As we mentioned above, the case of non-transversal heteroclinic cycles of type 1), as in Fig. 1(a), was studied in the paper [23] where, in fact, the RMD-Conjecture was proved for general one-parameter (reversible) unfoldings, under the generic condition…”

Section: Introductionmentioning

confidence: 99%

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Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Delshams

Гонченко²,

Gonchenko³

et al. 2012

Nonlinearity

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Abstract. We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.

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Section: Introductionmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 66%