Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Patterns of Synchrony in Neuronal Networks: The Role of Synaptic Inputs Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

Homoclinic Ω-Explosion: Hyperbolicity Intervals and Their Bifurcation Boundaries

verfasst von : Sergey Gonchenko, Oleg Stenkin

Erschienen in: Nonlinear Dynamics New Directions

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel On Topological and Hyperbolic Properties of Systems with Homoclinic Tangencies
Nächstes Kapitel Self-Organized Regularity in Long-Range Systems
Fußnoten
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
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
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
The number \(\bar k\) is chosen, in principle, depending on sizes of \(\Pi^+\) and Π i 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
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
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
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
An analog is the well-known “last bifurcation” in the Hénon family after which the nonwandering set becomes hyperbolic.
 
Literatur
1.
Shilnikov, L.P.: On a new type of bifurcations in multidimensional dynamical systems. Sov. Math. Dokl. 182(1), 53–56 (1969)MathSciNet
2.
Afraimovich, V.S., Shilnikov, L.P.: On accesible transitions from Morse–Smale systems to systems with many periodic motions. Russ. Acad. Sci. Izv. Math. 38(6), 1248–1288 (1974)
3.
Shilnikov L.P.: On birth of periodic orbits from an orbit bi-asymptotic to a saddle–saddle equilibrium. Sov. Math. Dokl. 170(1), 49–52 (1966)MathSciNet
4.
Palis, J.: A note on Ω-stability. Global Analysis. Proceedings of Symposia in Pure Mathematics, vol. 14, pp. 221–222. American Mathematical Society, Providence (1970)
5.
Gavrilov N.K., Shilnikov, L.P.: On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. Part 1, Math. USSR Sb. 17, 467–485 (1972); Part 2, Math. USSR Sb. 19, 139–156 (1973)
6.
Gavrilov, N.K.: On three-dimensional dynamical systems having a structurally unstable homoclinic contour. Sov. Math. Notes 14(5), 687–696 (1973)MathSciNet
7.
Gonchenko, S.V.: Moduli of Ω-conjugacy of two-dimensional diffeomorphisms with a nontransversal heteroclinic cycle. Russ. Math. Sb. 187(9), 3–24 (1996)MathSciNet
8.
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.: On Newhouse regions of two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. Proc. Steklov Inst. Math. 216, 70–118 (1997)MathSciNet
9.
Newhouse, S., Palis, J.: Cycles and bifurcation theory. Asterisque 31, 44–140 (1976)MATH
10.
Palis, J., Takens, F.: Cycles and measure of bifurcation sets for two dimensional diffeomorphisms. Invent. Math. 82, 397–422 (1985)CrossRefMATHMathSciNet
11.
12.
Palis, J., Takens, F.: Hyperbolicity and the Creation of Homoclinic Orbit. Cambridge Studies in Advanced Mathematics, vol. 35. Cambridge University Press, Cambridge (1993)
13.
Alligood, K., Sander, E., Yorke, J.: Explosions: Global bifurcations at heteroclinic tangencies. Ergod. Theory Dyn. Syst. 22, 953–972 (2002)CrossRefMATHMathSciNet
14.
Stenkin, O.V., Shilnikov, L.P.: Homoclinic Ω-explosion and hyperbolicity domains. Russ. Math. Sb. 189(4), 125–144 (1998)MathSciNet
15.
Gonchenko, S.V., Shilnikov, L.P.: Arithmetic properties of topological invariants of systems with a structurally unstable homoclinic trajectory. Ukr. Math. J. 39, 21–28 (1987)CrossRefMathSciNet
16.
Gonchenko, S.V., Gonchenko, V.S.: On bifurcations of birth of closed invariant curves in the case of two-dimensional diffeomorphisms with homoclinic tangencies. Proc. Math. Steklov Inst. 244, 80–105 (2004)MathSciNet
17.
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.: On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I. Nonlinearity 21, 923–972 (2008)CrossRefMATHMathSciNet
18.
Gonchenko, S.V., Shilnikov, L.P.: Homoclinic tangencies. Thematic issue: Moscow-Izhevsk 524 p. (in Russian) (2007)
19.
Gonchenko, S.V., Shilnikov, L.P.: Invariants of Ω-conjugacy of diffeomorphisms with a nontransversal homoclinic orbit. Ukr. Math. J. 42(2), 134–140 (1990)CrossRefMATHMathSciNet
20.
Gonchenko, S.V., Shilnikov, L.P.: On moduli of systems with a nontransversal Poincaré homoclinic orbit. Russ. Acad. Sci. Izv. Math. 41(3), 417–445 (1993)MathSciNet
21.
Gonchenko, S.V., Sten’kin, O.V., Turaev, D.V.: Complexity of homoclinic bifurcations and Ω-moduli. Int. J. Bifurc. Chaos 6(6), 969–989 (1996)CrossRefMATHMathSciNet
22.
Shilnikov, L.P.: On a Poincaré-Birkhoff problem. Math. USSR Sb. 3, 91–102 (1967)CrossRef
23.
Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V., Chua, L.O.: Methods of qualitative theory in nonlinear dynamics. World Scientific, Singapore, Part I (1998), Part II (2001)
24.
Gonchenko, S.V.: Nontrivial hyperbolic subsets of multidimensional systems with a nontransversal homoclinic curve. In: Methods of Qualitative Theory of Differential Equations, pp. 89–102. Gorky State University, Yekaterinberg (in Russian) (1984)
25.
Grines, V.Z.: On topological conjugacy of diffeomorphisms of a two-manifold on one-dimensional basic sets. Part I, Trudy Moskov. Mat. Obšč. 32, 35–60 (1975); Part II, Trudy Moskov. Mat. Obšč. 34, 243–252 (1977)
26.
27.
28.
Gonchenko, S.V., Turaev, D.V., Shilnikov, L.P.: Homoclinic tangencies of any order in Newhouse regions. J. Math. Sci. 105, 1738–1778 (2001)CrossRef
Metadaten
Titel
Homoclinic Ω-Explosion: Hyperbolicity Intervals and Their Bifurcation Boundaries
verfasst von
Sergey Gonchenko
Oleg Stenkin
Copyright-Jahr
2015
Buch
Nonlinear Dynamics New Directions

Print ISBN: 978-3-319-09863-0

Electronic ISBN: 978-3-319-09864-7

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-09864-7_3

Neuer Inhalt

    Bildnachweise