Some remarks on the abundance of stable periodic orbits inside homoclinic lobes. (English) Zbl 1244.37039

Summary: We consider a family \(F_{\varepsilon }\) of area-preserving maps (APMs) with a hyperbolic point \(H_{\varepsilon }\) whose invariant manifolds form a figure-eight and we study the abundance of elliptic periodic orbits visiting homoclinic lobes (EPL), a domain typically dominated by chaotic behavior. To this end, we use the Chirikov separatrix map (SM) as a model of the return to a fundamental domain containing lobes. We obtain an explicit estimate, valid for families \(F_{\varepsilon }\) with central symmetry and close to an integrable limit, of the relative measure of the set of parameters \(\varepsilon \) for which \(F_{\varepsilon }\) has EPL trajectories. To get this estimate we look for EPL of the SM with the lowest possible period. The analytical results are complemented with quantitative numerical studies of the following families \(F_{\varepsilon }\) of APMs:
\(\bullet \) The SM family, and we compare our analytical results with the numerical estimates.
\(\bullet \) The standard map (STM) family, and we show how the results referring to the SM model apply to the EPL visiting the lobes that the invariant manifolds of the STM hyperbolic fixed point form.
\(\bullet \) The conservative Hénon map family, and we estimate the number of a particular type of symmetrical EPL related to the separatrices of the 4-periodic resonant islands.
The results obtained can be seen as the quantitative analogs to those in [C. Simó and D. Treschev, Discrete Contin. Dyn. Syst., Ser. B 10, No. 2–3, 681–698 (2008; Zbl 1153.37406)], although here we deal with the a priori stable situation instead.


37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37D10 Invariant manifold theory for dynamical systems


Zbl 1153.37406


Full Text: DOI


[1] Simó, C.; Vieiro, A., Dynamics in chaotic zones of area preserving maps: close to separatrix and global instability zones, Physica D, 240, 8, 732-753, (2011) · Zbl 1217.37051
[2] Neishtadt, A.; Sidorenko, V.; Treschev, D., Stable periodic motions in the problem on passage through a separatrix, Chaos, 7, 1, 2-11, (1997) · Zbl 1002.34029
[3] Neishtadt, A.; Vasiliev, A.; Simó, C.; Treschev, D., Stability islands in domains of separatrix crossings in slow – fast Hamiltonian systems, Proc. Steklov inst. math., 259, 236-247, (2007) · Zbl 1153.37397
[4] Neishtadt, A.; Vasiliev, A., On the absence of stable periodic orbits in domains of separatrix crossings in nonsymmetric slow – fast Hamiltonian systems, Chaos, 17, 4, 043104, (2007) · Zbl 1163.37355
[5] Neishtadt, A.; Simó, C.; Treschev, D.; Vasiliev, A., Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow – fast systems, Discrete contin. dyn. syst. ser. B, 10, 2-3, 621-650, (2008) · Zbl 1149.37032
[6] Chirikov, B.V., A universal instability of many-dimensional oscillator system, Phys. rep., 52, 264-379, (1979)
[7] Rom-Kedar, V.; Zaslavsky, G., Islands of accelerator modes and homoclinic tangles, Chaos, 9, 3, 697-705, (1999) · Zbl 0987.37049
[8] Contopoulos, G.; Grousouzakou, E.; Polymilis, C., Distribution of periodic orbits and the homoclinic tangle, Celestial mech. dynam. astronom., 64, 363-381, (1996) · Zbl 0885.58067
[9] Simó, C.; Treschev, D.V., Stability islands in the vicinity of separatrices of near-integrable maps, Discrete contin. dyn. syst. ser. B, 10, 2-3, 681-698, (2008) · Zbl 1153.37406
[10] Chierchia, L.; Gallavotti, G., Drift and diffusion in phase space, Annales de l’I.H.P., 60, 1-144, (1994) · Zbl 1010.37039
[11] Baldomà, I.; Fontich, E., ()
[12] Neishtadt, A.; Simó, C.; Vasiliev, A., Geometrical and statistical properties induced by separatrix crossings in volume-preserving systems, Nonlinearity, 16, 2, 521-557, (2002) · Zbl 1038.37017
[13] C. Simó, D.V. Treschev, Evolution of the last invariant curve in a family of area preserving maps. http://www.maia.ub.es/dsg/2006/index.html.
[14] Simó, C.; Vieiro, A., Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps, Nonlinearity, 22, 5, 1191-1245, (2009) · Zbl 1181.37077
[15] C. Simó, A. Vieiro, Dynamical description of the domain of stability of area preserving maps. Work in progress, 2011.
[16] Zaslavsky, G.M.; Filonenko, N.N., Stochastic instability of trapped particles and conditions of applicability of the quasi-linear approximation, Soviet phys. JETP, 27, 851-857, (1968)
[17] Piftankin, G.N.; Treschev, D.V., Separatrix maps in Hamiltonian systems, Russian math. surveys IOP, 2, 62, 219-322, (2007) · Zbl 1151.37050
[18] Broer, H.; Simó, C.; Tatjer, J.C., Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11, 667-770, (1998) · Zbl 0937.37013
[19] Kuznesov, L.; Zaslavsky, G.M., Hidden renormalization group for the near-separatrix Hamiltonian dynamics, Phys. rep., 288, 457-485, (1997)
[20] Kuznesov, L.; Zaslavsky, G.M., Scaling invariance of the homoclinic tangle, Phys. rev. E, 66, 1-7, (2002)
[21] White, R.B.; Zaslavsky, G.M., Near threshold anomalous transport in the standard map, Chaos, 8, 4, 758-767, (1998) · Zbl 0987.37023
[22] Zaslavsky, G.M., Dynamical traps, Physica D, 168-169, 292-304, (2002) · Zbl 1019.37049
[23] The PARI Group, Bordeaux. PARI/GP, version 2.3.4, 2008. Available from http://pari.math.u-bordeaux.fr/.
[24] Treschev, D.; Zubelevich, O., Introduction to the perturbation theory of Hamiltonian systems, () · Zbl 0942.37042
[25] W. Feller, An Introduction to Probability Theory and its Applications, volume I-II, John Wiley & Sons Inc., New York, NY., 1950-1966.
[26] Chirikov, B.V., Chaotic dynamics in Hamiltonian systems with divided phase space, (), 29-46
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