메뉴 건너뛰기




Volumn 12, Issue 3, 2013, Pages 1280-1329

Interacting global invariant sets in a planar map model of wild chaos

Author keywords

Critical set; Lorenz like attractor; Noninvertible planar map; Robust homoclinic tangencies; Stable and unstable sets

Indexed keywords

CRITICAL SET; HOMOCLINIC TANGENCIES; LORENZ-LIKE ATTRACTOR; NONINVERTIBLE PLANAR MAP; STABLE AND UNSTABLE SETS;

EID: 84884918410     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/120902860     Document Type: Article
Times cited : (14)

References (69)
  • 3
    • 0000119416 scopus 로고
    • On structurally unstable attracting limit sets of Lorenz attractor type
    • V. S. Afrajmovich, V. V. Bykov, and L. P. Shilnikov, On structurally unstable attracting limit sets of Lorenz attractor type, Trans. Mosc. Math. Soc., 44 (1983), pp. 153-216.
    • (1983) Trans. Mosc. Math. Soc , vol.44 , pp. 153-216
    • Afrajmovich, V.S.1    Bykov, V.V.2    Shilnikov, L.P.3
  • 4
    • 0141730339 scopus 로고    scopus 로고
    • On the fractal structure of basin boundaries in two-dimensional noninvertible maps
    • A. Agliari, L. Gardini, and C. Mira, On the fractal structure of basin boundaries in two-dimensional noninvertible maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), pp. 1767-1785.
    • (2003) Internat. J. Bifur. Chaos Appl. Sci. Engrg , vol.13 , pp. 1767-1785
    • Agliari, A.1    Gardini, L.2    Mira, C.3
  • 5
    • 33745292753 scopus 로고    scopus 로고
    • Crossing bifurcations and unstable dimension variability
    • K. T. Alligood, E. Sander, and J. A. Yorke, Crossing bifurcations and unstable dimension variability, Phys. Rev. Lett., 96 (2006), 244103.
    • (2006) Phys. Rev. Lett , vol.96 , pp. 244103
    • Alligood, K.T.1    Sander, E.2    Yorke, J.A.3
  • 6
    • 67349115475 scopus 로고    scopus 로고
    • Hyperbolic sets exhibiting C1-persistent homoclinic tangency for higher dimensions
    • M. Asaoka, Hyperbolic sets exhibiting C1-persistent homoclinic tangency for higher dimensions, Proc. Amer. Math. Soc., 136 (2008), pp. 677-686.
    • (2008) Proc. Amer. Math. Soc , vol.136 , pp. 677-686
    • Asaoka, M.1
  • 7
    • 77951441095 scopus 로고    scopus 로고
    • Erratum to "Hyperbolic sets exhibiting C1-persistent homoclinic tangency for higher dimensions
    • M. Asaoka, Erratum to "Hyperbolic sets exhibiting C1-persistent homoclinic tangency for higher dimensions," Proc. Amer. Math. Soc., 138 (2010), p. 1533.
    • (2010) Proc. Amer. Math. Soc , vol.138 , pp. 1533
    • Asaoka, M.1
  • 10
    • 15744371234 scopus 로고    scopus 로고
    • Real polynomial diffeomorphisms with maximal entropy: Tangencies
    • E. Bedford and J. Smillie, Real polynomial diffeomorphisms with maximal entropy: Tangencies, Ann. of Math. (2), 160 (2004), pp. 1-26.
    • (2004) Ann. of Math , vol.160 , Issue.2 , pp. 1-26
    • Bedford, E.1    Smillie, J.2
  • 11
    • 0000997063 scopus 로고    scopus 로고
    • The numerical computation of homoclinic orbits for maps
    • W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps, SIAMJ. Numer. Anal., 34 (1997), pp. 1207-1236.
    • (1997) SIAMJ. Numer. Anal , vol.34 , pp. 1207-1236
    • Beyn, W.-J.1    Kleinkauf, J.-M.2
  • 12
    • 44149109395 scopus 로고    scopus 로고
    • Robust heterodimensional cycles and C1-generic dynamics
    • C. Bonatti and L. Diaz, Robust heterodimensional cycles and C1-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), pp. 469-525.
    • (2008) J. Inst. Math. Jussieu , vol.7 , pp. 469-525
    • Bonatti, C.1    Diaz, L.2
  • 13
    • 33746377160 scopus 로고    scopus 로고
    • Dynamics beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective
    • Springer-Verlag, Berlin
    • C. Bonatti, L. J. Diaz, and M. Viana, Dynamics beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia Math. Sci. 102, Springer-Verlag, Berlin, 2005.
    • (2005) Encyclopaedia Math. Sci , pp. 102
    • Bonatti, C.1    Diaz, L.J.2    Viana, M.3
  • 16
    • 47249159390 scopus 로고    scopus 로고
    • The boundary of hyperbolicity for Henon-like families
    • Y. Cao, S. Luzzatto, and I. Rios, The boundary of hyperbolicity for Henon-like families, Ergodic Theory Dynam. Systems, 28 (2008), pp. 1049-1080.
    • (2008) Ergodic Theory Dynam. Systems , vol.28 , pp. 1049-1080
    • Cao, Y.1    Luzzatto, S.2    Rios, I.3
  • 17
    • 0032214566 scopus 로고    scopus 로고
    • Basin properties in two-dimensional noninvertible maps
    • J. C. Cathala, Basin properties in two-dimensional noninvertible maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), pp. 2147-2189.
    • (1998) Internat. J. Bifur. Chaos Appl. Sci. Engrg , vol.8 , pp. 2147-2189
    • Cathala, J.C.1
  • 18
    • 8744230577 scopus 로고    scopus 로고
    • MatCont: A Matlab package for numerical bifurcation analysis of ODEs
    • A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, MatCont: A Matlab package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), pp. 141-164.
    • (2003) ACM Trans. Math. Software , vol.29 , pp. 141-164
    • Dhooge, A.1    Govaerts, W.2    Kuznetsov, Y.A.3
  • 19
    • 33846058378 scopus 로고    scopus 로고
    • Global bifurcations of the Lorenz manifold
    • E. J. Doedel, B. Krauskopf, and H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinearity, 19 (2006), pp. 2947-2972.
    • (2006) Nonlinearity , vol.19 , pp. 2947-2972
    • Doedel, E.J.1    Krauskopf, B.2    Osinga, H.M.3
  • 20
    • 81055148298 scopus 로고    scopus 로고
    • Global invariant manifolds in the transition to preturbulence in the Lorenz system
    • E. J. Doedel, B. Krauskopf, and H. M. Osinga, Global invariant manifolds in the transition to preturbulence in the Lorenz system, Indag. Math. (N.S.), 22 (2011), pp. 222-240.
    • (2011) Indag. Math. (N.S.) , vol.22 , pp. 222-240
    • Doedel, E.J.1    Krauskopf, B.2    Osinga, H.M.3
  • 21
    • 3042531966 scopus 로고    scopus 로고
    • Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse
    • J. P. England, B. Krauskopf, and H. M. Osinga, Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse, SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 161-190.
    • (2004) SIAM J. Appl. Dyn. Syst , vol.3 , pp. 161-190
    • England, J.P.1    Krauskopf, B.2    Osinga, H.M.3
  • 23
    • 85097995833 scopus 로고    scopus 로고
    • Mathematical foundations of neuroscience
    • Springer, New York
    • G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, Interdiscip. Appl. Math. 35, Springer, New York, 2010.
    • (2010) Interdiscip. Appl. Math , pp. 35
    • Ermentrout, G.B.1    Terman, D.H.2
  • 27
    • 43049144272 scopus 로고    scopus 로고
    • On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. i
    • S. V. Gonchenko, L. P. Shilnikov, and D. V. Turaev, On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I, Nonlinearity, 21 (2008), pp. 923-972.
    • (2008) Nonlinearity , vol.21 , pp. 923-972
    • Gonchenko, S.V.1    Shilnikov, L.P.2    Turaev, D.V.3
  • 28
    • 60349108289 scopus 로고    scopus 로고
    • On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors
    • S. V. Gonchenko, L. P. Shilnikov, and D. V. Turaev, On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regul. Chaotic Dyn., 14 (2009), pp. 137-147.
    • (2009) Regul. Chaotic Dyn , vol.14 , pp. 137-147
    • Gonchenko, S.V.1    Shilnikov, L.P.2    Turaev, D.V.3
  • 30
    • 0003372424 scopus 로고
    • A strange, strange attractor
    • J. E. Marsden and M. McCracken, eds., Springer-Verlag, New York
    • J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation Theorem and Its Applications, J. E. Marsden and M. McCracken, eds., Springer-Verlag, New York, 1976, pp. 368-381.
    • (1976) The Hopf Bifurcation Theorem and Its Applications , pp. 368-381
    • Guckenheimer, J.1
  • 33
    • 0002685374 scopus 로고
    • Recurrence and discrete dynamic systems
    • Springer, Berlin
    • I. Gumowski and C. Mira, Recurrence and Discrete Dynamic Systems, Lecture Notes in Math. 809, Springer, Berlin, 1980.
    • (1980) Lecture Notes in Math , pp. 809
    • Gumowski, I.1    Mira, C.2
  • 34
    • 0002591468 scopus 로고
    • A two-dimensional mapping with a strange attractor
    • M. Henon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), pp. 69-77.
    • (1976) Comm. Math. Phys , vol.50 , pp. 69-77
    • Henon, M.1
  • 35
    • 54949103538 scopus 로고    scopus 로고
    • Bifurcations of the global stable set of a planar endomorphism near a cusp singularity
    • C. A. Hobbs and H. M. Osinga, Bifurcations of the global stable set of a planar endomorphism near a cusp singularity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), pp. 2207-2222.
    • (2008) Internat. J. Bifur. Chaos Appl. Sci. Engrg , vol.18 , pp. 2207-2222
    • Hobbs, C.A.1    Osinga, H.M.2
  • 36
    • 0003406478 scopus 로고    scopus 로고
    • Introduction to the modern theory of dynamical systems
    • Cambridge University Press, Cambridge, UK
    • A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., Cambridge University Press, Cambridge, UK, 1996.
    • (1996) Encyclopedia Math. Appl.
    • Katok, A.1    Hasselblatt, B.2
  • 37
    • 0034239239 scopus 로고    scopus 로고
    • A method to calculate basin bifurcation sets for a twodimensional noninvertible map
    • H. Kitajima, H. Kawakami, and C. Mira, A method to calculate basin bifurcation sets for a twodimensional noninvertible map, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), pp. 2001-2014.
    • (2000) Internat. J. Bifur. Chaos Appl. Sci. Engrg , vol.10 , pp. 2001-2014
    • Kitajima, H.1    Kawakami, H.2    Mira, C.3
  • 38
    • 0002623916 scopus 로고    scopus 로고
    • Unstable dimension variability: A source of nonhyperbolicity in chaotic systems
    • E. J. Kostelich, I. Kan, C. Grebogi, E. Ott, and J. A. Yorke, Unstable dimension variability: A source of nonhyperbolicity in chaotic systems, Phys. D, 109 (1997), pp. 81-90.
    • (1997) Phys. D , vol.109 , pp. 81-90
    • Kostelich, E.J.1    Kan, I.2    Grebogi, C.3    Ott, E.4    Yorke, J.A.5
  • 39
    • 0000010680 scopus 로고    scopus 로고
    • Growing 1D and quasi-2D unstable manifolds of maps
    • B. Krauskopf and H. Osinga, Growing 1D and quasi-2D unstable manifolds of maps, J. Comput. Phys., 146 (1998), pp. 404-419.
    • (1998) J. Comput. Phys. , vol.146 , pp. 404-419
    • Krauskopf, B.1    Osinga, H.2
  • 40
    • 0000725929 scopus 로고    scopus 로고
    • Investigating torus bifurcations in the forced van der Pol oscillator
    • (Minneapolis, MN, 1997), IMA Vol. Math. Appl. 119, Springer, New York
    • B. Krauskopf and H. Osinga, Investigating torus bifurcations in the forced van der Pol oscillator, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems (Minneapolis, MN, 1997), IMA Vol. Math. Appl. 119, Springer, New York, 2000, pp. 199-208.
    • (2000) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems , pp. 199-208
    • Krauskopf, B.1    Osinga, H.2
  • 41
    • 54949090132 scopus 로고    scopus 로고
    • Unfolding the cusp-cusp bifurcation of planar endomorphisms
    • B. Krauskopf, H. M. Osinga, and B. B. Peckham, Unfolding the cusp-cusp bifurcation of planar endomorphisms, SIAM J. Appl. Dyn. Syst., 6 (2007), pp. 403-440.
    • (2007) SIAM J. Appl. Dyn. Syst , vol.6 , pp. 403-440
    • Krauskopf, B.1    Osinga, H.M.2    Peckham, B.B.3
  • 42
    • 0000906011 scopus 로고
    • How often are chaotic saddles nonhyperbolic
    • Y.-C. Lai, C. Grebogi, J. A. Yorke, and I. Kan, How often are chaotic saddles nonhyperbolic?, Nonlinearity, 6 (1993), pp. 779-797.
    • (1993) Nonlinearity , vol.6 , pp. 779-797
    • Lai, Y.-C.1    Grebogi, C.2    Yorke, J.A.3    Kan, I.4
  • 43
    • 0038333607 scopus 로고    scopus 로고
    • Complex patterns on the plane: Different types of basin fractalization in a two-dimensional mapping
    • R. Lopez-Ruiz and D. Fournier-Prunaret, Complex patterns on the plane: Different types of basin fractalization in a two-dimensional mapping, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), pp. 287-310.
    • (2003) Internat. J. Bifur. Chaos Appl. Sci. Engrg , vol.13 , pp. 287-310
    • Lopez-Ruiz, R.1    Fournier-Prunaret, D.2
  • 44
    • 0000241853 scopus 로고
    • Deterministic nonperiodic flow
    • E. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), pp. 130-141.
    • (1963) J. Atmospheric Sci , vol.20 , pp. 130-141
    • Lorenz, E.1
  • 45
    • 84891567501 scopus 로고    scopus 로고
    • Nonlinear laser dynamics: From quantum dots to cryptography
    • John Wiley & Sons, New York
    • K. Ludge and H. Schuster, Nonlinear Laser Dynamics: From Quantum Dots to Cryptography, Annu. Rev. Nonlinear Dynam. Complexity, John Wiley & Sons, New York, 2012.
    • (2012) Annu. Rev. Nonlinear Dynam. Complexity
    • Ludge, K.1    Schuster, H.2
  • 50
    • 79958795117 scopus 로고    scopus 로고
    • There are no C1-stable intersections of regular Cantor sets
    • C. G. Moreira, There are no C1-stable intersections of regular Cantor sets, Acta Math., 206 (2011), pp. 311-323.
    • (2011) Acta Math , vol.206 , pp. 311-323
    • Moreira, C.G.1
  • 52
    • 0000935850 scopus 로고
    • Nondensity of axiom A(a) on s2, in global analysis
    • (Berkeley, CA, 1968),., Providence, RI
    • S. E. Newhouse, Nondensity of axiom A(a) on S2, in Global Analysis (Berkeley, CA, 1968), Proc. Sympos. Pure Math. XIV, Amer. Math. Soc., Providence, RI, 1970, pp. 191-202.
    • (1970) Proc. Sympos. Pure Math. XIV, Amer. Math. Soc , pp. 191-202
    • Newhouse, S.E.1
  • 53
    • 51249181043 scopus 로고
    • The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms
    • S. E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math., 50 (1979), pp. 101-151.
    • (1979) Inst. Hautes Etudes Sci. Publ. Math , vol.50 , pp. 101-151
    • Newhouse, S.E.1
  • 54
    • 0032008390 scopus 로고    scopus 로고
    • An algorithm for the computation of preimages in noninvertible mappings
    • C.-H. Nien and F. J. Wicklin, An algorithm for the computation of preimages in noninvertible mappings, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), pp. 415-422.
    • (1998) Internat. J. Bifur. Chaos Appl. Sci. Engrg , vol.8 , pp. 415-422
    • Nien, C.-H.1    Wicklin, F.J.2
  • 57
    • 79959269369 scopus 로고    scopus 로고
    • An example of C1-generically wild homoclinic classes with index deficiency
    • K. Shinohara, An example of C1-generically wild homoclinic classes with index deficiency, Nonlinearity, 24 (2011), pp. 1961-1974.
    • (2011) Nonlinearity , vol.24 , pp. 1961-1974
    • Shinohara, K.1
  • 58
    • 84859089159 scopus 로고    scopus 로고
    • On the index problem of C1-generic wild homoclinic classes in dimension three
    • K. Shinohara, On the index problem of C1-generic wild homoclinic classes in dimension three, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), pp. 913-940.
    • (2011) Discrete Contin. Dyn. Syst. Ser. A , vol.31 , pp. 913-940
    • Shinohara, K.1
  • 59
    • 0004032638 scopus 로고
    • the collaboration of A. Fathi and R. Langevin, translated from the French by J. Christy, Springer-Verlag, New York
    • M. Shub, Global Stability of Dynamical Systems, with the collaboration of A. Fathi and R. Langevin, translated from the French by J. Christy, Springer-Verlag, New York, 1987.
    • (1987) Global Stability of Dynamical Systems
    • Shub, M.1
  • 60
    • 33645644841 scopus 로고
    • Hyperbolicity conditions for the Lorenz model
    • J. G. Sinai and E. B. Vul, Hyperbolicity conditions for the Lorenz model, Phys. D, 2 (1981), pp. 3-7.
    • (1981) Phys. D , vol.2 , pp. 3-7
    • Sinai, J.G.1    Vul, E.B.2
  • 61
    • 84968514104 scopus 로고
    • Differentiable dynamical systems
    • S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), pp. 747-817.
    • (1967) Bull. Amer. Math. Soc , vol.73 , pp. 747-817
    • Smale, S.1
  • 62
    • 0003293377 scopus 로고
    • The lorenz equations: Bifurcations, chaos, and strange attractors
    • Springer-Verlag, New York
    • C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Appl. Math. Sci. 41, Springer-Verlag, New York, 1982.
    • (1982) Appl. Math. Sci , pp. 41
    • Sparrow, C.1
  • 64
    • 0007679077 scopus 로고    scopus 로고
    • Elliptic islands appearing in near-ergodic flows
    • D. Turaev and V. Rom-Kedar, Elliptic islands appearing in near-ergodic flows, Nonlinearity, 11 (1998), pp. 575-600.
    • (1998) Nonlinearity , vol.11 , pp. 575-600
    • Turaev, D.1    Rom-Kedar, V.2
  • 65
    • 0032220873 scopus 로고    scopus 로고
    • An example of a wild strange attractor
    • D. V. Turaev and L. P. Shilnikov, An example of a wild strange attractor, Mat. Sb., 189 (1998), pp. 291-314.
    • (1998) Mat. Sb , vol.189 , pp. 291-314
    • Turaev, D.V.1    Shilnikov, L.P.2
  • 66
    • 41549128876 scopus 로고    scopus 로고
    • Pseudo-hyperbolicity and the problem on periodic perturbations of Lorenz-like attractors
    • D. V. Turaev and L. P. Shilnikov, Pseudo-hyperbolicity and the problem on periodic perturbations of Lorenz-like attractors, Russian Dokl. Math., 77 (2008), pp. 17-21.
    • (2008) Russian Dokl. Math , vol.77 , pp. 17-21
    • Turaev, D.V.1    Shilnikov, L.P.2
  • 67
    • 0009132956 scopus 로고
    • The abundance of hyperbolicity in the C1 topology
    • R. Ures, The abundance of hyperbolicity in the C1 topology, Ann. Sci. Ecole Norm. Sup. (4), 28 (1995), pp. 747-760.
    • (1995) Ann. Sci. Ecole Norm. Sup , vol.28 , Issue.4 , pp. 747-760
    • Ures, R.1
  • 68
    • 0002771881 scopus 로고    scopus 로고
    • Multidimensional nonhyperbolic attractors
    • M. Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Etudes Sci. Publ. Math., 85 (1997), pp. 63-96.
    • (1997) Inst. Hautes Etudes Sci. Publ. Math , vol.85 , pp. 63-96
    • Viana, M.1
  • 69
    • 84859772535 scopus 로고    scopus 로고
    • How to find a codimension-one heteroclinic cycle between two periodic orbits
    • W. Zhang, B. Krauskopf, and V. Kirk, How to find a codimension-one heteroclinic cycle between two periodic orbits, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), pp. 2825-2851.
    • (2012) Discrete Contin. Dyn. Syst. Ser. A , vol.32 , pp. 2825-2851
    • Zhang, W.1    Krauskopf, B.2    Kirk, V.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.