Citation: |
[1] |
V. I. Arnold, V. S. Afraimovich, Y. S. Iljashenko and L. P. Shilnikov, Bifurcation Theory and Catastrophe Theory, Dynamical Systems V. Encyclopaedia of Mathematical Sciences, Springer-Verlag, New York, 1994.doi: 10.1007/978-3-642-57884-7. |
[2] |
M. A. D. Aguiar, S. B. Castro and I. S. Labouriau, Simple Vector Fields with Complex Behaviour, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 369-381.doi: 10.1142/S021812740601485X. |
[3] |
M. A. D. Aguiar, I. S. Labouriau and A. A. P. Rodrigues, Switching near a heteroclinic network of rotating nodes, Dyn. Syst., 25 (2010), 75-95.doi: 10.1080/14689360903252119. |
[4] |
A. Arnéodo, P. Coullet and C. Tresser, A possible new mechanism for the onset of turbulence, Phys. Lett. A, 81 (1981), 197-201.doi: 10.1016/0375-9601(81)90239-5. |
[5] |
A. Arnéodo, P. Coullet and C. Tresser, Possible new strange attractors with spiral structure, Comm. Math. Phys., 79 (1981), 573-579.doi: 10.1007/BF01209312. |
[6] |
G. R. Belitskii, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, Funkcional. Anal. i Prilozen, 7 (1973), 17-28. |
[7] |
J. A. Beloqui, Módulo de Estabilidade Para Campos Vectoriais em Variedades Tridimensionais, Ph.D Thesis, IMPA-Brasil, 1981. |
[8] |
C. Bonatti and E. Dufraine, Equivalence topologique de connexions de selles en dimension 3, Ergodic Theory Dynam. Systems, 23 (2003), 1347-1381.doi: 10.1017/S0143385703000130. |
[9] |
V. V. Bykov, Orbit structure in a neighbourhood of a separatrix cycle containing two saddle-foci, Amer. Math. Soc. Transl, Serie 2, 200 (2000), 87-97. |
[10] |
J. C. Ceballos and R. Labarca, A note on modulus of stability for cycles of the complex type, Phys. D, 55 (1992), 37-44.doi: 10.1016/0167-2789(92)90186-Q. |
[11] |
B. Deng, The Shilnikov Problem, Exponential Expansion, Strong $\lambda$-Lemma, $C^1$ Linearisation and Homoclinic Bifurcation, Journal of Differential Equations, 79 (1989), 189-231.doi: 10.1016/0022-0396(89)90100-9. |
[12] |
B. Deng, On Shilnikov's Homoclinic Saddle-Focus Theorem, Journal of Differential Equations, 102 (1993), 305-329.doi: 10.1006/jdeq.1993.1031. |
[13] |
B. Deng, Exponential expansion with principal eigenvalues, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 1161-1167.doi: 10.1142/S0218127496000655. |
[14] |
E. Dufraine, Some topological invariants for three-dimensional flows, Chaos, 11 (2001), 443-448.doi: 10.1063/1.1385918. |
[15] |
E. Dufraine, Un critére d'existence d'invariant pour la conjugaison de difféomorphismes et de champs de vecteurs, C.R. Acad. Sci. Paris, 334 (2002), 53-58.doi: 10.1016/S1631-073X(02)02207-0. |
[16] |
M. Field, Lectures on Bifurcations, Dynamics and Symmetry, Pitman Research Notes in Mathematics Series, 356, Longman, Harlow, 1996. |
[17] |
A. C. Fowler, Homoclinic bifurcations in $n$ dimensions, Stud. Appl. Math., 83 (1990), 193-209. |
[18] |
A. Gaunersdorfer, Time averages for heteroclinic attractors, SIAM J. Appl. Math., 52 (1992), 1476-1489.doi: 10.1137/0152085. |
[19] |
P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits, J. Stat. Phys., 35 (1984), 645-696.doi: 10.1007/BF01010828. |
[20] |
S. V. Gonchenko, L. P. Shilnikov, O. V. Stenkin and D. V. Turaev, Bifurcations of systems with structurally unstable homoclinic orbits and moduli of $\Omega$-equivalence, Comput. Math. Appl., 34 (1997), 111-142.doi: 10.1016/S0898-1221(97)00121-1. |
[21] |
P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Math. Mexicana, 5 (1960), 220-241. |
[22] |
M. R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-233. |
[23] |
M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, 1974. |
[24] |
A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbit to a saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050.doi: 10.1088/0951-7715/15/4/304. |
[25] |
A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, Handbook of Dynamical Systems, 3 (2010), 379-524.doi: 10.1016/S1874-575X(10)00316-4. |
[26] |
M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory and Dynam. Sys., 15 (1995), 121-147.doi: 10.1017/S0143385700008270. |
[27] |
M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry. II, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 1177-1197.doi: 10.1017/S0308210500003693. |
[28] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998. |
[29] |
I. S. Labouriau and A. A. P. Rodrigues, Global generic dynamics close to symmetry, Journal of Differential Equations, 253 (2012), 2527-2557.doi: 10.1016/j.jde.2012.06.009. |
[30] |
I. S. Labouriau and A. A. P. Rodrigues, Partial symmetry breaking and heteroclinic tangencies, in Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., 54,, Springer-Verlag, 2013, 281-299.doi: 10.1007/978-3-642-38830-9_17. |
[31] |
I. M. Ovsyannikov and L. P. Shilnikov, On systems with saddle-focus homoclinic curve, Math. USSR Sbornik, 73 (1992), 415-443.doi: 10.1070/SM1987v058n02ABEH003120. |
[32] |
J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Dynamical systems, Vol. III, Warsaw. Soc. Math. France (Astérisque), 51 (1978), 335-346. |
[33] |
A. A. P. Rodrigues, Persistent switching near a heteroclinic model for the geodynamo problem, Chaos, Solitons & Fractals, 47 (2013), 73-86.doi: 10.1016/j.chaos.2012.12.005. |
[34] |
A. A. P. Rodrigues, Repelling dynamics near a Bykov cycle, J. Dynam. Differential Equations, 25 (2013), 605-625.doi: 10.1007/s10884-013-9289-2. |
[35] |
A. A. P. Rodrigues and I. S. Labouriau, Spiralling dynamics near a heteroclinic network, Phys. D, 268 (2014), 34-49.doi: 10.1016/j.physd.2013.10.012. |
[36] |
A. A. P. Rodrigues, I. S. Labouriau and M. A. D. Aguiar, Chaotic double cycling, Dyn. Syst., 26 (2011), 199-233.doi: 10.1080/14689367.2011.557179. |
[37] |
V. S. Samovol, Linearization of a system of differential equations in the neighbourhood of a singular point, Dokl. Akad. Nauk SSSR, 206 (1972), 545-548. |
[38] |
L. P. Shilnikov, Some cases of generation of periodic motion from singular trajectories, Math. USSR Sbornik, 61 (1963), 443-466. |
[39] |
L. P. Shilnikov, A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl., 160 (1965), 558-561. |
[40] |
L. P. Shilnikov, The existence of a denumerable set of periodic motions in four dimensional space in an extended neighbourhood of a saddle-focus, Sov. Math. Dokl., 172 (1967), 54-57. |
[41] |
L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics I, World Scientific Publishing Co., Inc., River Edge, NJ, 1998.doi: 10.1142/9789812798596. |
[42] |
L. Shilnikov, A. Shilnikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics II, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.doi: 10.1142/9789812798558. |
[43] |
F. Takens, Heteroclinic attractors: Time averages and moduli of topological conjugacy, Bol. Soc. Brasil. Mat., 25 (1994), 107-120.doi: 10.1007/BF01232938. |
[44] |
Y. Togawa, A modulus of $3$-dimensional vector fields, Ergod. Theory Dyn. Syst., 7 (1987), 295-301.doi: 10.1017/S0143385700004028. |
[45] |
C. Tresser, About some theorems by L. P. Shilnikov, Ann. Inst. H. Poincaré, 40 (1984), 441-461. |
[46] |
S. J. van Strien, One Parameter Families of Vector Fields, Bifurcations near Saddle-Connections, Ph.D. Thesis, Utrecht University, 1982. |
[47] |
S. Wiggins, Introduction in Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, TAM 2, New York, 1990.doi: 10.1007/978-1-4757-4067-7. |