Repelling Dynamics Near a Bykov Cycle

Journal of Dynamics and Differential Equations

Volumn 25, Issue 3, 2013, Pages 605-625

  Rodrigues, Alexandre A P ^a  

^a UNIVERSITY OF PORTO   (Portugal)

Author keywords

Bykov cycle; C1 vector fields; Generalized Conley Morse conditions; Repelling dynamics

Indexed keywords

EID: 84883458671     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1007/s10884-013-9289-2     Document Type: Article

References (33)

1. Aguiar, M., Castro, S. B., Labouriau, I. S.: Simple vector fields with complex behaviour. Int. J. Bifurc. Chaos 16(2), 369-381 (2006).
2. Aguiar, M. A. D., Labouriau, I. S., Rodrigues, A. A. P.: Swicthing near a heteroclinic network of rotating nodes. Dyn Syst Int J 25(1), 75-95 (2010).
3. Brannath, W.: Heteroclinic networks on the tetrahedron. Nonlinearity 7, 1367-1384 (1994).
4. Birkhoff, G. D.: Dynamical systems. Am. Math. Soc. Colloq. Publ. 9, 295 (1927).
5. Birkhoff, G. D.: Nouvelles recherches sur les systèmes dynamiques. Memorie Pont. Acad. Sci. Novo. Lyncaei 53(1), 85-216 (1935).
6. Bowen, R.: A horseshoe with positive measure. Invent. Math. 29, 203-204 (1975).
7. Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics. Springer, New York (1975).
8. Buescu, J.: Exotic Attractors, Progress in Mathematics, Vol. 153. Birkhauser Verlag, Basel (1997).
9. Bykov, V. V.: Orbit structure in a neighbourhood of a separatrix cycle containing two saddle-foci. Am. Math. Soc. Transl. 200, 87-97 (2000).
10. 2-symmetric electronic oscillator. (I) Analysis. Nonlinear Dyn. 28, 53-69 (2002).
11. Field M.: Lectures on Bifurcations, Dynamics and Symmetry. Pitman Research Notes in Mathematics Series, Vol. 356. Longman, New York (1996).
12. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, Vol. 42. Springer, New York (1983).
13. Homburg, A. J.: Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbit to a saddle-focus equilibria. Nonlinearity 15, 411-428 (2002).
14. Homburg, A. J., Sandstede, B.: Homoclinic and heteroclinic bifurcations in vector fields. In: Handbook of Dynamical Systems, vol. 3, pp. 379-524. North Holland, Amsterdam (2010).
15. 3. J. Differ. Equ. 208, 147-175 (2008).
16. Koon, W. S., Lo, M., Marsden, J., Ross, S.: Heteroclinic connections between periodic orbits and resonance transition in celestial mechanics. In: Control and Dynamical Systems Seminar. California Institute of Technology, Pasadena (1999).
17. Krupa, M., Melbourne, I.: Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergod. Theory Dyn. Syst. 15, 121-147 (1995).
18. Krupa, M., Melbourne, I.: Asymptotic stability of heteroclinic cycles in systems with symmetry, II. Proc. R. Soc. Edinb. 134A, 1177-1197 (2004).
19. Labouriau, I. S., Rodrigues, A. A. P.: Global generic dynamics close to symmetry. J. Differ. Edu. 253(8), 2527-2557 (2012).
20. Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, 177-195 (1985).
21. Mora, L., Viana, M.: Abundance of strange attractors. Acta Math. 171, 1-71 (1993).
22. Moser, J.: Stable and Random Motions in Dynamical Systems. Princeton University Press, Princeton (1973).
23. Podvigina, O., Ashwin, P.: On local attraction properties and a stability index for heteroclinic connections. Nonlinearity 24, 887-929 (2011).
24. Poincaré, H.: Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13, 1-270 (1890).
25. Rodrigues, A. A. P.: Persistent switching near a heteroclinic model for the geodynamo problem. Chaos Solitons Fractals 47, 73-86 (2013).
26. Rodrigues, A. A. P., Labouriau, I. S.: Spiraling sets near a heteroclinic network, submitted.
27. Rodrigues, A. A. P., Labouriau, I. S., Aguiar, M. A. D.: Chaotic double cycling. Dyn. Syst. Int. J. 26(2), 199-233 (2011).
28. Samovol, V. S.: Linearization of a system of differential equations in the neighbourhood of a singular point. Sov. Math. Dokl. 13, 1255-1959 (1972).
29. Shilnikov, L. P.: Some cases of generation of periodic motion from singular trajectories. Math. USSR Sbornik 61(103), 443-466 (1963).
30. Shilnikov, L. P.: A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl. 6, 163-166 (1965).
31. Shilnikov, L. P.: The existence of a denumerable set of periodic motions in four dimensional space in an extended neighbourhood of a saddle-focus. Sovit. Math. Dokl. 8(1), 54-58 (1967).
32. Smale, S.: Diffeomorphisms with many periodic orbits. In: Cairus, S. (ed.) Differential Combinatorial Topology, pp. 63-86. Princeton University Press, Princeton (1960).
33. Wiggins, S.: Introduction in Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990).