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## SHILNIKOV BIFURCATION: STATIONARY QUASI-REVERSAL BIFURCATION (2007)

### Citations

500 |
Pattern formation outside of equilibrium,
- Cross, Hohenberg
- 1993
(Show Context)
Citation Context ... to have a qualitative theory of dynamical systems [Guckenheimer & Holmes, 1983], that is, this study permits to describe in a universal way phenomena which belong to different fields [Coullet, 1985; =-=Cross & Hohenberg, 1993-=-]. In one parameter families of dissipative dynamical systems — in codimension one — only two local bifurcations occur generically for equilibrium points: the saddle-node and the Hopf bifurcations. Th... |

263 | Qualitative Universality for a Class of Nonlinear Transformations," - Feigenbaum - 1978 |

136 | Theory of Oscillations - Andronov, Vitt, et al. - 1966 |

83 |
A simple global characterization for normal forms of singular vector fields,
- Elphick, Tirapegui, et al.
- 1987
(Show Context)
Citation Context ...eigenfunction. The reduced linear operator L is then given by a simple Jordan block, L = 0 1 00 0 1 0 0 0 and the homological operator that characterizes the central manifold is [Arnold, 1980; =-=Elphick et al., 1987-=-] H(L) = L − LijAj ∂ ∂Ai , where A = {x, y, z} are the variables that characterize the central manifold [Elphick et al., 1987]. The above operator characterizes the dynamics around the bifurcation. It... |

66 | Iterations d’endomorphismes et groupe de renormalization, - Tresser, Coullet |

63 |
Chapitres Supplementaires de la Theorie des Equations Differentielles Ordinaires, Editions Mir,
- Arnold
- 1980
(Show Context)
Citation Context ...qual to the dimension of the system. In this kind of system the instabilities in one parameter families of equilibrium points are: (a) The stationary instability denoted by (02) in Arnold’s notation [=-=Arnold, 1980-=-], which we use from now on, corresponding to a resonance at zero frequency; and (b) The confusion of frequencies (iΩ2) or 1:1 resonance [Rocard, 1943], which corresponds to a resonance at a finite fr... |

30 | A case of the existence of a countable number of periodic motions,” - Shi’lnikov - 1965 |

9 | 2001] “The stationary instability in quasi-reversible systems and the November 13 - Clerc, Coullet, et al. |

7 | 1999a] “Lorenz bifurcation: Instabilities in quasi-reversible systems - Clerc, Coullet, et al. |

6 | Dissipation-induced instabilities in an optical cavity laser: A mechanical analog near the 1:1 resonance, Phys - Clerc, Marsden |

6 | 1966], “A thermally excited non-linear oscillator,” Astrophys - Moore, Spiegel |

6 | 1943] Dynamique Générale des Vibrations (Masson et cie - Rocard |

5 | 1999b] “The Maxwell–Bloch description of 1/1 resonances - Clerc, Coullet, et al. |

3 | 1971] “Aperiodic behaviour of a non-linear oscillator - Baker, Moore, et al. |

3 | 1980] “Ordinary differential equations with strange attractor - Marzec, Spiegel |

2 | 1985] “Nonlinear phenomena in dissipative systems - Coullet |

1 |
Topological defects and Melnikov’s theory,” Phys
- Coullet, Elphick
- 1987
(Show Context)
Citation Context ...explicitly obtain homoclinic or heteroclinic solutions, this is quite possible in time reversible dynamical systems. Hence, in quasireversible systems by means of a persistence or Melmikov condition [=-=Coullet & Elphick, 1987-=-], one can grasp the homoclinic and heteroclinic solutions [Clerc et al., 1999, 2000, 2001]. The persistence of the three-dimensional homoclinic solution guarantees the existence of chaos when the Shi... |

1 | 1987] “Homoclinic explosion in the vicinity of bifurcation at the triple-zero eigenvalue,” Phys - Pismen |