# American Institute of Mathematical Sciences

March  2007, 6(1): 1-21. doi: 10.3934/cpaa.2007.6.1

## Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle

 1 Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek, Belgium, Belgium 2 Université de Bourgogne, I.M.B., U.M.R. 5584 du C.N.R.S., Agoralaan–gebouw D, 21078-Dijon Cedex, France

Received  January 2006 Revised  June 2006 Published  December 2006

It is known that perturbations from a Hamiltonian 2-saddle cycle $\Gamma$can produce limit cycles that are not covered by the Abelian integral, even when the Abelian integral is generic. These limit cycles are called alien limit cycles. In this paper, extending the results of [6] and [2], we investigate the number of alien limit cycles in generic multi-parameter rigid unfoldings of the Hamiltonian 2-saddle cycle, keeping one connection unbroken at the bifurcation.
Citation: Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1
 [1] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [2] Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381 [3] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [4] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276 [5] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [6] Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049 [7] Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218 [8] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [9] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [10] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [11] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 [12] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [13] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [14] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [15] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [16] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [17] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [18] Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344 [19] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [20] Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

2019 Impact Factor: 1.105