American Institute of Mathematical Sciences

Advanced
January  2001, 7(1): 203-218. doi: 10.3934/dcds.2001.7.203

Saddle-node bifurcation of homoclinic orbits in singular systems

Flaviano Battelli 1,

1. 

Dipartimento di Matematica "V. Volterra", Facolta' di Ingegneria-Università, Via Brecce Bianche, 1, 60131 Ancona, Italy

Received  April 2000 Published  November 2000

We consider the singularly perturbed system $\dot\xi = f_0(\xi) + \varepsilon f_1(\xi,\eta,\varepsilon)$, $\dot\eta = \varepsilon g(\xi,\eta,\varepsilon )$ where $\xi\in\Omega\subset\mathbb R^n$, $\eta\in\mathbb R$ and $\varepsilon\in\mathbb R$ is a small real parameter. We assume that $\dot\xi = f_{0}(\xi)$ has a non degenerate heteroclinic solution $\g(t)$ and that the Melnikov function $\int_{-\infty}^{+\infty} \psi^{*}(t) f_{1}(\g(t),\alpha,0)\dt$ has a double zero at some point $\alpha_{0}$. Using a functional analytic approach we show that if a suitable second order Melnikov function is not zero, the above system has, in a neighborhood of $\{\gamma(t)\}\times\mathbb R$, two heteroclinic orbits for $\varepsilon$ on one side of $\varepsilon=0$ and none for $\varepsilon$ on the other side. We also study the transversality of the intersection of the center-stable and the center-unstable manifolds along these orbits.
Keywords: Homoclinic orbits, transversality, Melnikov functions, Lyapunov-Schmidt method., saddle-node bifurcation.
Mathematics Subject Classification: 34C23, 34C37, 58F3.
Citation: Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203
[1]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[2]

Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923
[3]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
[4]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[5]

Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069
[6]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201
[7]

Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21
[8]

Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215
[9]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
[10]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[11]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[12]

Flaviano Battelli, Ken Palmer. A remark about Sil'nikov saddle-focus homoclinic orbits. Communications on Pure & Applied Analysis, 2011, 10 (3) : 817-830. doi: 10.3934/cpaa.2011.10.817
[13]

Flaviano Battelli, Claudio Lazzari. On the bifurcation from critical homoclinic orbits in n-dimensional maps. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 289-303. doi: 10.3934/dcds.1997.3.289
[14]

José Mujica, Bernd Krauskopf, Hinke M. Osinga. A Lin's method approach for detecting all canard orbits arising from a folded node. Journal of Computational Dynamics, 2017, 4 (1&2) : 143-165. doi: 10.3934/jcd.2017005
[15]

E. Canalias, Josep J. Masdemont. Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 261-279. doi: 10.3934/dcds.2006.14.261
[16]

Kazuyuki Yagasaki. Application of the subharmonic Melnikov method to piecewise-smooth systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2189-2209. doi: 10.3934/dcds.2013.33.2189
[17]

Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i
[18]

Martin Wechselberger, Warren Weckesser. Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 829-850. doi: 10.3934/dcdss.2009.2.829
[19]

Qian Liu, Xinmin Yang, Heung Wing Joseph Lee. On saddle points of a class of augmented lagrangian functions. Journal of Industrial & Management Optimization, 2007, 3 (4) : 693-700. doi: 10.3934/jimo.2007.3.693
[20]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883

2017 Impact Factor: 1.179

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences