Codimension 2 bifurcations of double homoclinic loops☆
Section snippets
Setting of the problem
In recent years, a great number of papers have been devoted to the problem of the homoclinic or heteroclinic loop bifurcations (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and the further references therein). But few concerned the bifurcation of double homoclinic loops except [8]. In [8] the authors gave the number of limit cycles near double homoclinic loops under perturbations in planar Hamiltonian systems. In this
Preliminary results and bifurcation equations
Suppose that U is small enough, we can introduce successively three transformations such that system (1.1) has the following normal form in U:In fact, the first Cr transformation is used to straighten Ws and Wu, the second coordinate change is used to straighten the strong stable manifold Wss and the strong unstable manifold Wuu of O, and the last one is
Large homoclinic loop bifurcations
In this section, we study the existence, uniqueness and incoexistence of the double homoclinic, large 1-hom and large 1-per orbit. Firstly, we have the following result concerned with the uniqueness and the incoexistence. Theorem 3.1 Suppose that - and hold. Then, for small enough, system (1.1) has at most one large loop consisting of two homoclinic orbits, one large 1-homoclinic orbit or one large 1-per orbit in a small neighborhood of Γ, and these orbits can not coexist. Proof Change time t to −
Inexistence of large 2-hom and large 2-per orbit
In this section, we study the inexistence of large 2-hom and large 2-per orbits under the condition . Like before, we set τ3 and τ4 the times from to and from to , respectively, with and and . Then, we can easily see that there is an one to one correspondence between these large 2-hom or 2-per orbits and the solutions ofwith ,
References (21)
- et al.
Prediction of homoclinic bifurcation: the elliptic averaging method
Chaos, Solitons & Fractals
(2000) - et al.
Bifurcations for heteroclinic orbits of a periodic motion and a saddle-focus and dynamical chaos
Chaos, Solitons, & Fractals
(1998) - et al.
Homoclinic bifurcations in heterogeneous market models
Chaos Solitons & Fractals
(2003) Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations
J Differen Equat
(1995)- et al.
Bifurcation of limit cycles and separatrix loops in singular Lienard systems
Chaos, Solitons & Fractals
(2004) - et al.
Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation
Chaos, Solitons & Fractals
(2004) Global aspects of homoclinic bifurcations of three-dimensional saddles
Chaos, Solitons & Fractals
(1993)- et al.
Homoclinic bifurcation at resonant eigenvalues
J Dyn Syst Diff Eqs
(1990) Homoclinic solutions for autonomous dynamical systems in arbitrary dimension
SIAM J Math Anal
(1992)- et al.
On the number of limit cycles in double homoclinic bifurcations
Sci Chin Ser A
(2000)
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2023, Journal of Applied Analysis and ComputationBifurcations of twisted double homoclinic loops with resonant condition
2016, Journal of Nonlinear Science and ApplicationsThe twisting bifurcations of double homoclinic loops with resonant eigenvalues
2013, Abstract and Applied Analysis
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Supported by National Natural Science Foundation of PR China (Nos. 10371040, 10671069).