Bifurcation Analysis of Homoclinic Flips at Principal Eigenvalues Resonance

Abstract

One orbit flip and two inclination flips bifurcation is considered with resonant principal eigenvalues. We introduce a local active coordinate system to establish bifurcation equation and obtain the conditions when the original homoclinic orbit is kept or broken. We also prove the existence and the existence regions of double 1-periodic orbit bifurcation. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately, and are well located.

Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

“…It is well known that there are always two and −1 transformations successively, also by the stable (or unstable) manifold theorem in [15], to straighten the local manifolds loc and loc as loc = { ∈ , = = 0} and loc = { ∈ , = V = 0}, respectively, loc = { ∈ , = = = 0} (resp., loc = { ∈ , = = V = 0}); see [8][9][10]. Notice that now Γ ∩ loc = { ∈ , = ( ), = V = 0} and Γ ∩ loc = loc , where = ( , , , V) ∈ R 4 and (0) = (0) = 0.…”

“…Recently, Zhang et al in [8][9][10] studied a kind of multiple flips homoclinic resonant bifurcation and got the existence of some saddle-node bifurcations and homoclinic-doubling bifurcations. Meanwhile Geng et al in [11], Lu et al in [12], and Liu in [13] discussed, respectively, a heterodimensional cycle flip or accompanied by transcritical bifurcation; they found the double and triple periodic orbit bifurcations and gave also some coexistence conditions for homoclinic orbits and periodic orbits.…”

“…(resp., ∈ 10 4 ). (2) For 3 = 0, system (1) has exactly one (resp., not any) 1-periodic orbit near Γ as ∈ 01 1 (resp., ∈ 11 1 ∩ 01 14 ).…”