Codimension 3 Homoclinic Bifurcation of Orbit Flip With Resonant Eigenvalues Corresponding to the Tangent Directions

Abstract

Bifurcations of homoclinic orbit with orbit-flip and resonant eigenvalues corresponding to the tangent directions are investigated in a four-dimensional system. The existence, number, coexistence and incoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are given, and the bifurcation surfaces and the existence regions are also located.

Bifurcations of heterodimensional cycles with one orbit flip and one inclination flip

“…Proof Here we only prove (14) because the proof of (15) is similar. In order to verify (14), we only need to prove φ j *…”

“…Taking into account the normal forms (9) and (10), and using the formula of variation of constants, we can easily get that system (1) has the following solution in the neighborhood U 1 (see [14,17]):…”

“…In recent years, homoclinic and heteroclinic bifurcation theory has attracted a lot of attention due to its important role in applications (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and references therein) to many physical phenomena. Specially, there has been a surging interest in the bifurcations of homoclinic and heteroclinic loop with resonance, orbit flip or inclination flip, etc.…”

“…Specially, there has been a surging interest in the bifurcations of homoclinic and heteroclinic loop with resonance, orbit flip or inclination flip, etc. [12,14,18,20]. A heteroclinic loop is said to be a heterodimensional cycle if it is formed by two or more normally hyperbolic invariant manifolds which have different dimensions of the unstable manifolds and two or more heteroclinic orbits connecting them; that is to say, a heterodimensional cycle is a special heteroclinic loop, which has activated interest of many scholars [1,3,10,11,17].…”

“…Proof (1) Assume (29) has a solution satisfying s 1 = 0, then in the region w 14 2 M 1 2 μ < 0, we have…”