Linearly independent homoclinic bifurcations parameterized by a small function

Blazquez, 1986, Bifurcation from a homoclinic orbit in parabolic differential equations, Proc. Roy. Soc. Edinburgh Ser. A, 103, 265, 10.1017/S0308210500018916 Blazquez, 1986, Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal., 10, 1277, 10.1016/0362-546X(86)90066-0 Chow, 1982 Chow, 1980, An example of bifurcation to homoclinic orbits, J. Differential Equations, 37, 551, 10.1016/0022-0396(80)90104-7 Feckam, 2000, Bifurcation from homoclinic to periodic solutions in singular ordinary differential equations, J. Math. Anal. Appl., 246, 245, 10.1006/jmaa.2000.6791 Gruendler, 1992, Homoclinic solutions for autonomous systems in arbitrary dimension, SIAM J. Math. Anal., 23, 702, 10.1137/0523036 Gruendler, 1995, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbation, J. Differential Equations, 122, 1, 10.1006/jdeq.1995.1136 Guckenheimer, 1983 Hale, 1985, Perturbation of homoclinic and subharmonics in Duffing's equation, Nonlinear Anal., 9, 181, 10.1016/0362-546X(85)90071-9 He, 2005, Subharmonic bifurcations in a perturbed nonlinear oscillation, Nonlinear Anal., 61, 1057, 10.1016/j.na.2005.01.095 Palmer, 1984, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55, 225, 10.1016/0022-0396(84)90082-2 Zhang, 2004, On nulls of perturbed Fredholm operators and degenerate homoclinic bifurcations, Sci. China Ser. A, 47, 617, 10.1360/02ys0193 Zhang, 1996, Bounded solution for non-autonomous parabolic equations, Dynam. Stab. Systems, 11, 109, 10.1080/02681119608806219 Zhang, 1989, Bifurcation of homoclinics in a nonlinear oscillation, Acta Math. Sinica (N.S.), 5, 170, 10.1007/BF02107670