Dynamics near homoclinic orbits to a saddle in four-dimensional systems with a first integral and a discrete symmetry

The paper establishes Theorem A (single homoclinic loop) and the h=0 classification (by invoking [BLT22]) via a decomposition of the Poincaré map into local (near the saddle) and global (along the homoclinic) pieces, with estimates obtained from normal forms and a Shilnikov boundary-value argument. It proves: (i) for h<0 there is a unique hyperbolic periodic orbit L_h in a small neighborhood, with 2D W^s/u(L_h), and all other orbits exit; (ii) for h=0 and λ2≠2λ1, the non-escaping orbits are precisely those in W^s_loc(Γ)∪W^s_{U0}(O) (forward) and W^u_loc(Γ)∪W^u_{U0}(O) (backward), with the cd and bd sign criteria deciding whether the local sets are just Γ or C^1 2D sheets; (iii) for h>0 all orbits leave the neighborhood. These are stated explicitly in the paper’s statements and proved in Sections 4–5 for the single loop, with Theorem 1 (from [BLT22]) handling h=0 . The candidate solution follows the same blueprint: it constructs the Poincaré map as a composition of a local Dulac-type map (with exponent ρ=λ1/λ2=γ) and a global map with linear part M=[a b; c d], reproduces the same three conclusions for h<0, h=0 (with the same sign conditions), and h>0. Minor differences are only technical (Dulac-map vs. Shilnikov BVP derivations), not substantive. The assumptions on a,b,c,d match Theorem A (all nonzero) and the h=0 classification is correctly tied to the nonresonant case λ2≠2λ1 per Theorem 1 from [BLT22] ; the negative, zero, and positive energy cases are handled exactly as in the paper’s arguments, including the cone and hyperbolicity conclusions for h≠0 (unique L_h for h<0, no bounded sets for h>0) .