Spatially localized structures in lattice dynamical systems

The paper proves two main results: (i) the existence and organization of symmetric homoclinic orbits via submanifolds Γ1, Γ2 that are C^k–close to a lifted heteroclinic locus Γlift with errors O(η^{2N+(j−1)}), implying snaking for 1-loops and isolas for 0-loops; and (ii) symmetry-breaking: near a quadratic tangency, pairs of asymmetric homoclinic branches bifurcate in pitchforks at µ = µ0 + O(η^N), are bounded by pitchforks near oppositely curved saddle-nodes on the symmetric branch, and otherwise occur as isolas. These are stated in Theorems 2.4 and 2.5 and proved through local straightening near u*, entry–exit estimates (Lemma 3.2), quotienting by the phase map q, and Z2-symmetric matching functions (Lemma 4.4, Lemma 4.5, Lemma 4.6) . The candidate solution reproduces the same architecture: local reversible coordinates, exponential entry–exit bounds with η ∈ (0,1), construction of matching equations for on-site/off-site symmetry, C^k closeness O(η^{2N}) and O(η^{2N+1}), the 1-loop vs 0-loop dichotomy, and a Z2-normal form for asymmetric branches with pitchforks at µ0 + O(η^N). The only notable omission is an explicit statement of the paper’s additional nondegeneracy Hypothesis 5 used to guarantee that asymmetric branches are either smooth isolas or smooth curves bounded precisely by pitchforks (Lemma 4.5) . With that minor caveat, the model’s proof strategy and conclusions align closely with the paper’s results.