Dynamics of C1-perturbed non-autonomous systems with strong cone conditions and its applications

The paper proves a robust dichotomy for differences of orbits near a compact invariant set (Theorem 2.1), the almost 1-cover property of minimal sets when codim H = 1 (Theorem 2.2), and a trichotomy for omega-limit sets (Theorem 2.3), all under strong cone conditions (H1)–(H5). It does so by introducing a difference cocycle T, establishing k-exponential separation along nested cones, and deriving uniform contraction outside the outer cone, with robustness under small C1 perturbations. The candidate model’s solution follows essentially the same blueprint: define the linear difference cocycle, invoke k-exponential separation/dominated-splitting, use robustness to pass properties to nearby semiflows, and then deduce the dichotomy, avoidance of H, almost 1-covering, and trichotomy. The main technical differences are stylistic: the model leans on abstract dominated-splitting/Sacker–Sell language, whereas the paper constructs the cone layers and the needed contraction directly via Proposition 3.1 and Lemma 6.2. No substantive contradictions were found, and the claims align closely with the paper’s statements and proofs (cf. the setup of (H1)–(H5) and T in (2.2)–(2.3), Theorem 2.1, Proposition 3.1, and the trichotomy Theorem 2.3). See the paper’s statements and proofs for these items: assumptions and T in (2.2)–(2.3) and (H1)–(H5) , the robust cone trapping vs exponential-decay estimates in Proposition 3.1 and its use in Theorem 2.1(1)–(3) , auxiliary facts on the difference cocycle and exponential separation in Section 6 (e.g., (6.1)–(6.2)) , the perturbation lemma (Lemma 6.5) , and the almost 1-cover and trichotomy results (Theorems 2.2, 2.3) .