{"id":"9ce87efb-61f3-42a2-8fd8-378b3211c288","arxiv_id":"2501.01279","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For contact Hamiltonian systems, if solution semigroups converge to ordered weak KAM solutions, action-minimizing semi-infinite and heteroclinic orbits asymptotic to the associated Mane slices exist.","lead":"This mathematics paper proves conditions under which orbits of dissipative contact Hamiltonian systems travel toward or between special invariant sets called Mane slices, without requiring the system to be monotone. It matters because such systems arise in celestial mechanics and optimal control, and the work extends a standard variational framework to a broader class of models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.4's expansiveness estimate treats v_+ as a fixed point of T^-_t, but v_+ ∈ S_+ is only fixed by T^+; inequality (4.15) is unjustified, so the lower bound t_ε → ∞ and the heteroclinic construction (B2) are unsupported.","rationale":"The reader's weakest assumption identifies the same internal error: Lemma 4.4 applies the Lipschitz estimate for T^-_t to v_+ as if v_+ were a fixed point of T^-_t. I checked the surrounding text: Proposition 7.2(4) is the only source cited for (4.15), and it cannot yield the claimed inequality unless T^-_{t_ε} v_+ = v_+. The paper never proves this, and the model analysis in Section 6 shows that forward weak KAM solutions are generally not backward-invariant under T^-_t. Since the entire construction of the heteroclinic orbit Z_0 in (B2) depends on t_ε → ∞, the proof of the central theorem is incomplete. This is a genuine gap, not a stylistic or consensus issue. I also note that Theorem A and the general framework may be salvageable: Theorem A's proof uses only T^-_t convergence and the global characteristic method, and the model system analysis appears self-contained. The verdict should remain REJECT as the main advertised result (heteroclinic orbits) is not supported as written.","tokens_in":37926,"tokens_out":8907,"duration_ms":76890,"concrete_test":"For the concrete non-monotone model (6.10) in Section 6.3, compute T^-_1 v_+ at a fixed time, say t=1, using the variational formula (2.4) or by numerical integration of the Hamilton-Jacobi evolution, and compare the result with v_+. If T^-_1 v_+ ≠ v_+, then the equality T^-_{t_ε} v_+ = v_+ used to derive (4.15) is false, and Lemma 4.4's lower bound on t_ε is not justified. As a complementary check, compute t_ε numerically as ε→0; if it approaches a finite value rather than diverging to +∞, the construction of the heteroclinic orbit in (B2) fails as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Lemma 4.4 (Section 4.2), the authors claim by the expansiveness estimate that ||T^-_{t_ε} φ_ε − v_+||_∞ ≤ e^{λ t_ε} ||φ_ε − v_+||_∞. Proposition 7.2(4) gives ||T^-_{t_ε} φ_ε − T^-_{t_ε} ψ||_∞ ≤ e^{λ t_ε} ||φ_ε − ψ||_∞ for any ψ, so the displayed inequality is valid only if T^-_{t_ε} v_+ = v_+. But v_+ is assumed to be a forward weak KAM solution, i.e., T^+_t v_+ = v_+; no statement in the paper implies that v_+ is a fixed point of the backward semigroup T^-_t. In fact, for the model (1.2) with sign-changing λ, T^-_t u_+ typically converges to a different weak KAM solution (Section 6.2). Without (4.15), the lower bound t_ε ≥ (1/λ) ln(ε_0/ε) in (4.17) has no basis. This is load-bearing: Lemma 4.4 is exactly what forces the shift t_ε to diverge as ε→0, which is required for the shifted orbits Z_ε(t) to converge to a complete orbit Z_0 defined on all of R and hence for the heteroclinic orbit in (B2) to exist. The gap is internal to the proof, not a disagreement with consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the dynamics of contact Hamiltonian systems satisfying Tonelli conditions (H1)-(H2) and uniform Lipschitz dependence in u (H3), without the monotonicity assumptions of the authors' earlier work. Using the action-function and semigroup framework developed in references [36]-[38], the authors prove a global characteristic method and then use it to establish Theorem A: uniform convergence of the backward solution semigroup T^-_t φ to a backward weak KAM solution u_- implies existence of a semi-infinite orbit starting on the 1-graph of φ whose ω-limit set is contained in the associated Mane slice ᵎc_{u_-}. They then state Theorem B: if both T^-_t φ and T^+_t φ converge to ordered weak KAM solutions v_+ < u_-, then there exists a complete orbit with α-limit in ᵎc_{v_+} and ω-limit in ᵎc_{u_-}. The proof of Theorem B(B2) proceeds by approximating φ, producing shifted orbits, and using an expansiveness estimate to force the shift t_ε to tend to infinity. The paper concludes with applications to a model Hamiltonian H = F(x,p) + λ(x)u, giving a classification of global action-minimizing orbits and an example of a heteroclinic orbit that does not lie on the zero energy level.","tokens_in":38248,"tokens_out":6895,"duration_ms":63827,"significance":"If the results are correct, the paper is significant: it extends the variational (weak KAM/Aubry-Mather) approach to non-monotone contact Hamiltonian systems, constructs semi-infinite and heteroclinic orbits for such systems, and provides a concrete model illustrating that connecting orbits may have nonzero energy. Strengths include a clear statement of the variational framework, a global characteristic method, and a nontrivial application to a sign-changing λ model. The reliance on previously published semigroup results is standard and does not, by itself, constitute circularity. However, the central construction in Theorem B(B2) rests on Lemma 4.4, and the proof of that lemma contains a serious technical error; as written, the main claim of Theorem B is not established.","major_comments":[{"comment":"The proof of Lemma 4.4 misapplies the expansiveness estimate from Proposition 7.2(4). The inequality stated in (4.15) is ||T^-_{t_ε} φ_ε - v_+||_∞ ≤ e^{λ t_ε} ||φ_ε - v_+||_∞. Proposition 7.2(4) gives ||T^-_{t_ε} φ_ε - T^-_{t_ε} ψ||_∞ ≤ e^{λ t_ε} ||φ_ε - ψ||_∞, so the displayed inequality is valid only if T^-_{t_ε} v_+ = v_+. However, v_+ is assumed to be a forward weak KAM solution, i.e., T^+_t v_+ = v_+, and no statement in the paper (nor any property of the backward semigroup T^-_t) implies that v_+ is fixed by T^-_t. In fact, for the model (1.2) with sign-changing λ, forward and backward weak KAM solutions are typically different, as the paper itself indicates in Section 6.2. Without (4.15), the lower bound t_ε ≥ (1/λ) ln(ε_0/ε) in (4.17) has no basis. This lower bound is load-bearing: it is exactly what forces the shift t_ε to diverge as ε → 0, which is required for the translated orbits Z_ε(t) to converge to a complete orbit Z_0 defined on all of R. Consequently, the construction of the heteroclinic orbit in (B2) is unsupported.","section":"Section 4.2, Lemma 4.4 and Eq. (4.15)"},{"comment":"Because Lemma 4.4 is not justified, the existence of the complete orbit Z_0 in (4.20) and the conclusions of Proposition 4.7 (the ω-limit and α-limit inclusions) do not follow from the given arguments. This also affects later statements that rely on Theorem B, in particular Theorem 6.8(3) and the Example 6.9, where a heteroclinic orbit between the two fixed points is asserted on the basis of (B2). The gap is internal to the proof; it is not a matter of an alternative method that the paper already contains. A correct proof of the required lower bound on t_ε, or a substantially different argument for the completeness of the limiting orbit, is needed to repair the central claim.","section":"Section 4.2, proof of Theorem B(B2) and Section 6.3"}],"minor_comments":[{"comment":"In the displayed computation near Eq. (2.11), the symbol '˙q0(τ)' appears instead of '˙x0(τ)'. This is a typographical error in the expression for the Lagrangian along the first characteristic segment.","section":"Section 2.2, proof of Theorem 2.9"},{"comment":"The notation u_- is used both for a general backward weak KAM solution and for the specific limit u_- := lim_{t→∞} T^-_t u_+ introduced in (6.5). This overloaded notation can confuse the reader, especially since Theorem 6.6 and the following applications use u_- in both senses. Please clarify by using a distinct symbol for the minimal element of (S_-, ⪯).","section":"Section 6.2, Theorem 6.6 and surrounding text"}],"recommendation":"major_revision","confidential_remarks":"The main gap is in Lemma 4.4, which is central to Theorem B. If the authors can supply a correct argument for the divergence t_ε → ∞ (or otherwise establish the completeness of the limiting orbit), the paper could be publishable. The reliance on the authors' previous works is acceptable as external, published results. The paper is within the scope of the journal, but in its current form the main theorem is not proven."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a genuinely useful core: the global characteristics method (Theorem 2.11) and Theorem A, which constructs semi-infinite orbits asymptotic to a Máne slice from convergence of the backward semigroup. That part is new, extends the authors' monotone theory to non-monotone systems, and the proof appears sound. The model analysis in Section 6, especially the classification of global action minimizers and the example of heteroclinic orbits off the null energy level, is also interesting and well-informed.\n\nThe trouble is Theorem B, specifically (B2). The proof relies on Lemma 4.4 to force the shift time t_ε to go to infinity. That lemma uses the expansiveness estimate\n\n‖T^-_{t_ε} φ_ε − v_+‖ ≤ e^{λ t_ε} ‖φ_ε − v_+‖.\n\nThis inequality is not justified. Proposition 7.2(4) gives the bound for ‖T^-_t φ − T^-_t ψ‖, not for ‖T^-_t φ − ψ‖. To get the displayed line one would need T^-_{t_ε} v_+ = v_+. But v_+ is a forward weak KAM solution; it is fixed by T^+, not by T^-. The paper nowhere shows v_+ ∈ S_-, and in fact for the sign-changing model in Section 6.2, T^-_t u_+ converges to the minimal backward solution, not to u_+ itself. So (4.15) is false in general. Without it, the lower bound t_ε ≥ (1/λ) ln(ε_0/ε) does not follow, and the shifted orbits need not converge to a complete orbit. The heteroclinic orbit advertised in (B2) is unsupported.\n\nThis is not a cosmetic gap. It sits at the load-bearing point of the paper's main new claim. The paper might be repairable—one could try to control the difference between T^-_t v_+ and v_+ using the convergence assumptions, or use a different comparison argument—but that is not what the paper does.\n\nFor all that, I would not desk-reject this. Theorem A, the characteristics method, and the model analysis have independent value, and the flaw is specific and identifiable. A serious referee would catch it and the authors would need to fix it. So yes, send it out. Just go in expecting the central claim to need substantial rework.","headline":"The core characteristics method and Theorem A are solid, but the Lemma 4.4 argument for Theorem B's heteroclinic orbits is genuinely wrong, so the paper's headline result is unproven as written.","tokens_in":38815,"tokens_out":3720,"would_cite":false,"duration_ms":35463,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37J50","35F21","37J55","37C29"],"pacs":[],"model":"deepseek-v4-flash","headline":"Assuming ordered convergence of the forward and backward solution semigroups for a common initial function, the paper constructs a complete orbit of the contact Hamiltonian flow whose past limit lies on the lower Mane slice and whose…","keywords":["contact Hamiltonian systems","weak KAM solutions","Mane slices","action minimizing orbits","heteroclinic orbits","Hamilton-Jacobi equations","viscosity solutions","variational construction"],"falsifier":"For a Hamiltonian satisfying (H1)-(H3) with a forward limit $v_+$ that is not a backward weak KAM solution, compute $T^-_t v_+$; if the inequality $\\|T^-_{t_\\varepsilon} \\varphi_\\varepsilon - v_+\\|_\\infty \\le e^{\\lambda t_\\varepsilon} \\varepsilon$ used in Lemma 4.4 fails for some small $\\varepsilon$, the proof of the heteroclinic construction loses its lower bound on $t_\\varepsilon$ and the construction in (B2) would need a different estimate.","tokens_in":37665,"feed_emoji":"🔁","tokens_out":9646,"duration_ms":85335,"temperature":0.7,"pith_summary":"This paper studies the dynamics of contact Hamiltonian systems without the monotonicity assumption used in the authors' earlier work, focusing on action-minimizing orbits. It proves that when the backward solution semigroup of the associated Hamilton-Jacobi equation converges to a backward weak KAM solution $u_-$, there is a semi-infinite orbit starting from the 1-jet of the initial data that limits onto the Mane slice $\\widetilde{\\mathcal{N}}_{u_-}$. If, in addition, the forward semigroup converges to a forward weak KAM solution $v_+$ with $v_+ < u_-$ pointwise, the paper constructs a complete orbit whose $\\alpha$-limit lies in $\\widetilde{\\mathcal{N}}_{v_+}$ and whose $\\omega$-limit lies in $\\widetilde{\\mathcal{N}}_{u_-}$, and shows no reverse connection exists. Applying this to a decoupled model $H = F(x,p) + \\lambda(x)u$ with sign-changing $\\lambda$, the authors obtain a classification of global action-minimizing orbits and an example of heteroclinic orbits that do not lie on the zero-energy level.","feed_headline":"Semigroup convergence builds asymptotic orbits in contact systems","feed_subtitle":"The forward and backward limits of one initial datum force a connecting orbit","key_machinery":"The load-bearing object is the Mane slice $\\widetilde{\\mathcal{N}}_u$, a compact $\\Phi^t_H$-invariant set obtained by intersecting backward or forward images of the 1-pseudograph of a weak KAM solution $u$; it is the action-minimizing set associated with that solution. The mechanism is a global characteristics method (Theorem 2.11) extending classical characteristics to all times: for each $(x,t)$ there is $Z_0 \\in \\mathcal{J}^1_\\varphi$ whose orbit segment satisfies $u(\\tau) = T^-_\\tau \\varphi(x(\\tau))$ and $p(\\tau) = \\partial_x T^-_\\tau \\varphi(x(\\tau))$, so the orbit stays on the graph of the viscosity solution. This converts semigroup convergence into pre-compact families of orbit segments, whose limits are the desired semi-infinite and heteroclinic orbits. The action functions $h_{x_0,u_0}$ and their Markov property supply the variational estimates.","core_discovery":"The central discovery is a variational duality: large-time convergence of the solution semigroups governing the evolutionary Hamilton-Jacobi equation forces the existence of asymptotic orbits of the contact flow, even though the flow has no monotonicity. Theorem A shows that uniform convergence $T^-_t \\varphi \\to u_-$ implies that some $Z \\in \\mathcal{J}^1_\\varphi$ has $\\omega(Z) \\subset \\widetilde{\\mathcal{N}}_{u_-}$, and that the pseudograph $\\mathcal{J}^1_{u_-}$ is contained in the forward saturation of $\\mathcal{J}^1_\\varphi$. Theorem B shows that if the same initial datum gives forward convergence to $v_+$ and backward convergence to $u_-$ with $v_+ < u_-$, then a complete orbit exists with $\\alpha(Z) \\subset \\widetilde{\\mathcal{N}}_{v_+}$ and $\\omega(Z) \\subset \\widetilde{\\mathcal{N}}_{u_-}$; part (B1) rules out the reversed ordering. The proof uses a global characteristics method: for every terminal point $(x,t)$ one can find an orbit of the flow starting on $\\mathcal{J}^1_\\varphi$ that stays on the graph of the viscosity solution $U(x,t) = T^-_t \\varphi(x)$ with $p = \\partial_x U$, so semigroup convergence can be converted into orbit convergence.","pith_inferences":["The paper leaves implicit that the heteroclinic construction in Theorem B depends on the forward limit $v_+$ being compatible with the backward semigroup through an exponential estimate; if that compatibility fails, the proof of (B2) would need a different time-scale argument even if the theorem itself remains true.","The nonzero-energy heteroclinics suggest that for non-monotone contact flows the action variable $u$, rather than the energy, is the natural ordering coordinate; a testable consequence is that connecting orbits can be found at energy levels away from zero whenever the two Mane slices have different action values.","The global characteristics method may transfer to other evolution equations whose Lax-Oleinik-type semigroups converge to ordered limits, producing asymptotic orbits for non-conservative dynamics beyond contact Hamiltonian systems."],"forward_implications":["If $T^-_t \\varphi$ converges uniformly to $u_-$, then $\\omega(Z) \\subset \\widetilde{\\mathcal{N}}_{u_-}$ for some $Z$ in the 1-jet of $\\varphi$; convergence of the PDE alone forces a genuine orbit of the contact flow asymptotic to the Mane slice.","If both semigroups converge from the same initial datum and $v_+ < u_-$, a complete orbit connecting $\\widetilde{\\mathcal{N}}_{v_+}$ to $\\widetilde{\\mathcal{N}}_{u_-}$ exists, and no orbit can connect in the opposite order.","The connecting orbits built in (B2) are global action minimizers; in the model Hamiltonian $H = p^2 + \\sin x \\cdot u - \\tfrac14$ they can have nonzero energy and therefore are not semi-static.","For the model $H = F(x,p) + \\lambda(x)u$ with sign-changing $\\lambda$, global action-minimizing orbits are classified by the initial action value relative to the two extremal weak KAM solutions: initial data above the upper solution converge forward to $\\widetilde{\\mathcal{N}}_{\\bar u_-}$, below the lower solution escape in action, and between them give heteroclinic connections."],"supporting_citations":[{"why":"Supplies the variational representation of the solution semigroups and the action functions $T^\\pm_t \\varphi(x) = \\inf/\\sup h_{x_0,\\varphi(x_0)}(x,t)$ used throughout.","marker":"[37]"},{"why":"Provides the weak KAM solutions, calibrated curves, and the fixed-point characterization of the backward and forward semigroups that define the Mane slices.","marker":"[38]"},{"why":"The authors' first paper on monotone contact Hamiltonian systems, whose attractor picture and notation the present work extends to non-monotone cases.","marker":"[25]"},{"why":"Introduces the toy model $H = F(x,p) + \\lambda(x)u$ and the concrete example used in Section 6.3 to exhibit nonzero-energy heteroclinic orbits.","marker":"[26]"},{"why":"Supplies the characterization of limit sets of forward and backward action-minimizing orbits, including the time-free minimization property used in Proposition 5.6.","marker":"[42]"},{"why":"Gives the large-time convergence results for the model system that identify the maximal and minimal weak KAM solutions used in Theorem 6.8 and Theorem C.","marker":"[31]"},{"why":"Provides the implicit variational principle for contact Hamiltonian systems on which the action functions and the global characteristics method are built.","marker":"[36]"}],"fun_headline_variants":["Semigroup convergence forces asymptotic orbits in contact flows","No monotonicity needed: semigroup limits yield contact orbits","Semigroup limits create heteroclinic orbits in contact systems","Asymptotic orbits via Hamilton-Jacobi semigroup convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the key time-scale lemma in Theorem B treats the forward weak KAM limit $v_+$ as a fixed point of the backward solution semigroup when applying the exponential estimate; the assumptions only give that $v_+$ is fixed by the forward semigroup, so this premise is not guaranteed and the lower bound on $t_\\varepsilon$ depends on it.","fun_headline_variants_meta":{"raw":{"variants":["Semigroup convergence forces asymptotic orbits in contact flows","No monotonicity needed: semigroup limits yield contact orbits","Semigroup limits create heteroclinic orbits in contact systems","Asymptotic orbits via Hamilton-Jacobi semigroup convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000881,"raw_usage":{"total_tokens":3803,"prompt_tokens":938,"completion_tokens":2865,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":2807}},"tokens_in":554,"tokens_out":2865,"duration_ms":20455,"temperature":1.0,"reasoning_tokens":2807,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:32:00.531375+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a Hamiltonian satisfying (H1)-(H3) with a forward limit $v_+$ that is not a backward weak KAM solution, compute $T^-_t v_+$; if the inequality $\\|T^-_{t_\\varepsilon} \\varphi_\\varepsilon - v_+\\|_\\infty \\le e^{\\lambda t_\\varepsilon} \\varepsilon$ used in Lemma 4.4 fails for some small $\\varepsilon$, the proof of the heteroclinic construction loses its lower bound on $t_\\varepsilon$ and the construction in (B2) would need a different estimate.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the variational representation of the solution semigroups and the action functions $T^\\pm_t \\varphi(x) = \\inf/\\sup h_{x_0,\\varphi(x_0)}(x,t)$ used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the weak KAM solutions, calibrated curves, and the fixed-point characterization of the backward and forward semigroups that define the Mane slices."},{"cited_title":"Monotone systems","cited_arxiv_id":null,"evidence_quote":"The authors' first paper on monotone contact Hamiltonian systems, whose attractor picture and notation the present work extends to non-monotone cases."},{"cited_title":"Minimax The- ory Appl","cited_arxiv_id":null,"evidence_quote":"Introduces the toy model $H = F(x,p) + \\lambda(x)u$ and the concrete example used in Section 6.3 to exhibit nonzero-energy heteroclinic orbits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the characterization of limit sets of forward and backward action-minimizing orbits, including the time-free minimization property used in Proposition 5.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the large-time convergence results for the model system that identify the maximal and minimal weak KAM solutions used in Theorem 6.8 and Theorem C."},{"cited_title":"Nonlinearity 30 (2017), 492-515","cited_arxiv_id":null,"evidence_quote":"Provides the implicit variational principle for contact Hamiltonian systems on which the action functions and the global characteristics method are built."}],"review_version":1}