{"id":"5f2c7bd5-a337-44ba-a8c3-5261162e3f40","arxiv_id":"1908.03789","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The authors construct a switched linear system with two self-excited double-scroll attractors and a numerically observed hidden attractor, proposing that breaking heteroclinic-like connections is the design mechanism.","lead":"This paper builds a piecewise linear dynamical system in which two ordinary chaotic attractors and one hidden attractor coexist. It argues that hidden attractors appear when the connecting paths between the ordinary attractors are broken.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Trapping-region proof fails at final comparison (Eqs. 145 vs 148) and rests on unverified 3D Assumption 4; hidden-attractor claim needs direct 3D validation.","rationale":"The reader's weakest_assumption (Assumption 4) is real: the 270-degree radius-growth estimate is validated only on the planar system (77), and the full 3D flow includes changes of equilibrium, the z1 contraction, and switching at SW12/SW23 that could alter radial expansion. I did not find that concern fatal by itself, because Section 5's inequalities are meant only as heuristics and the hidden attractor is also supported by direct simulation. However, a second independent problem strengthens the conditional verdict: the final comparison in the trapping proof is numerically in the wrong direction on the printed bounds. At the parameter values, the exact radii probably still satisfy the needed inequality once alpha is inserted, since the alpha=1 term reduces the q2 radius from 7*gamma/15 to 7*gamma/15 - alpha/3, so the claim may be repairable; but as written, the analytical argument has a gap. The single trajectory in Fig. 13b cannot by itself certify an attractor, since the same paper shows transients lasting thousands of time units for other gamma values. These points do not disprove the hidden attractor; they mean the conditional verdict is appropriate, with a direct 3D check required before acceptance.","tokens_in":27784,"tokens_out":10034,"duration_ms":97571,"concrete_test":"Test the trapping claim in the full 3D system (1),(2),(4),(23),(107) at a=0.2, b=5, c=-7, alpha=1, gamma=10: sample at least 10^4 initial conditions on a grid in R2 (in z^(2) coordinates), integrate with a tolerance-controlled solver to T=10^5, and record whether any trajectory reaches I1/I2 or converges to one of the two self-excited double-scroll attractors instead of remaining in R1 union R2. Separately, measure the maximal radius growth in z^(2) coordinates over the first 270-degree rotation in the full 3D flow and compare it with the Assumption 4 bound exp(3*a*pi/(4*b)); if the measured growth exceeds the 2D bound, the trapping argument fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 5's case for a hidden attractor turns on showing that the modified region R1 traps trajectories, so none can be routed through I1/I2 to the self-excited attractors. The final verification does not deliver this. Using Eqs. (140)-(148), the exaggerated escape radius is bounded by gamma*sqrt(193/225)*exp(a*258.2317*pi/(180*b)) <= 1.1091*gamma, while the minimum radius of the R1 segment pa1z2-pa2z2 is bounded below by gamma*sqrt((4/5)^2+(23/30)^2) approximately 1.1081*gamma. The paper writes '1.1092*gamma approximately 1.1081*gamma', but the required conclusion is 1.1091*gamma < 1.1081*gamma; the printed numbers point the other way. So the trapping argument has an actual numerical gap, separate from the acknowledged lack of a formal proof. Moreover the estimate depends on Assumption 4 (radius growth over 270 degrees in the full P1 union P2 flow equals the 2D rotation bound), which is validated only on the planar hysteresis model (77), not on (1),(2),(4),(23),(107). If the true 3D radial growth is larger, trajectories from R2 can reach I2 and fall into a self-excited attractor; the single trajectory in Fig. 13b, t in [50000,50100], is not enough to rule this out, especially given the long transients documented in Sec. 3.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a constructive approach for generating hidden attractors in piecewise-linear switched systems. Starting from a two-atom system with heteroclinic chaos (Section 2), the authors add two more equilibria (Section 3) to obtain two self-excited double-scroll attractors, and then modify the switching surface SW23 to x1=0 (Section 5) with the aim of blocking the routes to the self-excited attractors and thereby creating a hidden attractor. The main claim is that the system given by (1), (2), (4), (23), and (107) with a=0.2, b=5, c=-7, alpha=1, gamma=10 possesses a hidden attractor coexisting with two self-excited double-scroll attractors. The argument is a mix of analytic estimates for trapping regions R1 and R2, a set of parameter assumptions, and numerical simulations, with the final confirmation resting on the sentence 'Simulation experiments verify the conjecture on the emergence of the hidden attractor.'","tokens_in":28143,"tokens_out":3594,"duration_ms":39215,"significance":"If the central claim were rigorously established, the paper would offer a useful geometric design principle for multistable PWL systems with coexisting hidden and self-excited attractors, and it would connect hidden attractors to the breaking of large-scale heteroclinic-like orbits. The paper has notable strengths: an explicit and simple PWL construction, exact linear-flow formulas in Eq. (21), explicit parameter ranges in Assumptions 2 and 3, and numerical illustrations for several parameter regimes. However, the analytic trapping proof has a numerical gap at the final comparison, and the key Assumption 4 is validated only on a reduced planar model rather than on the original 3D system. These issues leave the principal hidden-attractor claim supported mainly by simulation, so the paper is currently more a numerical demonstration with a heuristic framework than a fully established result.","major_comments":[{"comment":"The trapping verification for the region R1 does not deliver the claimed conclusion. The escape-radius bound is estimated as 1.1091*gamma in Eq. (145), while the minimum radius of the segment pa1z2-pa2z2 is bounded below by 1.1081*gamma in Eq. (148). The paper then writes '1.1092*gamma approximately 1.1081*gamma', but the required conclusion is 1.1091*gamma < 1.1081*gamma, and the printed numerical bounds go in the opposite direction. Since this comparison is exactly what is supposed to show that trajectories starting in R1 cannot reach the subsets I1/I2 and hence cannot be routed to the self-excited attractors, the gap is load-bearing for the hidden-attractor claim.","section":"Section 5, Eqs. (143)-(148)"},{"comment":"Assumption 4 states that the increment in radius for a 270-degree rotation in the full 3D system is approximately the same as for a rotation around a single equilibrium in the 2D projection, but it is validated only on the reduced planar hysteresis system (77), not on the original system (1), (2), (4), (23), (107). All subsequent inequalities defining the trapping region R1 and the absence of escape routes depend on this approximation. If the true 3D radial growth is larger, trajectories from R2 could reach I2 and fall into a self-excited attractor. The paper should either prove a rigorous bound for the full 3D flow or provide direct 3D numerical verification for the parameter set used in Section 5.","section":"Section 4, Assumption 4, p. 24"},{"comment":"The derivation of the gamma interval Gamma mixes t<0 and t>0 analyses in a way that needs clarification. For the point pa on the heteroclinic orbit joining x_eq2 and x_eq3, the requirement is that the trajectory remains in P2 for all t<0, but the maximum of z_3^{(2)}(t) is then computed for t>0 using Eqs. (40)-(42). The same issue applies to the computation for pb. Since the interval (gamma_L, gamma_U) is used to assert the existence of six heteroclinic orbits and to distinguish the regimes in Figures 3 and 4, this inconsistency should be resolved before the heteroclinic-orbit count is relied upon.","section":"Section 3, Proposition 3.1, p. 13-14"},{"comment":"The evidence for the hidden attractor is a single trajectory segment for t in [50000, 50100] with initial condition x0=(0,0,0). Given the long transients documented earlier in the paper (about 350 a.u. for gamma=100 and about 3090 a.u. for gamma=1000), this one segment does not by itself rule out eventual convergence to one of the self-excited attractors. A direct longer-time integration, a basin-of-attraction computation, or an explicit invariant region in the full 3D system is needed to substantiate the claim that the attractor is genuinely hidden and persistent.","section":"Section 5, Fig. 13b"}],"minor_comments":[{"comment":"There are several typographical errors: 'seft-excited' should be 'self-excited', 'biestability' should be 'bistability', and 'two self-excited attractor' should be 'two self-excited attractors'. The abstract also states that the approach 'consists of the coexistence' of attractors, which should be rephrased for clarity.","section":"Abstract"},{"comment":"In Eq. (107) the switching surfaces are written with the union symbol, e.g., 'SW12 = cl(P1) union cl(P2)', whereas the earlier definition in Section 2 correctly uses the intersection cl(P1) intersect cl(P2). This appears to be a typographical error.","section":"Eq. (107)"},{"comment":"The proof of Proposition 2.2 is only the sentence 'A direct consequence of the proposition 2.1'. Since the proposition is used to justify the multiscroll construction in Section 3, a real proof or a clear statement that it is a conjecture would be more appropriate.","section":"Proposition 2.2"},{"comment":"The sentence '1.1092*gamma approximately 1.1081*gamma' is misleading: the left-hand side is not 1.1092 but 1.1091, and the two numbers are not approximately equal in the sense required for the strict inequality. This should be corrected together with the underlying bound.","section":"Section 5, sentence after Eq. (148)"}],"recommendation":"major_revision","confidential_remarks":"The paper is related to the authors' previous work, especially Ref. [24], and the novelty is mostly the modified switching surface and the geometric interpretation. The main risk is not the use of simulation per se but that the analytical trapping proof fails numerically and that the key Assumption 4 is only verified on a reduced model. If the authors can fix the inequality in Eqs. (145)-(148) or replace it with a valid bound, and provide direct 3D validation of Assumption 4 and of the hidden-attractor persistence, the paper could become publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: this is a real construction paper, not a proof paper. The new piecewise linear system with two coexisting self-excited double-scroll attractors and a numerically observed hidden attractor is a legitimate contribution to the hidden-attractor design literature, and the proposed mechanism—breaking heteroclinic-like orbits by relocating the switching surface—is plausible and worth taking seriously. But the supporting argument for the hidden attractor is not airtight, and the final numerical comparison in Section 5 has a small but real gap.\n\nThe paper does useful work before that. The heteroclinic-orbit constructions in Sections 2 and 3 are mostly sound, and the interval for gamma in Proposition 3.1 is derived explicitly from the linear flows. The step from six heteroclinic orbits to two self-excited attractors and then to a hidden attractor by moving SW23 is geometrically clear. The self-citations to the group's earlier work are appropriate; the new construction is not a repackaging.\n\nThe soft spots are in proportion. The biggest one is the final trapping-region verification. The authors need to show that trajectories in R1 cannot escape to the self-excited attractors. Their own inequalities give an exaggerated escape radius bounded by about 1.1091γ and a minimum trapping-segment radius of about 1.1081γ. They write these as approximately equal, but the required conclusion is 1.1091γ < 1.1081γ, and the printed numbers point the other way. That is a genuine numerical gap, separate from the acknowledged lack of a formal existence proof. On top of that, Assumption 4—that the 270-degree radius growth in the full 3D system matches the 2D rotation bound—is validated only on the planar hysteresis model, not on the actual system. If true 3D growth is larger, trajectories from R2 could reach I2 and fall into a self-excited attractor. The single trajectory in Fig. 13b, over t in [50000,50100], is not enough to rule out long transients of the kind documented earlier in the paper. Smaller issues: Proposition 2.2 is asserted without proof, and the gamma-interval derivation mixes t<0 and t>0 analyses without much comment.\n\nWho is this for? People working on hidden-attractor generation, multistability, and switched PWL systems. They will find the geometric design idea useful even if the proof stays incomplete. It deserves serious refereeing, but with a clear ask: fix the inequality or provide a direct 3D check of Assumption 4, and if neither works, frame the hidden attractor as a well-supported conjecture rather than a verified claim.","headline":"Genuine construction paper with a plausible hidden-attractor mechanism, but the trapping-region proof has a numerical gap and rests on an assumption not validated in 3D.","tokens_in":28621,"tokens_out":2139,"would_cite":true,"duration_ms":25462,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34A36","34C28","37C29","37D45"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a hidden attractor emerges in a piecewise linear switched system when the switching plane between two self-excited double-scroll attractors is moved so that trajectories that would be funneled into those attractors…","keywords":["hidden attractors","self-excited attractors","piecewise linear systems","multistability","heteroclinic orbits","switched systems","double-scroll attractors","chaos"],"falsifier":"Integrate the full system (1), (2), (4), (23) and (107) with $a=0.2$, $b=5$, $c=-7$, $\\alpha=1$, $\\gamma=10$ from $x_0=(0,0,0)$ for time well beyond the reported window; if the trajectory eventually converges to one of the two self-excited double-scroll attractors rather than continuing to cross $R_1$ and $R_2$, then the claimed hidden attractor is not an attractor. A second check: compare the radius growth over a 270-degree arc in the full system with the prediction of the reduced system (77); a mismatch larger than the inequalities allow would break the trapping proof.","tokens_in":27547,"feed_emoji":"🌀","tokens_out":7975,"duration_ms":69079,"temperature":0.7,"pith_summary":"The paper reports a geometric recipe for making hidden attractors appear in piecewise linear (PWL) switched systems. Starting from a system whose four equilibria support heteroclinic orbits and two self-excited double-scroll attractors, the authors move the middle switching surface to a new plane. This removes the intersection between the funneling regions and the subsets of initial conditions that lead to the self-excited attractors. For parameters $a=0.2$, $b=5$, $c=-7$, $\\alpha=1$ and $\\gamma=10$, they find a hidden attractor coexisting with the two self-excited attractors. The relevance is that hidden attractors, usually hard to locate, are here produced by a deliberate geometric modification of the switching surface.","feed_headline":"Moving one switching plane exposes a hidden attractor","feed_subtitle":"A piecewise-linear system with two double-scroll attractors also hosts a third, invisible one.","key_machinery":"The load-bearing object is the switching surface $SW_{23}$, moved from the plane $2x_1-x_3=0$ to the plane $x_1=0$ while the other two switching surfaces stay in place. On this surface the authors define two trapping regions $R_1$ and $R_2$ by four corner points each, chosen so that the funneling subsets $I_1$ and $I_2$ (the neighborhoods of the heteroclinic connection points $p_a$ and $p_c$) no longer intersect them. The proof that trajectories in $R_1$ reach $R_2$ or a self-excited attractor rests on comparing radii of rotation in the $z^{(2)}$ coordinate system, using the explicit linear diagonalization $A=QEQ^{-1}$ with eigenvalues $c$, $a\\pm ib$, and on the approximation that a 270-degree rotation in the full three-dimensional system grows in radius about as much as a rotation around a single equilibrium in the two-dimensional projection.","core_discovery":"The central claim is that the system defined by (1), (2), (4), (23) and (107) with $a=0.2$, $b=5$, $c=-7$, $\\alpha=1$ and $\\gamma=10$ possesses a hidden attractor that coexists with two self-excited double-scroll attractors. The hidden attractor arises because the modified switching surface $SW_{23}=\\{x\\in\\mathbb{R}^3: x_1=0\\}$ makes the regions $R_1$ and $R_2$ on that surface disjoint from the subsets $I_1$ and $I_2$ that would otherwise funnel trajectories into the self-excited attractors. The paper verifies this by constructing trapping regions on $SW_{23}$, proving that trajectories starting in $R_1$ reach $R_2$ or a self-excited attractor and vice versa, and showing numerically that for $\\gamma=10$ the claimed hidden attractor persists over long time windows.","pith_inferences":["A testable extension is to scan $\\gamma$ continuously in the same four-equilibrium system: the paper's inequalities predict that the hidden attractor should appear only above a threshold where the funneling intersections vanish, and disappear again if $\\gamma$ becomes so large that the rotational-radius approximation degrades.","A direct numerical check of radius growth in the full three-dimensional system versus the reduced two-dimensional system (77) would settle whether Assumption 4 holds beyond the tested parameter range; if it fails, $R_1$ may leak trajectories into the self-excited basins.","The same 'break the heteroclinic-like orbit' idea could be applied to chains of more than four equilibria, potentially producing several hidden attractors arranged between multiple self-excited attractors.","Because the hidden attractor's basin of attraction contains no equilibria, the geometric construction may be adaptable to systems without equilibria, which the paper notes as a feasible direction."],"forward_implications":["For the parameter set $a=0.2$, $b=5$, $c=-7$, $\\alpha=1$, $\\gamma=10$, the system has a hidden attractor plus two self-excited double-scroll attractors, so multistability occurs without any equilibria inside the hidden attractor's basin.","Moving only the middle switching surface to $x_1=0$ preserves the two heteroclinic loops and therefore preserves the two self-excited attractors while enabling the hidden one.","For sufficiently large $\\gamma$, the intersections $N(p_a)\\cap R_1$ and $N(p_c)\\cap R_2$ are empty, so no trajectory is forced into a self-excited attractor; the paper's inequalities give explicit parameter conditions of this kind.","Trajectories starting in $R_1$ must reach $R_2$ or a self-excited attractor, and symmetrically for $R_2$, so any hidden attractor must pass through both regions.","The construction indicates a geometric route to multistable PWL systems with coexistence of hidden and self-excited attractors, and possibly to systems with no equilibria at all."],"supporting_citations":[{"why":"Supplies the definition of hidden attractors and the analytical-numerical localization algorithm the paper builds on.","marker":"[2]"},{"why":"Reports a multistable PWL system with two self-excited double-scroll attractors and one double-scroll hidden attractor, the starting point of this work.","marker":"[24]"},{"why":"Provides the survey definition of hidden versus self-excited attractors used throughout.","marker":"[3]"},{"why":"Documents multistability scenarios in unstable dissipative systems that motivate the coexistence question.","marker":"[1]"},{"why":"Defines multiscroll attractors in switching systems, used to classify the quad-scroll and double-scroll attractors.","marker":"[25]"},{"why":"Shows generation of multiscroll hidden attractors in PWL systems without equilibria, the design context for the present construction.","marker":"[20]"},{"why":"Establishes PWL systems without equilibria with chaotic attractors, background for the no-equilibrium direction.","marker":"[9]"}],"fun_headline_variants":["Hidden attractor emerges from broken heteroclinic orbit","Ruptured orbit reveals invisible attractor in switched system","Coexisting attractors: hidden and self-excited in PWL system","Breaking heteroclinic orbits spawn hidden attractors","Hidden attractor lurks beside two self-excited scrolls"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The trapping-region construction rests on the approximation that a 270-degree rotation in the full three-dimensional system grows in radius about the same as a rotation around a single equilibrium in a two-dimensional projection, checked only on a reduced two-dimensional system with a hysteresis function, together with the parameter regime $b/a\\ge 25$, $2\\ge |c/b|\\ge 7/5$, $\\gamma/\\alpha\\ge 10$.","fun_headline_variants_meta":{"raw":{"variants":["Hidden attractor emerges from broken heteroclinic orbit","Ruptured orbit reveals invisible attractor in switched system","Coexisting attractors: hidden and self-excited in PWL system","Breaking heteroclinic orbits spawn hidden attractors","Hidden attractor lurks beside two self-excited scrolls"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000711,"raw_usage":{"total_tokens":3202,"prompt_tokens":952,"completion_tokens":2250,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":2166}},"tokens_in":568,"tokens_out":2250,"duration_ms":15739,"temperature":1.0,"reasoning_tokens":2166,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:02:10.691811+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Integrate the full system (1), (2), (4), (23) and (107) with $a=0.2$, $b=5$, $c=-7$, $\\alpha=1$, $\\gamma=10$ from $x_0=(0,0,0)$ for time well beyond the reported window; if the trajectory eventually converges to one of the two self-excited double-scroll attractors rather than continuing to cross $R_1$ and $R_2$, then the claimed hidden attractor is not an attractor. A second check: compare the radius growth over a 270-degree arc in the full system with the prediction of the reduced system (77); a mismatch larger than the inequalities allow would break the trapping proof.","supporting_citations":[{"cited_title":"Localization of hidden Chua’s attractors","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of hidden attractors and the analytical-numerical localization algorithm the paper builds on."},{"cited_title":"Onta˜ n´ on-Garc´ ıa and E","cited_arxiv_id":null,"evidence_quote":"Reports a multistable PWL system with two self-excited double-scroll attractors and one double-scroll hidden attractor, the starting point of this work."},{"cited_title":"Kuznetsov, Gennady A","cited_arxiv_id":null,"evidence_quote":"Provides the survey definition of hidden versus self-excited attractors used throughout."},{"cited_title":"Anzo-Hern´ andez, Hector Eduardo Gilardi-Vel´ azquez, and E","cited_arxiv_id":null,"evidence_quote":"Documents multistability scenarios in unstable dissipative systems that motivate the coexistence question."},{"cited_title":"Sol´ ıs-Perales, and Ricardo Femat","cited_arxiv_id":null,"evidence_quote":"Defines multiscroll attractors in switching systems, used to classify the quad-scroll and double-scroll attractors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows generation of multiscroll hidden attractors in PWL systems without equilibria, the design context for the present construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes PWL systems without equilibria with chaotic attractors, background for the no-equilibrium direction."}],"review_version":1}