{"id":"8b771738-3602-4812-bba8-8d8ed65a8d61","arxiv_id":"2411.13140","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper introduces computable indicators R_K and I_K that measure exponential convergence rate and ultimate error ball for MIMO-PI controllers, and uses them to optimize controller gains.","lead":"This paper proposes a quantitative robustness indicator for multi-input multi-output proportional-integral (MIMO-PI) controllers on nonlinear systems with disturbances, and an optimization method to tune the controller gains. A generalist might care because it offers a systematic, model-based recipe for tuning a widely used controller class on complex systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2 is falsified by a scalar counterexample: local Hurwitz stability of A_K(0) does not imply the claimed global exponential convergence, so the indicator R_K, I_K does not quantify what is claimed.","rationale":"The reader's weakest_assumption correctly identified the local-to-global leap in Theorem 2: the Lyapunov conditions are verified only at Omega = {0}, yet Remark 2 asserts global exponential stabilization. My stress-test agrees and goes further: the claim is not just under-proved but false. The scalar counterexample satisfies every stated assumption of Theorem 2 -- smoothness, differentiability at 0, a stable A_K(0) -- while the trajectory from x(0)=2 diverges because the cubic nonlinearity dominates far from the origin. This directly invalidates the claimed guarantee that the state converges exponentially to the ball B(0, 2 L_d L_f I_K) for arbitrary initial conditions. The robust convergence indicator R_K and I_K, computed from the local linearization A_K(0), therefore does not provide the advertised robustness certificate for the nonlinear system. The paper's EVP formulation and GA optimization are interesting engineering tools, and the simulations show plausible behavior for the aircraft model, but the theoretical foundation stated in Theorem 2 is not sound. A revision could rescue a local or semi-global version by adding assumptions that the Lyapunov inequality holds on a forward-invariant region containing the initial conditions, or by imposing global one-sided Lipschitz conditions; however, as written the central claim is false, not merely incomplete. I recommend REJECT rather than CONDITIONAL because the universal global statement is contradicted by an elementary example, and the indicator's validity is the paper's main contribution.","tokens_in":19122,"tokens_out":11238,"duration_ms":115256,"concrete_test":"Simulate the scalar example: plant x_dot = -2x + x^3 - u, u = x + 4z, z_dot = x, with x(0)=2, z(0)=0, d=0. Verify that A_K(0) = [[-3,-4],[1,0]] is Hurwitz, so Theorem 2 conditions hold. If the simulated x(t) grows without bound (e.g., exceeds 10 within t <= 1) rather than converging exponentially to 0, the global convergence claim is falsified. An analytic check: for large x the term x^3 dominates the linear and integral feedback, so the origin cannot be globally asymptotically stable.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim is that if A_K(0) is Hurwitz, the nonlinear closed loop converges globally and exponentially to B(0, 2 L_d L_f I_K). The proof applies Theorem 1 with Omega = {0}, but Theorem 1 requires the Jacobian bound and the Lyapunov matrix inequality to hold on the region Omega traversed by trajectories. At Omega = {0} the inequality is checked at a single point, so the Lyapunov derivative is uncontrolled away from the origin; local linear stability cannot imply global convergence. This is not merely a gap: the statement is false. Consider the scalar plant x_dot = -2x + x^3 - u with PI control u = x + 4z, z_dot = x, and d = 0. Then A_K(0) = [[-3, -4], [1, 0]], whose eigenvalues have real part -1.5, so condition (29) holds; f is smooth and ||d_dot|| <= L_d for any L_d > 0. Theorem 2 predicts exponential convergence to 0 for every initial state, including x(0)=2, z(0)=0. But the closed loop is x_dot = -3x + x^3 - 4z; the cubic term dominates and x(t) diverges, contradicting the theorem. A secondary inconsistency: the constant L_f used in Eq. (39) and in I_K is taken from the original plant Jacobian, whereas Theorem 1 applied to the augmented s-system requires the Jacobian norm of A_K(0); as written the bound is dimensionally and conceptually mismatched.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a quantitative robustness indicator for MIMO-PI controllers applied to general perturbed nonlinear systems. It first states a Lyapunov-based theorem (Theorem 1) on exponential convergence of trajectories to a 'global random attractor', then derives a velocity-form augmented system for the closed loop with a MIMO-PI controller and defines two indicators: R_K, the exponential convergence rate, and I_K, the radius of the ultimate attractor. The indicators are computed through an eigenvalue problem (EVP), and an optimization model is proposed to tune the controller gains subject to input-magnitude and input-rate constraints. The claims are supported by simulations on a Duffing oscillator and on a fixed-wing kinematic model, comparing controllers with different R_K and I_K values.","tokens_in":19469,"tokens_out":13365,"duration_ms":138864,"significance":"If the global exponential-convergence claim were correct, the paper would offer a practical, optimization-ready tuning metric for MIMO-PI controllers, going beyond SISO/LTI tuning rules. The algebraic derivation of R_K and I_K from the Lyapunov equation and the EVP formulation is a strength, and the comparative simulations show a plausible empirical trend in which controllers with larger R_K have better transient error metrics. However, the central theoretical claim is not justified and is in fact false: local Hurwitz stability of the augmented linearization does not imply global exponential convergence to the claimed attractor. The paper also contains a dimensional inconsistency between the attractor bound and the definition of I_K. As a result, the proposed indicator cannot currently be accepted as a quantitative robustness guarantee for the nonlinear system.","major_comments":[{"comment":"Theorem 1 concludes 'for any initial value x0' convergence to the global random attractor, but conditions 1) and 2) are only assumed on a set Omega, and the proof does not show that the trajectory x(t) remains in Omega or that Omega is invariant. The proof uses L_f(Omega) and epsilon(Omega) at the current state, which is legitimate only while x(t) is in Omega. Consequently, the global conclusion does not follow from a local verification at Omega={0}. This is the same local-to-global logical gap that later invalidates Theorem 2.","section":"Section 2.2, Theorem 1"},{"comment":"Theorem 2 is false as stated. The proof applies Theorem 1 with Omega={0}, but verifying the Jacobian bound and the Lyapunov inequality at the single point s=0 does not control the Lyapunov derivative away from the origin. Equality (29), i.e., Re[A_K(0)]<0, therefore does not imply the exponential convergence claimed in Eq. (30). A counterexample within the theorem's assumptions is the scalar plant xdot=x^3-u with the PI law u=x+4z, zdot=x. Then A_K(0)=[[-1,-4],[1,0]] is Hurwitz (eigenvalues -0.5 +/- i*sqrt(15)/2), so Eq. (29) holds; taking d=0 and x(0)=2, z(0)=0 gives xdot(0)=6, and the cubic term drives the trajectory away from the origin rather than to the claimed attractor. Since Theorem 2 is the basis for the indicators R_K and I_K, this is a load-bearing error.","section":"Section 4.1, Theorem 2 and Remark 2"},{"comment":"The Lipschitz constant used in the attractor bound is inconsistent with Theorem 1. Applied to the augmented system in Eq. (35), Theorem 1 requires the Jacobian norm of f_s at the origin, namely ||A_K(0)||_2, but Eq. (39) uses the plant constant L_f from Eq. (27). Correspondingly, the definition of I_K in Eq. (31) uses ||A_K(0)^{-1}||_2 without the factor ||A_K(0)||_2. The bound in Eq. (30) should therefore contain the product ||A_K(0)||_2 ||A_K(0)^{-1}||_2 (together with the Lyapunov condition number), not L_f ||A_K(0)^{-1}||_2. This changes the numerical value of the claimed attractor radius and the interpretation of I_K.","section":"Section 4.1, Eq. (31) and Eq. (39)-(40)"},{"comment":"The optimization model enforces the input magnitude and input-rate constraints only at the initial time through Eq. (55). It does not guarantee u_min <= u(t) <= u_max or dot u_min <= dot u(t) <= dot u_max for all t along the trajectory. The claim that the optimized controller satisfies the input constraints is therefore not established by the optimization; the simulations in Figure 4 show the computed gains respect the constraints in the tested cases, but this is not a certification.","section":"Section 4.4, Eq. (54)-(55)"}],"minor_comments":[{"comment":"The assumption that an exact inverse observer h^{-1} exists and provides the exact state for every output is very strong; the controller in Eq. (25) is state feedback. The authors should explicitly state this limitation, since the simulations use full-state feedback.","section":"Section 3, after Eq. (24)"},{"comment":"The phrase 'eigenvalue 0 corresponds to the single characteristic factor' in Lemma 1 is unclear; for the purposes of this paper it suffices to state that A is Hurwitz.","section":"Section 2.2, Lemma 1"},{"comment":"The genetic-algorithm hyperparameters (population size, generation count, crossover and mutation rates) and the chosen value of I^* are not reported, so the optimization results in Eq. (72)-(73) are not reproducible.","section":"Section 4.4, Eq. (54)"},{"comment":"The role of the parameter epsilon in Table 2 and in the perturbation Delta K = -epsilon(I_p, I_i) should be defined in the text; the reader currently has to infer it from the surrounding discussion.","section":"Table 2 and Section 5.2.2"},{"comment":"The term 'global random attractor' is used informally: Definition 2 defines a deterministic invariant set, while Definition 3 refers to a stochastic setting, but Theorem 1 concerns a deterministic bounded disturbance. The terminology should be made precise so the reader knows whether the result is a deterministic bound or a stochastic attractor statement.","section":"Definitions 2-3"},{"comment":"References [20] and [23] are the same article (Cheng Zhao and Lei Guo, 'PID controller design for second order nonlinear uncertain systems'); the duplicate citation should be removed.","section":"References"}],"recommendation":"reject","confidential_remarks":"The paper's central theorem is falsified by a simple scalar counterexample within the theorem's own assumptions. A local reformulation with a certified region of attraction would require substantial changes to the indicator claims and to the optimization model; as it stands, the main contribution is not established. The empirical correlation between R_K and transient error metrics is interesting, but it does not rescue the theoretical claim. I recommend rejection unless the authors can provide a genuinely global or invariant-region-based Lyapunov analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe paper's central theorem (Theorem 2) is not true as stated, and the counterexample in the stress-test note lands. Take the scalar plant x_dot = -2x + x^3 - u with PI control u = x + 4z, z_dot = x. The augmented Jacobian A_K(0) has eigenvalues with real part -1.5, so condition (29) holds, but the closed loop with x(0)=2, z(0)=0 diverges because the cubic term dominates. So local Hurwitz stability of A_K(0) does not imply the claimed global exponential convergence, and the indicator R_K/I_K does not quantify what the theorem promises.\n\nThat said, the paper is not without value. The idea of mapping MIMO-PI robustness to the spectrum of the velocity-form augmented matrix is new, and the EVP formulation for R_K, the relation to I_K, and the GA-based optimization model (54) are concrete and reproducible. The simulations on the aircraft kinematic model behave sensibly, and the local Lyapunov algebra in the proof is correct. For a locally exponentially stable linearization with mild nonlinearities, the proposed indicators could be a useful heuristic for gain tuning.\n\nThe soft spots are serious. The proof applies Theorem 1 with Omega={0}, but Theorem 1 explicitly needs the Lyapunov inequality and Jacobian bound to hold on the region traversed by trajectories. Checking at a single point says nothing about the rest of the state space; Remark 1's claim that Omega={0} suffices is simply false. There is also a technical mismatch: L_f in Eq. (39) comes from the original plant Jacobian, whereas Theorem 1 applied to the augmented s-system needs the Jacobian norm of A_K(0). The input constraints are enforced only at t0, so the optimization affords no guarantee over time, and the comparisons only use perturbations of the single optimized controller, not independent baselines.\n\nWho this paper is for: researchers working on practical PID/PI tuning for nonlinear MIMO systems would get ideas from the R_K/I_K machinery, but anyone citing Theorem 2 as a guarantee would be misled. The paper needs major revision—either restrict the claim to a genuine local result with a Lyapunov function on a neighborhood, or add global sector or quadratic conditions that make the convergence actual. I would send it to referees, because the idea is novel and the counterexample is instructive, but I would not accept it in anything like this form. Clear recommendation: major revision, and make sure the reviewers have the counterexample in hand.","headline":"The core theorem overreaches: local Hurwitz stability of A_K(0) does not imply global exponential convergence, and the counterexample is valid—still, the R_K/I_K tuning machinery is a novel and potentially useful heuristic.","tokens_in":19964,"tokens_out":3733,"would_cite":false,"duration_ms":34865,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93D15","93D21","93C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A computable robustness indicator predicts when MIMO-PI controllers force errors into an exponentially shrinking ball, even with bounded disturbances.","keywords":["MIMO-PI control","robust stabilization","exponential convergence","global random attractor","velocity form","eigenvalue problem","disturbed nonlinear systems","gain tuning"],"falsifier":"Run a nonlinear simulation of the closed-loop system with a fixed Hurwitz $A_K(0)$, a nonzero initial condition far from the origin, and a bounded random perturbation $\\dot d$ with $\\|\\dot d\\| \\le L_d$. If the error norm $\\|s(t)\\|_2$ either exceeds the predicted radius $2 L_d L_f I_K$ after a long time or decays slower than the predicted exponential rate $R_K$ (e.g., by fitting an exponential envelope to $\\|s(t)\\|$), the theorem's global claim is false. To be conclusive, test initial conditions at several distances and disturbance amplitudes $L_d$; the theory says convergence to the ball must occur from any initial state, so a single counterexample trajectory suffices.","tokens_in":18919,"feed_emoji":"🎛️","tokens_out":2248,"duration_ms":23065,"temperature":0.7,"pith_summary":"This paper tries to answer a practical question: given a multi-input multi-output nonlinear system with bounded disturbances, how can you tell whether a proportional-integral controller will stabilize it, and how fast and how tightly? The authors claim that everything hinges on one matrix: the Jacobian of the velocity-form system at the origin, $A_K(0)$, which is built from the plant's Jacobians and the controller gains $K_P, K_I$. If that matrix is Hurwitz, they assert the error state $s(t) = (\\dot x(t), x(t))$ converges exponentially to a ball of radius $2 L_d L_f I_K$ at rate $R_K$, where $R_K$ and $I_K$ are explicit formulas in the gains and a Lyapunov matrix $P$. They then formulate $R_K$ and $I_K$ as the solution of a semidefinite program, and use them as objectives to tune the gains under input constraints. If the claim holds, it gives engineers a parameter-free, optimization-ready robustness certificate for a standard control architecture in a nonlinear setting.","feed_headline":"One matrix test certifies MIMO-PI robustness","feed_subtitle":"If the velocity-form Jacobian at the origin is stable, errors converge exponentially to a computable ball","key_machinery":"The load-bearing object is the augmented matrix $A_K(0)$ from the velocity-form linearization: $A_K(0) = \\begin{pmatrix} \\frac{\\partial f}{\\partial x}(0) + \\frac{\\partial f}{\\partial u}(0) K_P & \\frac{\\partial f}{\\partial u}(0) K_I \\\\ I_n & 0 \\end{pmatrix}$. This matrix captures the effect of both proportional and integral gains on the coupled dynamics of $(\\dot x, x)$. The argument rests on showing that if this matrix is Hurwitz and the plant is $\\beta_f$-smooth with $\\|\\dot d\\| \\le L_d$, then the Lyapunov function $V(s) = f_s(s)^T P f_s(s)$ yields a dissipation inequality that bounds the trajectory in terms of $R_K$ and $I_K$.","core_discovery":"The central discovery is that the robustness of a MIMO-PI controller for a perturbed nonlinear system can be quantified by two scalar indicators extracted from the origin of the velocity-form augmented system. Specifically, for $\\dot x = f(x,u) + d$ with $\\|\\dot d\\| \\le L_d$, the controller $u = K_P x + K_I \\int_0^t x\\,dt$ leads to the augmented dynamics $\\dot s = A_K(s)s + d_s$ where $s = (\\dot x, x)$ and $A_K(0)$ is the Jacobian of the velocity form at the origin. The paper's Theorem 2 states that if $\\mathrm{Re}[A_K(0)] < 0$, then there exists a positive-definite $P$ and $\\varepsilon(0)>0$ such that the state converges exponentially to the ball $B(0, 2 L_d L_f I_K)$ with rate $R_K = \\varepsilon(0)/\\lambda_{\\max}(P)$ and $I_K = \\|A_K(0)^{-1}\\|_2 \\lambda_{\\max}^2(P)/(\\varepsilon(0)\\lambda_{\\min}(P))$. These indicators are then computed by solving a convex eigenvalue problem (EVP) for the scaled Lyapunov matrix $Q_K$, and the gains are optimized by maximizing $R_K$ subject to $I_K \\le I^*$ and input constraints. The paper validates the theory on a Duffing oscillator and on a fixed-wing aircraft kinematic model under sinusoidal disturbances, showing that larger $R_K$ correlates with lower ITAE, overshoot, and post-stabilization variance.","pith_inferences":["A natural testable extension is to check whether the same $R_K$/$I_K$ framework applies when the disturbance bound is on $\\|d\\|$ rather than $\\|\\dot d\\|$; Theorem 1 uses $L_d$ for $d$, while Theorem 2 requires $\\|\\dot d\\| \\le L_d$, which may be overly restrictive for constant or step disturbances.","Because the proof only verifies the Lyapunov inequality at the origin, a more careful region-of-attraction analysis would be needed to guarantee that trajectories starting far away actually stay within the region where $A_K(s)$ remains stable; without that, the 'global' statement is only local in practice.","The indicator $I_K$ depends on $\\|A_K(0)^{-1}\\|_2$, so the ball radius grows when $A_K(0)$ is nearly singular. This suggests a design trade-off that the paper does not fully exploit: explicitly shaping the eigenvalues of $A_K(0)$ (e.g., via LMI-based pole placement) could directly reduce $I_K$ for a fixed $R_K$.","The experiments vary $K$ by a single scalar perturbation $\\varepsilon$; a more comprehensive validation would sample the gain space randomly around $K^*$ and test whether $R_K$ and $I_K$ remain monotone predictors of the performance metrics, which is exactly what the indicators claim to do."],"forward_implications":["If $R_K$ and $I_K$ are valid predictors, gain tuning for MIMO-PI controllers on disturbed nonlinear plants becomes a convex optimization problem in $Q_K$ plus a scalar search, rather than a black-box heuristic search.","The paper's Theorem 2 implies that any gains making $A_K(0)$ Hurwitz guarantee exponential convergence to a bounded region, so robustness can be certified without simulating the full nonlinear closed-loop system.","The optimization model in Eq. (54) shows that the maximal convergence rate can be pursued while enforcing actuator limits at the initial instant, which is directly applicable to flight control and similar input-constrained problems.","The indicators $R_K$ and $I_K$ provide a quantitative trade-off: larger rate $R_K$ does not automatically yield a smaller ultimate ball $I_K$; the relationship involves the condition number of $Q_K$, so tuning should consider both metrics.","The Duffing and aircraft experiments suggest that the indicators can rank controller gains by transient and steady-state error in a prescriptive way, making them usable as surrogate objectives in auto-tuning tools."],"supporting_citations":[{"why":"Provides the prior sufficient conditions for exponential stabilization with PID gains tied to a Lipschitz constant, which this paper aims to supersede with a computable indicator.","marker":"[10]"},{"why":"Extended PID with high-order error derivatives; the paper positions itself as addressing the robustness indicator gap left by such extended PID methods.","marker":"[11]"},{"why":"Gives PID parameter ranges for second-order nonlinear systems; used as a baseline for the robustness-condition literature.","marker":"[23]"},{"why":"Supplies Lemma 1 (matrix stability and Lyapunov equation), the basis for constructing $P$ in the proof of Theorem 2.","marker":"[24]"},{"why":"Weyl's inequality, used in Lemma 3 to bound the difference of eigenvalues of Jacobians, which is part of the $\\beta_f$-smoothness machinery.","marker":"[25]"},{"why":"Velocity-form linearization for nonlinear systems; this is the conceptual basis for writing the augmented $A_K(0)$ in the velocity coordinates.","marker":"[26]"},{"why":"Provides the fixed-wing aircraft kinematic model used in the simulation validation of the MIMO-PI controller.","marker":"[28]"},{"why":"Supplies the Duffing oscillator model used to validate Theorem 1 in the numerical experiments.","marker":"[27]"}],"fun_headline_variants":["Jacobian test certifies MIMO-PI exponential robustness","Scalar robustness indicator guides MIMO-PI gain selection","Velocity-form Jacobian stability yields guaranteed convergence","MIMO-PI: two scalars predict robust exponential error decay"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main load-bearing premise is that the stability of the linearized augmented matrix $A_K(0)$ at the origin guarantees the claimed exponential convergence of the nonlinear closed-loop system for all initial conditions, even though the Lyapunov inequality is only verified at the single point $\\Omega = \\{0\\}$.","fun_headline_variants_meta":{"raw":{"variants":["Jacobian test certifies MIMO-PI exponential robustness","Scalar robustness indicator guides MIMO-PI gain selection","Velocity-form Jacobian stability yields guaranteed convergence","MIMO-PI: two scalars predict robust exponential error decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000499,"raw_usage":{"total_tokens":2488,"prompt_tokens":1036,"completion_tokens":1452,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":652,"completion_tokens_details":{"reasoning_tokens":1386}},"tokens_in":652,"tokens_out":1452,"duration_ms":11269,"temperature":1.0,"reasoning_tokens":1386,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:47:58.650972+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a nonlinear simulation of the closed-loop system with a fixed Hurwitz $A_K(0)$, a nonzero initial condition far from the origin, and a bounded random perturbation $\\dot d$ with $\\|\\dot d\\| \\le L_d$. If the error norm $\\|s(t)\\|_2$ either exceeds the predicted radius $2 L_d L_f I_K$ after a long time or decays slower than the predicted exponential rate $R_K$ (e.g., by fitting an exponential envelope to $\\|s(t)\\|$), the theorem's global claim is false. To be conclusive, test initial conditions at several distances and disturbance amplitudes $L_d$; the theory says convergence to the ball must occur from any initial state, so a single counterexample trajectory suffices.","supporting_citations":[{"cited_title":"Towards a theoretical foundation of pid control for uncertain nonlinear systems","cited_arxiv_id":null,"evidence_quote":"Provides the prior sufficient conditions for exponential stabilization with PID gains tied to a Lipschitz constant, which this paper aims to supersede with a computable indicator."},{"cited_title":"Control of nonlinear uncertain systems by extendedpid","cited_arxiv_id":null,"evidence_quote":"Extended PID with high-order error derivatives; the paper positions itself as addressing the robustness indicator gap left by such extended PID methods."},{"cited_title":"Pid controller design for second order nonlinear uncertain systems","cited_arxiv_id":null,"evidence_quote":"Gives PID parameter ranges for second-order nonlinear systems; used as a baseline for the robustness-condition literature."},{"cited_title":"Gain-scheduled and nonlinear systems: dynamicanalysisbyvelocity-basedlinearizationfamilies","cited_arxiv_id":null,"evidence_quote":"Velocity-form linearization for nonlinear systems; this is the conceptual basis for writing the augmented $A_K(0)$ in the velocity coordinates."},{"cited_title":"Predictive path-following controlforfixed-winguavsusingtheqlmpcframeworkinthepresence ofwinddisturbances.In AIAASCITECH2024FORUM .AIAA,2024","cited_arxiv_id":null,"evidence_quote":"Provides the fixed-wing aircraft kinematic model used in the simulation validation of the MIMO-PI controller."},{"cited_title":"The duffing oscillator equation and its applications in physics.MATHEMATICAL PROBLEMS IN ENGINEERING, 2021, NOV 30 2021","cited_arxiv_id":null,"evidence_quote":"Supplies the Duffing oscillator model used to validate Theorem 1 in the numerical experiments."}],"review_version":1}