Switch System

The paper correctly writes the two subsystems in cylindrical coordinates, identifies the common periodic orbit Gamma = {r=1, z=0}, and shows the averaged outer dynamics have transverse Jacobian diag(-4, -4), hence a stable periodic orbit for the averaged system (equations (15)–(16) and discussion) . It also formulates a general upper-triangular family (equations (17)–(18)) and notes that b_i does not affect the transverse eigenvalues, so average stability depends only on the averages of a_i and c_i . However, the paper’s claim that fast switching yields stability (and slow switching yields instability) is presented only by heuristic averaging and simulation (Section 3; equations (7)–(10) and the figures), with no rigorous closeness-to-average or Poincaré multiplier analysis, and with an overstatement that slow switching implies instability based on a single trajectory . In contrast, the model computes the exact monodromy (Poincaré) matrix of the switched system linearized along Gamma in the outer region: for each half-dwell, the (rho,z) dynamics are upper triangular with A1 = [[-10, -1],[0, 2]] and A2 = [[2, 1],[0, -10]] (paper’s (12) and (14)), whose exponentials multiply over one full switch to P(tau) = e^{-4 tau} I_2, giving transverse multipliers e^{-4 tau} < 1 for all tau > 0. This proves local exponential orbital stability of Gamma for the switched system directly (no averaging assumption) and generalizes cleanly to the family {S_i}, where the monodromy remains upper triangular with eigenvalues exp(a_bar tau) and exp(c_bar tau), independent of the b_i . The paper’s core claim (stability) is therefore under-justified, while the model’s solution supplies a correct, stronger proof; the paper’s added assertion that slow switching is “unstable” is not established and is likely false in the local (Floquet) sense, even if large excursions can occur for nonlocal initial conditions (the paper’s figure) . The paper’s concluding narrative about novelty and boundedness lacks proof as well .