CYCLICITY OF RIGID CENTERS ON CENTER MANIFOLDS OF THREE-DIMENSIONAL SYSTEMS

The paper proves that system (10) has a rigid center and that quadratic perturbations can yield 13 small-amplitude limit cycles (Theorem 14), via rank computations on the first 13 focal coefficients and a higher-order transversality argument, exactly as the model outlines. The only discrepancy is that the model miscounts the immediate rank-based bound: with rank 9 on the linear parts of the first 13 coefficients, Theorem 2 gives k−1 = 8 cycles, and the trace perturbation adds one more to make 9, not 10; the paper is explicit about this (rank 9 → 9 cycles with trace) before proceeding to 13 using higher-order developments. This slip does not affect the final conclusion of 13 cycles, which is consistent with the paper’s result. See Theorem 14 and its proof details, including the rank statement and transversality verification .