Chebyshev-Certified Multi-Cycle Neural ODE

Use a one-dimensional or low-dimensional latent neural ODE containing an Abel-type oscillator branch. Choose f_k(t) as Chebyshev polynomials T_k(t/T), or another basis whose Chebyshev property is checked numerically on the relevant interval. Keep the observation map as a standard MLP, but constrain the latent drift coefficients to the selected basis. For each minibatch, sample an ordered amplitude grid rho_1 through rho_m, integrate each trajectory from -T to T, and compute R(rho_i) = x(T;rho_i) - rho_i. Add a root-placement loss at desired cycle amplitudes, plus a sign-margin loss that encourages opposite signs of the residual at the endpoints of intervals intended to contain cycles. Estimate the Melnikov function by evaluating the residual for several small perturbation values epsilon and fitting its leading power epsilon^r, or use automatic differentiation with respect to epsilon. Periodically bracket roots with bisection and refine them with Newton iteration. Reject roots whose estimated residual derivative has magnitude below a threshold gamma. The paper provides the zero-count and Melnikov mechanisms; the implementation must verify the basis property and the asymptotic expansion empirically. First experiment: compare an unconstrained latent neural ODE, a Fourier-basis model, and a four-function Chebyshev model on synthetic Abel trajectories with two known cycles. The predicted signature is a capacity boundary: for dM = 4, no more than three simple Melnikov roots should exist. A requested fourth root should disappear unless the model leaves the certified span. Also test that R(rho,epsilon) divided by epsilon^r converges linearly in epsilon to M_r(rho).