Generalize the vanishing of even couplings in first-order phase reduction beyond antisymmetric oscillators

Determine whether the result that the symmetric (even under the transformation X → −X) component of the coupling function makes no contribution at first order in the averaged phase reduction persists for oscillators whose velocity fields are not antisymmetric (i.e., do not satisfy F(X) = −F(−X)) and for systems exhibiting chaotic dynamics. Specifically, ascertain conditions under which even-order coupling terms vanish or contribute in the O(ε) phase-reduced equations for higher-order (many-body) interactions when individual unit dynamics lack antisymmetry or are chaotic.

Background

The paper proves that for limit-cycle oscillators whose vector fields are antisymmetric (F(X) = −F(−X)), the symmetric part of the interaction—such as even polynomial couplings—does not contribute to the first-order (O(ε)) phase-reduced model after averaging. This result relies on the Fourier structure of the limit cycle and phase sensitivity function, which contain only odd harmonics under the antisymmetry, causing symmetric coupling terms to average out.

However, many oscillators do not exhibit this antisymmetry, and some systems are chaotic rather than limit-cyclic. Whether an analogous cancellation of even coupling terms occurs in first-order phase reduction for these broader classes remains unresolved. Clarifying this would inform when even-order many-body couplings can be ignored in phase reductions and when they may materially affect the dynamics.