Dynamics of a hybrid vibro-impact oscillator: canonical formalism

The candidate solution reproduces the paper’s slow-flow invariant C(ν, ξ) and the LPT condition C=0 for zero initial conditions, and derives the same transition boundaries: fm(σ|ξ̃) in the linear and HVI regimes (matching Eq. (22)), the saddle-threshold fs(σ)=σ via the degenerate saddle at ξs=1/2 (Eq. (25)), and the unique coexistence point f*=σ* given in closed form (Eq. (26)). It also correctly interprets Type I as the ξ̃=1/2 case of Type II, with left/right branches governed by the maximum/saddle mechanisms (Fig. 14, Fig. 22). Differences are minor (e.g., writing fs as |σ| to emphasize f≥0, and asserting monotonicity when in the paper it is treated implicitly), so the methods and conclusions agree substantially with the paper’s analysis and results .