Extended invariant cones as Nonlinear Normal Modes of inhomogeneous piecewise linear systems

The paper’s modified invariant cone (MIC) formulation introduces the augmented homogeneous dynamics v̇ = ñ(α)v with v = [x; vδ], establishes the constrained MIC system F(x0, t−, t+, α, a) = 0, and shows how its solutions correspond to periodic motions that cross the switching hyperplane once per subinterval; it also derives the monodromy Φ = e^{t+ Ã+(α)} e^{t− Ã−(α)} (with identity saltation due to continuity) and extends the construction to the forced case via a further augmentation by (zc, zs) and a phase φ0, yielding Fext(X, Ω) = 0 and t− + t+ = 2π/Ω. These are precisely the steps the candidate solution gives, including the block-exponential/phi-function identity and the role of α = 0 along unforced periodic branches. The candidate’s argument mirrors the paper’s logic and scope (with standard caveats on transversality and non-grazing), so both are correct and essentially the same. Key elements appear explicitly in the paper: the augmented form and switching geometry (26)–(28) , the MIC reduction using exp([B c; 0 0]) = [e^B, φ1(B)c; 0, 1] (37) and the simplified F(x0, t−, t+, α, a) system (40)–(41)–(42)–(39) as written in the text around (36)–(39) , the implication α = 0 for periodic solutions in the unforced case , the monodromy expression (42) with identity saltation matrices due to continuity , and the forced augmentation (47)–(51) including φ0 and the period constraint t− + t+ = 2π/Ω .