THE HYBRID MATCHING OF HURWITZ SYSTEMS

The paper proves Theorem A using a displacement map Δ(x) and amplitude-independent flight times to classify all global behaviors under the crossing hypothesis. In the focus–focus case, it derives an explicit inequality (14) from which it concludes either GAS, global repeller, global center (balanced exponents), or a unique hyperbolic limit cycle; in the node cases, only finitely many impacts occur and trajectories converge to the origin. The candidate solution instead constructs an exact one-dimensional Poincaré return map P(u)=C u^{μ} on Σ1, with μ=rs and an explicit constant C from the two linear flights and the power-law jumps, and then classifies dynamics by iterating P. This yields the same trichotomy for μ=1 and the existence/uniqueness of a hyperbolic cycle for μ≠1, and furthermore identifies the cycle’s stability (attracting if μ<1, repelling if μ>1). These conclusions are consistent with, and refine, the paper’s statements. Hence both are correct, via different proofs. Key definitions of Σρ and ϕρ, the crossing hypothesis, Theorem A, the amplitude-independent flight times t±, and the displacement-map derivation (10)–(14) appear in the paper and match the model’s constructions.