A 4N − 1 lower bound for crossing limit cycles of discontinuous piecewise linear Liénard systems

For every integer N ≥ 2, we construct a discontinuous piecewise linear Liénard system ẋ = y − F(x), ẏ = −x, in which F has exactly N jump discontinuities, all at positive abscissae, and no other breakpoints, and which has at least 4N − 1 distinct hyperbolic crossing limit cycles. This improves by one the lower bound 4N − 2 of Chen, Jin, Li and Lu. Their inductive construction gives 3N − 3 simple zeros of a first-order averaged function. We show that a nonlinear term of the exact far-field return equation permits N + 2, instead of N + 1, cycles of large amplitude, and we choose a common parameter scaling under which the 3N − 3 cycles of bounded amplitude persist in the same system in which these N + 2 large cycles appear. Using the same transfer maps in separate constructions, we obtain for every N ≥ 4 systems with exactly N jumps that have an exterior crossing center although F is not even, and whose unique equilibrium is an attracting node. Near an explicit four-jump example we determine the center conditions and the local center ideal in the full fourteen-parameter family, and we prove that the local cyclicity at infinity is exactly six in the subfamily with balanced tails and exactly eight in the full family. For this example we also determine the finite boundary of the exterior annulus: it has two one-sided tangencies and contains no sliding segment, and at one tangency the Filippov inclusion also admits a forward solution converging to the node. Finally, we construct a one-parameter family of nonisolated composite periodic Filippov solutions, each with one attracting and one repelling sliding arc; they share an arc with the boundary and are not limit cycles.