The number of limit cycles of piecewise linear Liénard systems

What the math gives to ML

The paper provides a constructive mechanism for generating many coexisting limit cycles in planar piecewise-linear Lienard systems. Fold points and jump points in the restoring function control the number of cycles, with a construction producing n+m cycles when m of n breakpoints are jumps. The asymptotic slopes of the outermost linear pieces create a sharp transition at absolute slope 2 in the compactified phase portrait. This can transfer to continuous-time recurrent networks by replacing a generic latent drift with a piecewise-linear Lienard cell whose attractor structure is explicitly designed and experimentally verifiable.