Stability and Bifurcations of Planar Switched Linear and Homogeneous Systems
arXiv:2607.12189
2026
Dynamics
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper develops explicit worst-case switching criteria for uniform asymptotic stability of two-dimensional switched linear and homogeneous systems, together with bifurcation boundaries and local basin results for switched nonlinear systems. The transferable mechanism is to treat a residual network, recurrent model, or optimizer as a switched dynamical system and certify stability against adverse sequences of mode Jacobians rather than checking each mode independently. The most practical neural implementation is a worst-case Jacobian-product monitor combined with a nonlinear Jacobian-variation penalty, using the predicted transition at worst-case growth equal to one as a falsifiable stability boundary.
Ideas from this paper
✗ Failed on benchmark
2026
Model a residual network, recurrent update, or optimizer as a switched linearized system in which each layer type, token, data batch, or optimizer regime selects a matrix mode. Constrain the worst-case product growth over admissible switches, rather than merely constraining every individual Jacobian, so arbitrary mode sequences remain contractive.
Useful8/10
Difficulty6/10
Novelty7/10
△ Mechanism confirmed, baseline not beaten
2026
Use the switched nonlinear extension to distinguish stability of the linearized modes from stability of the full neural dynamics. Stabilize worst-case linear products and limit the variation of each nonlinear Jacobian inside a specified radius, yielding an explicit local basin estimate and a penalty that prevents mode interactions from destroying attraction.
Useful7/10
Difficulty5/10
Novelty8/10