The linear pantograph equation, introduced to model conduction of electric power in electric trains, has been studied from the Applied Mathematics, Analysis, Number Theory and more recently Probabilistic viewpoints. References span from 1971 through 2024.

The Alpha-Riccati equation has been introduced as a toy model for the understanding of probabilistic methods in PDE’s. When linearized, it can lead to the pantograph equation.

In this talk, I will review some basic results showing non-uniqueness of solutions that combine probabilistic and analysis methods, illustrate via numerical experiments a rich bifurcation pattern, present some open problems and, if time allows, discuss speculative paths for the relations between non-uniqueness in the linear and non-linear problem.

The talk is based on an ongoing collaboration with Radu Dascaliuc and Ed Waymire (OSU), Tuan Pham (BYU-HI) and Nicholas Hale and Andre’ Weideman (Stellenbosch U - South Africa).