Higher derivative discontinuous solutions to linear ordinary differential equations: A new route to complexity?

TL;DR

This paper introduces a new family of solutions with higher derivative discontinuities to a simple linear ODE, potentially offering insights into complex natural phenomena and symmetry breaking.

Contribution

It presents a novel class of solutions with derivative discontinuities for a scale-invariant linear ODE, extending to higher derivatives and breaking reflection symmetry.

Findings

Discontinuous solutions exist for specific even derivatives.

Solutions break the discrete parity symmetry.

Potential implications for understanding complexity in nature.

Abstract

We present a new one parameter family of second derivative discontinuous solutions to the simplest scale invariant linear ordinary differential equation. We also point out how the construction could be extended to generate families of higher derivative discontinuous solutions as well. The discontinuity can occur only for a subset of even order derivatives, viz.,2nd, 4th, 8th, 16th, ....The solutions are shown to break the discrete parity (reflection) symmetry of the underlying equation. These results are expected to gain significance in the contemporary search of a new {\em dynamical principle} for understanding complex phenomena in Nature.