V-shaped action functional with delay

TL;DR

This paper introduces a new V-shaped action functional with delay in symplectic geometry, bridging Rabinowitz and V-shaped functionals, with implications for gradient flow analysis.

Contribution

It defines a novel delayed action functional, establishes a smooth interpolation with existing functionals, and shows a bijection of gradient flow lines.

Findings

Established a smooth interpolation between functionals.

Proved a bijection between gradient flow lines.

Identified the critical points and actions are preserved.

Abstract

In this note we introduce the V-shaped action function with delay in a symplectization which is an intermediate action functional between Rabinowitz action functional and the V-shaped action functional. It lives on the same space as the V-shaped action functional but its gradient flow equation is a delay equation as in the case of Rabinowitz action functional. We show that there is a smooth interpolation between the V-shaped action functional and the V-shaped action functional with delay during which the critical points and its actions are fixed. On the other hand there is a bijection between gradient flow lines of the V-shaped action functional with delay and the ones of Rabinowitz action functional.