# Jacob's ladders, nonlinear interactions between $\zeta$-oscillating systems and corresponding constraints

## TL;DR

This paper introduces a new class of nonlinear interactions between $z$-oscillating systems, revealing asymptotic relationships in a functional algebra framework based on trigonometric functions.

## Contribution

It presents a novel class of nonlinear interactions and establishes asymptotic equivalences within a new functional algebra derived from $z$-oscillating systems.

## Key findings

- Cube of two-part form asymptotically equals another two-part form
- Main formula generated by subset of trigonometric functions
- Introduction of a new class of nonlinear interactions

## Abstract

In this paper we introduce new class of nonlinear interactions of $\zeta$-oscillating systems. The main formula is generated by corresponding subset of the set of trigonometric functions. Next, the main formula generates certain set of two-parts forms. For this set the following holds true: the cube of two-part form is asymptotically equal to other two-part form -- short functional algebra.