# How do Conservative Backbone Curves Perturb into Forced Responses? A Melnikov Function Analysis

## TL;DR

This paper provides a mathematical framework using Melnikov functions to predict how conservative backbone curves influence forced responses in weakly damped mechanical systems, clarifying when certain periodic orbits dominate the response.

## Contribution

It introduces a systematic analysis predicting which conservative periodic orbits form backbone curves in forced responses and establishes conditions for approximate methods and isolas.

## Key findings

- Predicts which conservative orbits serve as backbone curves
- Derives conditions for the validity of energy balance and force appropriation methods
- Provides criteria for the emergence of isolated response branches (isolas)

## Abstract

Weakly damped mechanical systems under small periodic forcing tend to exhibit periodic response in a close vicinity of certain periodic orbits of their conservative limit. Specifically, amplitude frequency plots for the conservative limit have often been noted, both numerically and experimentally, to serve as backbone curves for the near resonance peaks of the forced response. In other cases, such a relationship between the unforced and forced response was not observed. Here we provide a systematic mathematical analysis that predicts which members of conservative periodic orbit families will serve as backbone curves for the forced-damped response. We also obtain mathematical conditions under which approximate numerical and experimental approaches, such as energy balance and force appropriation, are justifiable. Finally, we derive analytic criteria for the birth of isolated response branches (isolas) whose identification is otherwise challenging from numerical continuation.