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A Method to Estimate a Neighborhood of a Periodic Orbit

Mario Cavani

TL;DR

This work addresses the problem of bounding and characterizing a neighborhood of a periodic orbit arising from parameter-dependent nonlinear ODEs. It combines the integral-averaging method to convert a non-autonomous or parameter-sensitive problem into an autonomous averaged system, with Hopf-bifurcation analysis in two dimensions to guarantee the existence of a small-amplitude periodic orbit, and then uses extended averaging to bound the orbit within an explicit annulus. For the HHW predator-prey model with equal voracity, the authors derive conditions under which the Hopf bifurcation yields an orbit that persists and is orbitally stable, locating it inside a computable annular region and providing amplitude and period estimates. The approach thus delivers quantitative, location-specific information about periodic orbits that extends beyond classical Hopf theory and offers a pathway to higher-dimensional generalizations. Overall, the method offers a practical framework for bounding periodic dynamics in parameter-dependent nonlinear systems with potential ecological applications.

Abstract

In this paper we describe a method to estimate a neighborhood containing a periodic orbit of a given system of two ordinary differential equations. By using the theory of integral averages, the system of differential equations can be transformed into an equivalent autonomous system which, by using the Hopf bifurcation theorem, the existence of a periodic solution of this autonomous nonlinear differential equations can be demonstrated. Using of this procedure it is possible to estimate an annular region where the orbit of the periodic solution is located. The method allows to improve the results that the Hopf Bifurcation provides on periodic solutions. In addition, some quantitative characteristics of the solution can be known, such as the amplitude, the period and, a region where the periodic orbit of the original system is located. The method is applied to a three-dimensional system of differential equations that models the competition of two predators and one prey, which under the assumption that the predators are equally voracious, property that in this case leads to a two-dimensional system, where all of the conditions of the method described here are easily applicable.

A Method to Estimate a Neighborhood of a Periodic Orbit

TL;DR

This work addresses the problem of bounding and characterizing a neighborhood of a periodic orbit arising from parameter-dependent nonlinear ODEs. It combines the integral-averaging method to convert a non-autonomous or parameter-sensitive problem into an autonomous averaged system, with Hopf-bifurcation analysis in two dimensions to guarantee the existence of a small-amplitude periodic orbit, and then uses extended averaging to bound the orbit within an explicit annulus. For the HHW predator-prey model with equal voracity, the authors derive conditions under which the Hopf bifurcation yields an orbit that persists and is orbitally stable, locating it inside a computable annular region and providing amplitude and period estimates. The approach thus delivers quantitative, location-specific information about periodic orbits that extends beyond classical Hopf theory and offers a pathway to higher-dimensional generalizations. Overall, the method offers a practical framework for bounding periodic dynamics in parameter-dependent nonlinear systems with potential ecological applications.

Abstract

In this paper we describe a method to estimate a neighborhood containing a periodic orbit of a given system of two ordinary differential equations. By using the theory of integral averages, the system of differential equations can be transformed into an equivalent autonomous system which, by using the Hopf bifurcation theorem, the existence of a periodic solution of this autonomous nonlinear differential equations can be demonstrated. Using of this procedure it is possible to estimate an annular region where the orbit of the periodic solution is located. The method allows to improve the results that the Hopf Bifurcation provides on periodic solutions. In addition, some quantitative characteristics of the solution can be known, such as the amplitude, the period and, a region where the periodic orbit of the original system is located. The method is applied to a three-dimensional system of differential equations that models the competition of two predators and one prey, which under the assumption that the predators are equally voracious, property that in this case leads to a two-dimensional system, where all of the conditions of the method described here are easily applicable.

Paper Structure

This paper contains 8 sections, 11 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Periodic Solutions
  4. The Hopf Bifurcation Theorem in ${\mathbb{R} }^{2}$
  5. The Integral Average Method
  6. Method to Find the Neighborhood
  7. Application of the Method
  8. Conclusions

Key Result

Lemma 1

A necessary and sufficient condition for a solution $x(t,\alpha )$ of the equation (eq1) to have period $T$ is,

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4
  • Theorem 5
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 10 more