Resonance properties and chaotic dynamics of a three-dimensional discrete logistic ecological system within the neighborhoods of bifurcation points

TL;DR

This paper analyzes the complex dynamics, including resonance and chaos, of a three-dimensional logistic ecological system near bifurcation points using advanced mathematical theories and numerical simulations.

Contribution

It provides a comprehensive classification of fixed points, stability analysis, and bifurcation types, including codimension-2 bifurcations, in a 3D ecological model.

Findings

Identification of various bifurcations near fixed points

Existence of chaos in the system confirmed by Marotto's criteria

Numerical validation of theoretical bifurcation and chaos results

Abstract

In this paper, we delve into the dynamical properties of a class of three-dimensional logistic ecological models. By using the complete discriminant theory of polynomials, we first give a topological classification for each fixed point and investigate the stability of corresponding system near the fixed points. Then employing the bifurcation and normal form theory, we discuss all possible codimension-1 bifurcations near the fixed points, i.e., transcritical, flip, and Neimark-Sacker bifurcations, and further prove that the system can undergo codimension-2 bifurcations, specifically 1:2, 1:3, 1:4 strong resonances and weak resonance Arnold tongues. Additionally, chaotic behaviors in the sense of Marotto are rigorously analyzed. Numerical simulations are conducted to validate the theoretical findings and illustrate the complex dynamical phenomena identified.