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

Yujiang Chen, Lin Li, Lingling Liu, Zhiheng Yu

TL;DR

The paper addresses the rich dynamics of a three-dimensional discrete-time ecological system, modeled by a map $F:\,\mathbb{R}_+^3\to\mathbb{R}_+^3$ with fixed points $O=(0,0,0)$, $E_1=(1-1/\mu,0,0)$, $E_2=(1/\beta,1-1/\mu-1/\beta,0)$, and $E_3$, by applying complete polynomial discriminant theory, center-manifold reductions, and normal-form analysis. It derives exhaustive codimension-1 bifurcations (transcritical, flip, Neimark-Sacker) near the fixed points and maps the ensuing dynamics, including strong resonances $1:2$, $1:3$, $1:4$ and weak-resonance Arnold tongues, complemented by Marotto chaos criteria. The contributions include rigorous bifurcation proofs, explicit normal-form unfoldings, and detailed resonance sequences, supported by numerical simulations that illustrate invariant circles, period-doubling, and chaotic regimes. The results enhance understanding of multi-species ecological interactions by revealing how parameter changes can drive transitions between extinction, persistent oscillations, and chaotic population dynamics, with implications for ecosystem management and resilience.

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.

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

TL;DR

The paper addresses the rich dynamics of a three-dimensional discrete-time ecological system, modeled by a map with fixed points , , , and , by applying complete polynomial discriminant theory, center-manifold reductions, and normal-form analysis. It derives exhaustive codimension-1 bifurcations (transcritical, flip, Neimark-Sacker) near the fixed points and maps the ensuing dynamics, including strong resonances , , and weak-resonance Arnold tongues, complemented by Marotto chaos criteria. The contributions include rigorous bifurcation proofs, explicit normal-form unfoldings, and detailed resonance sequences, supported by numerical simulations that illustrate invariant circles, period-doubling, and chaotic regimes. The results enhance understanding of multi-species ecological interactions by revealing how parameter changes can drive transitions between extinction, persistent oscillations, and chaotic population dynamics, with implications for ecosystem management and resilience.

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.
Paper Structure (14 sections, 14 theorems, 218 equations, 12 figures, 2 tables)

This paper contains 14 sections, 14 theorems, 218 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Qualitative properties of fixed points
  3. Codimension-1 bifurcations
  4. Transcritical bifurcation of $O$
  5. Transcritical and flip bifurcations of $E_1$
  6. Transcritical and flip bifurcations of $E_2$
  7. Neimark-Sacker bifurcation of $E_{2}$
  8. Codimension-2 bifurcations
  9. $1:2$ resonance at $E_2$
  10. $1:3$ resonance at $E_2$
  11. $1:4$ resonance at $E_2$
  12. Weak resonance Arnold tongues
  13. Existence of Marotto's Chaos
  14. Numerical simulations

Key Result

Proposition 1

For parameter $\Lambda:=(\lambda,\mu,\beta)\in \mathbb{R}_+^{3}$, mapping eq2.1 has at most four fixed points, i.e., a fixed point $O:(0,0,0)$, which always exists, two boundary fixed points $E_1:(1-1/\mu,0,0)$ if $\mu>1$ and $E_2:\left({1}/{\beta},1-1/\mu-1/\beta,0\right)$ if $\beta\geq \mu/(\mu-1) which exists in the case $\beta>\lambda\mu/(\lambda\mu - \lambda - \mu)$, $\lambda>\mu/(\mu-1)$ and

Figures (12)

  • Figure 1: Bifurcation diagram of system \ref{['1.2.9']}
  • Figure 2: Bifurcation diagram of system \ref{['1.3.10']}
  • Figure 3: Partitioning of $(a_0, b_0)$-plane of system \ref{['1.4.12']}
  • Figure 4: Bifurcation diagram of system \ref{['1.4.13']}
  • Figure 5: Flip bifurcation diagram and corresponding Lyapunov exponents diagram of system \ref{['eq2.1']} at fixed point $E_1$
  • ...and 7 more figures

Theorems & Definitions (28)

  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 18 more