Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global solution curves for first order periodic problems, with applications

Philip Korman, Dieter S. Schmidt

TL;DR

This paper develops a global continuation framework for periodic first-order ODEs of the form $u'+g(t,u)=f(t)$ by constructing a solution curve $u(t,\xi)$ parameterized by the average value $\xi$ of $u$, and solving a corresponding family with $f(t)=\mu+e(t)$ to obtain $\mu(\xi)$. Using a combination of a priori estimates, the Fredholm alternative, and the implicit function theorem, it proves existence, monotonicity, and global continuation of the solution curve, and shows how zeros of $\mu(\xi)$ yield $T$-periodic solutions of the homogeneous equation; Brezis–Nirenberg type results are re-derived and extended. The authors then apply this framework to limit cycles via a polar formulation $r'+g(\theta,r)=0$, establishing conditions for the existence of positive $2\pi$-periodic $r(\theta)$ and illustrating with Van der Pol, Sel'kov, and predator–prey systems, including regularization strategies for singular problems. Numerical continuation in $\xi$ is detailed for computing the global solution curves and limit cycles, with applications to a periodic population model with fishing and to optimal control questions about fishing strategy under periodic environmental forcing. Overall, the work provides a unified, computationally tractable approach to existence, stability, and numerical computation of periodic solutions and limit cycles in periodic and autonomous systems with broad applicability in biology and nonlinear dynamics.

Abstract

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for periodic problems of first order. The results are applied to a population model with fishing, and to the existence and stability of limit cycles. We also describe in detail our numerical computations of curves of periodic solutions, and of limit cycles.

Global solution curves for first order periodic problems, with applications

TL;DR

This paper develops a global continuation framework for periodic first-order ODEs of the form by constructing a solution curve parameterized by the average value of , and solving a corresponding family with to obtain . Using a combination of a priori estimates, the Fredholm alternative, and the implicit function theorem, it proves existence, monotonicity, and global continuation of the solution curve, and shows how zeros of yield -periodic solutions of the homogeneous equation; Brezis–Nirenberg type results are re-derived and extended. The authors then apply this framework to limit cycles via a polar formulation , establishing conditions for the existence of positive -periodic and illustrating with Van der Pol, Sel'kov, and predator–prey systems, including regularization strategies for singular problems. Numerical continuation in is detailed for computing the global solution curves and limit cycles, with applications to a periodic population model with fishing and to optimal control questions about fishing strategy under periodic environmental forcing. Overall, the work provides a unified, computationally tractable approach to existence, stability, and numerical computation of periodic solutions and limit cycles in periodic and autonomous systems with broad applicability in biology and nonlinear dynamics.

Abstract

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for periodic problems of first order. The results are applied to a population model with fishing, and to the existence and stability of limit cycles. We also describe in detail our numerical computations of curves of periodic solutions, and of limit cycles.
Paper Structure (6 sections, 13 theorems, 99 equations, 6 figures)

This paper contains 6 sections, 13 theorems, 99 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The global solution curve
  3. Existence of $T$-periodic solutions
  4. A further existence result, and applications to limit cycles
  5. Numerical computation of solutions
  6. Periodic population model with fishing

Key Result

Lemma 2.1

Assume that $f(t)$ is a continuously differentiable function of period $T$, and of zero average, i.e., $\int_0^T f(s) \, ds=0$. Then

Figures (6)

  • Figure 1: a) The curve of $2\pi$-periodic solutions of (\ref{['12c']}) for the Van der Pol equation. b) The domain for the $2\pi$-periodic solutions, with a periodic solution with $\xi=2.2$
  • Figure 2: The limit cycle for Sel'kov's system (\ref{['be3']})
  • Figure 3: a) The curve of $2\pi$-periodic solutions for the problem (\ref{['12c']}) corresponding to (\ref{['29a']}) b) The limit cycle for the predator-prey system (\ref{['29a']})
  • Figure 4: The curve of $2\pi$-periodic solutions of (\ref{['33']})
  • Figure 5: The two limit cycles of (\ref{['30']})
  • ...and 1 more figures

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • ...and 3 more