Rich dynamics in planar systems with heterogeneous nonnegative weights

Julián López-Gómez; Eduardo Muñoz-Hernández; Fabio Zanolin

doi:10.3934/cpaa.2023020

Article Contents

2023, Volume 22, Issue 4: 1043-1098. Doi: 10.3934/cpaa.2023020

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Rich dynamics in planar systems with heterogeneous nonnegative weights

1.
Universidad Complutense de Madrid, Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Plaza de las Ciencias 3, 28040 Madrid, Spain
2.
Università degli Studi di Udine, Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Via delle Scienze 2016, 33100 Udine, Italy

^*Corresponding author: Fabio Zanolin
^*Corresponding author: Fabio Zanolin

Received: July 2022

Early access: January 2023

Published: April 2023

This paper has been written under the auspices of the Ministry of Science and Innovation of Spain, under Research Grant PID2021-123343NB-I00, and of the IMI of Complutense University. The second author, ORCID 0000-0003-1184-6231, has been also supported by contract CT42/18-CT43/18 of Complutense University of Madrid

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Abstract

This paper studies the global structure of the set of nodal solutions of a generalized Sturm–Liouville boundary value problem associated to the quasilinear equation

$ -(\phi(u'))' = \lambda u + a(t)g(u), \quad \lambda\in {\mathbb R}, $

where $ a(t) $ is non-negative with some positive humps separated away by intervals of degeneracy where $ a\equiv 0 $. When $ \phi(s) = s $ this equation includes a generalized prototype of a classical model going back to Moore and Nehari [35], 1959. This is the first paper where the general case when $ {\lambda}\in\mathbb{R} $ has been addressed when $ a\gneq 0 $. The semilinear case with $ a\lneq 0 $ has been recently treated by López-Gómez and Rabinowitz [28,29,30].

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Mathematics Subject Classification: Primary: 34B08, 34B24, 35C15.

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Figure 1. Bifurcation diagram for $ p = 1 $, where $ {\sigma}_n\equiv {\sigma}_n[-D^2-a(t)] $

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Figure 2. Plots of $ \mathcal {T} $ according to Table 1

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Figure 3. Bifurcation diagrams for $ a\equiv\mu>0 $

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Figure 4. Two minimal global bifurcation diagrams

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Figure 5. The action of the Poincaré map

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Figure 6. The action of the flow induced by $ ({\mathcal L}_{0}) $

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Figure 7. The oriented regions $ \mathcal A^{\rm energy}_{i, j} $ (left) and $ \mathcal A^{\rm axis}_{i, j} $ (right)

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Figure 8. A closed orbit of $ ({\mathcal L}_{\lambda}) $ crossing $ {\mathcal A}_i $ and $ {\mathcal A}_{i+1} $

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Figure 9. Admissible transitions of $ \Psi_i\circ\Phi_i $ (right panel). In the left panel we have put in evidence the intermediate transitions across the sets $ \mathcal B^{\rm axis} $

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Table 1. Behavior of the period map $ \mathcal {T}(x_+) $

$ {\lambda} $	behavior of $ x_+ $	case $ p>1 $	case $ 0<p<1 $
$ {\lambda}\geq 0 $	$ x_+ $ increases	$ { \mathcal T}(x_+) $ decreases	$ { \mathcal T}(x_+) $ increases
	$ x_+\downarrow 0 $	$ { \mathcal T}(x_+)\uparrow 2\pi/\sqrt{{\lambda}} $	$ { \mathcal T}(x_+)\downarrow 0 $
	$ x_+\uparrow +\infty $	$ { \mathcal T}(x_+)\downarrow 0 $	$ { \mathcal T}(x_+)\uparrow 2\pi/\sqrt{{\lambda}} $
$ \lambda<0 $	$ x_+ $ increases	$ { \mathcal T}(x_+) $ decreases	$ { \mathcal T}(x_+) $ increases
	$ x_+\downarrow x^* $	$ { \mathcal T}(x_+)\uparrow +\infty $
	$ x_+\downarrow 0 $		$ { \mathcal T}(x_+)\downarrow 0 $
	$ x_+\uparrow +\infty $	$ { \mathcal T}(x_+)\downarrow 0 $
	$ x_+\uparrow {\omega}_+ $		$ { \mathcal T}(x_+)\uparrow +\infty $

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