Monotonicity of the period function and chaotic dynamics in a class of singular ODEs
type: Item
instance of: scholarly article
title: Monotonicity of the period function and chaotic dynamics in a class of singular ODEs
zbMATH Open document ID: 1492.34039
author: Lakshmi Burra
author: Fabio Zanolin
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-12-13T00:00:00Z
review text: This paper deals with a periodically perturbed singular Duffing equation of the type \[ u'' - \frac{1}{u^\alpha} = e(t), \] where \(\alpha \geq 1\) and \(e: \mathbb{R} \to \mathbb{R}\) is a locally integrable and periodic function. More precisely, the authors prove the existence of chaotic dynamics (in the coin-tossing sense), as well as the presence of infinitely many subharmonic solutions, when the forcing term \(e(t)\) is a stepwise function with large period. It is also pointed out that this result is stable under perturbations, meaning that the result still holds if the forcing term \(e(t)\) is close to a stepwise function as above, and a forcing term of the type \(cu'\), with \(c\) small, is introduced in the equation. The proof relies on the theory of topological horseshoes, in the framework of a so-called ``Linked Twist Maps'' geometry, along a line of research extensively developed by both the authors in the last years. As an auxiliary result for the proof, the authors also prove, using the celebrated Chicone's theorem, the strict monotonocity of the period map in the autonomous case, a fact which can be considered of independent interest. The paper contains several numerical simulations illustrating the complex behavior of the Poincaré map. Moreover, the final section provides an interesting discussion about the dynamics of the equation from various points of view.
reviewed by: Alberto Boscaggin
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37C60
Mathematics Subject Classification ID: 70K40
Mathematics Subject Classification ID: 34C28
Mathematics Subject Classification ID: 34B30
zbMATH DE Number: 7442658
zbMATH Keywords: nonautonomous Duffing equations
zbMATH Keywords: positive solutions
zbMATH Keywords: Poincaré map
zbMATH Keywords: subharmonic solutions
zbMATH Keywords: topological horseshoes
zbMATH Keywords: chaotic dynamics
MaRDI profile type: Publication
full work available at URL: j.jmaa.2021.125814
OpenAlex ID: W3212989884
cites work: Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions
cites work: Non-singular solutions of a Rayleigh-Plesset equation under a periodic pressure field
cites work: Q5236893
cites work: The local period function for Hamiltonian systems with applications
cites work: The monotonicity of the period function for planar Hamiltonian vector fields
cites work: Twist periodic solutions for differential equations with a combined attractive-repulsive singularity
cites work: Period function for a class of Hamiltonian systems
cites work: Infinitely many \(T\)-periodic solutions for a problem arising in nonlinear elasticity
cites work: Subshifts of Finite Type in Linked Twist Mappings
cites work: Q4878889
cites work: Subharmonic solutions of second order nonlinear equations: a time-map approach
cites work: Boundedness and stability for the damped and forced single well Duffing equation
cites work: Playing Around Resonance
cites work: Subharmonic Solutions for Some Second-Order Differential Equations with Singularities
cites work: Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth
cites work: A higher-dimensional Poincaré-Birkhoff theorem without monotone twist
cites work: A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows
cites work: First derivative of the period function with applications
cites work: An example of chaotic dynamics for the motion of a charged particle in a magnetic field
cites work: Q3871201
cites work: On the Definition of Chaos
cites work: On Periodic Solutions of Nonlinear Differential Equations with Singularities
cites work: Chaos in periodically perturbed planar Hamiltonian systems using linked twist maps
cites work: Chaotic dynamics of the Kepler problem with oscillating singularity
cites work: CHAOTIC DYNAMICS FOR MAPS IN ONE AND TWO DIMENSIONS: A GEOMETRICAL METHOD AND APPLICATIONS TO ECONOMICS
cites work: Stable and Random Motions in Dynamical Systems
cites work: Sur les périodes des solutions de l'équation différentielle x'' + g(x) = 0
cites work: Chaotic dynamics in a periodically perturbed Liénard system.
cites work: Q3074259
cites work: Chaotic dynamics in the Volterra predator-prey model via linked twist maps
cites work: Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities
cites work: Finding a horseshoe on the beaches of Rio
cites work: Twist Solutions of a Hill's Equation with Singular Term
cites work: Mathematical models with singularities. A zoo of singular creatures
cites work: Twist periodic solutions of repulsive singular equations
cites work: The period function of the generalized Lotka-Volterra centers
cites work: Lyapunov stability of periodic solutions of Brillouin type equations
cites work: Foundations of chaotic mixing
cites work: Q4359743
DOI: 10.1016/J.JMAA.2021.125814
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description: scientific article; zbMATH DE number 7442658