Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Pokrovskiĭ, A. V.; Rasskazov, Oleg; Visetti, Daniela

doi:10.3934/dcdsb.2007.8.943

Cited by 6 publications

(16 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More precisely, we prove in a rigorous way (i.e. without the need of computer assistance, differently from several works on related topics [12,14,53,55,98,109,129,165,169,173]) the presence of infinitely many periodic solutions for our systems, as well as of a chaotic behaviour in the sense of Definition 2.1.2 for the associated Poincaré map Ψ. In fact, a classical approach (see [79]) to show the existence of periodic solutions (harmonics or subharmonics) of non-autonomous differential systems like We notice that the kind of chaos in Definition 2.1.2, when considered in relation to the case of the Poincaré map, looks similar to other ones detected in the literature on complex dynamics for ODEs with periodic coefficients (see, for instance, [33,150]).…”

Section: Chapter 4 Examples From the Odesmentioning

confidence: 86%

“…We conclude by observing that the same arguments employed above seem to work also for more general time-dependent coefficients. However, a rigorous proof in such cases would require a more delicate analysis or possibly the aid of computer assistance (as in [12,53,109,129,165,169,173]) and this is beyond the aims of the present thesis.…”

Section: Suspension Bridges Modelmentioning

confidence: 97%

See 1 more Smart Citation

Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications

Pireddu

2009

Preprint

View full text Add to dashboard Cite

Ph.D. Thesis 1 21 This is the final thesis for the Ph.D. program in Mathematics and Physics -Section: Mathematics. The final defence has been held on June 4, 2009.2 Sinceri ringraziamenti al Prof. Zanolin per l'infinita disponibilità ed il prezioso e costante aiuto.The N-dimensional Euclidean space R N is endowed with the usual scalar product • , • , norm • and distance dist(• , •). In the case of R, the norm• will be replaced with the absolute value |In a normed space (X, • X ), we denote by B(x 0 , r) and B[x 0 , r] the open and closed balls centered in x 0 ∈ X with radius r > 0, i.e. B(x 0 , r) := {x ∈ X : x − x 0 X < r} and B[x 0 , r] := {x ∈ X : x − x 0 X ≤ r}. For M ⊆ X, we set B(M, r) := {x ∈ X : ∃ w ∈ M with x − w X < r}. The set B[M, r] is defined accordingly. If we wish to specify the dimension m of the ball, when it is contained in R m , we write B m (x 0 , r) and B m [x 0 , r] in place of B(x 0 , r) and B[x 0 , r]. We denote byGiven an open bounded subset Ω of R N , an element p of R N and a continuous function f : Ω → R N , we indicate with deg(f, Ω, p) the topological degree of the map f with respect to Ω and p. We say that degWe denote by C(D, R N ) the set of the continuous maps f : D → R N on a compact set D, endowed with the infinity norm |f | ∞ := max x∈D f (x) .

show abstract

Section: Chapter 4 Examples From the Odesmentioning

confidence: 86%

Section: Suspension Bridges Modelmentioning

confidence: 97%

Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications

Pireddu

2009

Preprint

View full text Add to dashboard Cite

show abstract

“…The application we consider in the present article deals with a simpler situation, namely the case of a Lipschitz continuous planar vector field which is smooth (indeed of class C ∞ ) except for a single vertical line which intersects transversally the homoclinic orbit. More in detail, we reconsider, in the light of the Melnikov method, a model previously studied by Pokrovskii, Rasskazov and Visetti in [27] dealing with the periodically perturbed Duffing-type equation…”

Section: Introductionmentioning

confidence: 99%

“…where s(x) is a truncated signum function, that is it coincides with x/| x | for |x| ≥ d > 0 and is linear for | x| ≤ d. As shown in [9,11] and also recalled in [27], equations of this form arise in the engineering literature as models of oscillators with stops. Also Holmes and Moon [28,29] showed how to produce a large variery of Duffing-type equations with different restoring forces, by combining mechanical and magnetic effects in buckled magnetoelastic oscillators.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Application of the Melnikov Method to a Piecewise Oscillator

Gjata

Zanolin

2023

Contemp. Math.

View full text Add to dashboard Cite

In this paper we present a new application of the Melnikov method to a class of periodically perturbed Duffing equations where the nonlinearity is non-smooth as otherwise required in the classical applications. Extensions of the Melnikov method to these situations is a topic with growing interests from the researchers in the past decade. Our model, motivated by the study of mechanical vibrations for systems with ''stops'', considers a case of a nonlinear equation with piecewise linear components. This allows us to provide a precise analytical representation of the homoclinic orbit for the associated autonomous planar system and thus obtain simply computable conditions for the zeros of the associated Melnikov function.

show abstract

Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm

et al. 2012

View full text Add to dashboard Cite

Seasonality is a complex force in nature that affects multiple processes in wild animal populations. In particular, seasonal variations in demographic processes may considerably affect the persistence of a pathogen in these populations. Furthermore, it has been long observed in computer simulations that under seasonal perturbations, a host-pathogen system can exhibit complex dynamics, including the transition to chaos, as the magnitude of the seasonal perturbation increases. In this paper, we develop a seasonally perturbed Susceptible-Infected-Recovered model of avian influenza in a seabird colony. Numerical simulations of the model give rise to chaotic recurrent epidemics for parameters that reflect the ecology of avian influenza in a seabird population, thereby providing a case study for chaos in a host- pathogen system. We give a computer-assisted exposition of the existence of chaos in the model using methods that are based on the concept of topological hyperbolicity. Our approach elucidates the geometry of the chaos in the phase space of the model, thereby offering a mechanism for the persistence of the infection. Finally, the methods described in this paper may be immediately extended to other infections and hosts, including humans.

show abstract

Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Cited by 6 publications

References 10 publications

Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications

Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications

An Application of the Melnikov Method to a Piecewise Oscillator

Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm

Contact Info

Product

Resources

About