Existence of a Reversible T-Point Heteroclinic Cycle in a Piecewise Linear Version of the Michelson System

Carmona, Victoriano; Fernández-Sánchez, Fernando; Teruel, Antonio E.

doi:10.1137/070709542

Cited by 37 publications

(33 citation statements)

References 26 publications

(24 reference statements)

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“…For example, piecewise smooth functions [5] are often used as caricatures of nonlinear functions [34]. A nonlinear function is replaced by piecewise linear approximations and a set of simpler linear problems is solved instead [6].…”

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confidence: 99%

On the Use of Blowup to Study Regularizations of Singularities of Piecewise Smooth Dynamical Systems in $\mathbb{R}^3$

Kristiansen¹,

Hogan²

2015

SIAM J. Appl. Dyn. Syst.

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Abstract.In this paper we use the blow up method of Dumortier and Roussarie [11, 12, 13], in the formulation due to Krupa and Szmolyan [24], to study the regularization of singularities of piecewise smooth dynamical systems [18] in R 3 . Using the regularization method of Sotomayor and Teixeira [37], first we demonstrate the power of our approach by considering the case of a fold line. We quickly recover a main result of Bonet and Seara [36] in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain non-resonance condition holds. Finally, we provide numerical evidence for the existence of secondary canards near resonance.

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confidence: 99%

On the Use of Blowup to Study Regularizations of Singularities of Piecewise Smooth Dynamical Systems in $\mathbb{R}^3$

Kristiansen¹,

Hogan²

2015

SIAM J. Appl. Dyn. Syst.

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“…In addition to this, BPAS turns out to be an appropriate tool to understand nonlinear phenomena. For instance, BPAS version of the well known traveling wave equation model of Michelson System [11], the Wien-bridge oscillator of [12], a biological network model by Imura et al [13], are examples of nonlinear systems modeled as BPAS.…”

Section: Introductionmentioning

confidence: 99%

Well posedness conditions for Bimodal Piecewise Affine Systems

Şahan

Eldem

2015

Systems & Control Letters

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“…In fact, a simple linear change of variables followed by the change of function x 2 → |x| allows to obtain system (2) from system (1), see [3]. Moreover, both systems are volume-preserving and time-reversible with respect to the involution R(x, y, z) = (−x, y, −z).…”

Section: Introductionmentioning

confidence: 99%

“…Following this idea, in [3] and [4] the authors studied some global connections of system         ẋ = y, y = z,…”

Section: Introductionmentioning

confidence: 99%

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

Carmona

Fernández-García

Fernández-Sánchez

et al. 2012

Nonlinear Analysis: Theory, Methods & Applications

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The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the wellknown Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameterized three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems.In the piecewise system, the noose bifurcation involves reversible periodic orbits that intersect the separation plane at two or four points. This work is focused on those reversible periodic orbits that intersect the separation ✩ This work was partially supported by the Ministerio de Ciencia e Innovación, under the projects MTM2009-07849, MTM2010-20907-C02-01 and MTM2011-22751 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (TIC-0130, P08-FQM-03770). The second author is supported by Ministerio de Educación, grant AP2008-02486.* Corresponding author Email addresses: vcarmona@us.es (V. Carmona), soledad@us.es (S. Fernández-García), fefesan@us.es (F. Fernández-Sánchez), egarme@us.es (E. García-Medina), antonioe.teruel@uib.es (A. E. Teruel) Preprint submitted to Nonlinear Analysis: Theory, Methods & ApplicationsFebruary 16, 2012 This is a preprint of: "Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations", Victoriano Carmona, Soledad Fernández-García, Fernando Fernández-Sánchez, Elisabeth Garcia-Medina, Antonio E. Teruel, Nonlinear Anal., vol. 75, 5866-5883, 2012. DOI: [10.1016/j.na.2012 plane twice (RP2-orbits). It is established that for every T between 2π and a critical point there exists a unique value of the parameter for which the system has a RP2-orbit with period T . Moreover, this critical value, that separates periodic orbits with two or four intersection points with the separation plane, corresponds to a RP2-orbit that crosses tangentially the separation plane.It is also proved that in a bifurcation diagram parameter versus period, the curve of this family of periodic orbits has a unique maximum point, which corresponds to the saddle-node bifurcation of periodic orbits that appears in the noose bifurcation.

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Existence of a Reversible T-Point Heteroclinic Cycle in a Piecewise Linear Version of the Michelson System

Cited by 37 publications

References 26 publications

On the Use of Blowup to Study Regularizations of Singularities of Piecewise Smooth Dynamical Systems in $\mathbb{R}^3$

On the Use of Blowup to Study Regularizations of Singularities of Piecewise Smooth Dynamical Systems in $\mathbb{R}^3$

Well posedness conditions for Bimodal Piecewise Affine Systems

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

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