Tangency Bifurcations of Global Poincaré Maps

Lee, Clare M.; Collins, Pieter; Krauskopf, Bernd; Osinga, Hinke M.

doi:10.1137/07069972x

Cited by 25 publications

(34 citation statements)

References 26 publications

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“…A such orbit is characterized by some extra loops around the one-dimensional manifold between two consecutive intersections with the separation plane and, as we can see in Figure 10(b), a natural way for them to disappear is the existence of tangencies with the separation plane (p 1 andp 3 ). Some results about tangencies and their importance can be found in [15]. In the particular case of the second curve of the zero set of E (that is, the first one of Figure 6(b) with period greater than 4π), it has been checked that there exist only a piece that gives RP2-orbits (similar to those shown in Figure 10(a)).…”

Section: Some Comments About Other Rp2-orbitsmentioning

confidence: 99%

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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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Section: Some Comments About Other Rp2-orbitsmentioning

confidence: 99%

“…Lemma 2 describes the zero set of function G given in (15). This set is formed by infinitely many simple, closed and isolated curves.…”

Section: Proof the Following Expressions For The Trigonometric Functmentioning

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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“…In piecewise linear systems we define the Poincaré map by means of the composition of some applications namely Poincaré half-maps (see [6,12,20,19,22,26]). If the considered system, in particular, is two-zonal, the Poincaré map is defined by composition of the left Poincaré half-map and the right Poincaré half-map, as we will do in a few lines.…”

Section: Construction Of the Melnikov Functionmentioning

confidence: 99%

Periodic orbits for perturbations of piecewise linear systems

Carmona

Fernández-García

Freire

2011

Journal of Differential Equations

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We consider the existence of periodic orbits in a class of threedimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.

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“…In this graph, a fold point corresponds to an orbit which is tangent to one of the Poincaré planes of intersection. A detailed account of the geometry of the manifold in the vicinity of such points can be found in Lee et al [10]. x 1 x 2 x 3 Fig.…”

Section: 1mentioning

confidence: 99%

On Matrix-Free Computation of 2D Unstable Manifolds

Kawahara¹,

Atsushi²

2011

SIAM J. Sci. Comput.

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Recently, a flexible and stable algorithm was introduced for the computation of 2D unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this manifold as the solutions of an appropriate boundary value problem. The boundary value problem is under determined and a one parameter family of solutions can be found by means of arclength continuation. This family of orbits covers a piece of the manifold. The quality of this covering depends on the way the boundary value problem is discretised, as do the tractability and accuracy of the computation. In this paper, we describe an implementation of the orbit continuation algorithm which relies on multiple shooting and Newton-Krylov continuation. We show that the number of time integrations necessary for each continuation step scales only with the number of shooting intervals but not with the number of degrees of freedom of the dynamical system. The number of shooting intervals is chosen based on linear stability analysis to keep the conditioning of the boundary value problem in check. We demonstrate our algorithm with two test systems: a low-order model of shear flow and a well-resolved simulation of turbulent plane Couette flow.

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Tangency Bifurcations of Global Poincaré Maps

Cited by 25 publications

References 26 publications

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

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

Periodic orbits for perturbations of piecewise linear systems

On Matrix-Free Computation of 2D Unstable Manifolds

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