On multi-parametric bifurcations in a scalar piecewise-linear map

Аврутин, Виктор; Schanz, Michael

doi:10.1088/0951-7715/19/3/001

Cited by 85 publications

(87 citation statements)

References 71 publications

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“…These bifurcations initially reported by Avrutin & Schanz (2006) are characterized by two manifolds in the three-dimensional parameter space: one-dimensional and two-dimensional. The bifurcation structures above the two-dimensional manifold are formed by the period-adding phenomenon and below this manifold by the phenomenon of period increment with the coexistence of attractors.…”

Section: Investigated Systemmentioning

confidence: 96%

The bandcount increment scenario. I. Basic structures

2008

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Bifurcation structures in two-dimensional parameter spaces formed only by chaotic attractors are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In the first part, we report a novel bifurcation scenario formed by crises bifurcations, which includes multi-band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain.

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Section: Investigated Systemmentioning

confidence: 96%

The bandcount increment scenario. I. Basic structures

2008

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“…The most efficient way to investigate these structures is to detect their organizing centers, which are given by bifurcations of higher codimension [7,8]. Typically, at such a point in the parameter space an infinite number of existence regions of periodic orbits emerges.…”

Section: Introductionmentioning

confidence: 99%

Breaking the continuity of a piecewise linear map

2012

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Abstract. Knowledge about the behavior of discontinuous piecewise-linear maps is important for a wide range of applications. An efficient way to investigate the bifurcation structure in 2D parameter spaces of such maps is to detect specific codimension-2 bifurcation points, called organizing centers, and to describe the bifurcation structure in their neighborhood. In this work, we present the organizing centers in the 1D discontinuous piecewise-linear map in the generic form, which can be used as a normal form for these bifurcations in other 1D discontinuous maps with one discontinuity. These organizing centers appear when the continuity of the system function is broken in a fixed point. The type of an organizing center depends on the slopes of the piecewise-linear map. The organizing centers that occur if the slopes have an absolute value smaller than one were already described in previous works, so we concentrate on presenting the organizing centers that occur if one or both slopes have absolute values larger than one. By doing this, we also show that the behavior for each organizing center can be explained using four basic bifurcation scenarios: the period incrementing and the period adding scenarios in the periodic domain, as well as the bandcount incrementing and the bandcount adding scenarios in the chaotic domain.AMS (2000) subject classification. 37E05, 37G10, 37G35. Keywords. Discontinuous piecewise-linear map, codimension-2 bifurcation, organizing center.Résumé. Les connaissances sur le comportement d'applications linéaires par morceaux discontinues sont importantes pour de nombreuses applications. Une méthode puissante pourétudier la structure de bifurcation dans les espaces de paramètre 2D de telles applications est de détecter des points de bifurcation spécifiques de codimension 2, appelés centres organisateurs, et de décrire la structure de bifurcation dans leur voisinage. Dans ce travail, nous présentons les centres organisateurs pour une application linéaire par morceaux discontinue 1D sous forme générique, ce qui peutêtre utilisé comme une forme normale pour ces bifurcations d'autres applications discontinues 1D avec une discontinuité. Ces centres organisateurs apparaissent lorsque la continuité de la fonction du système est brisée en un point fixe. Le type d'un centre organisateur dépend des pentes de l'application linéaire par morceaux. Les centres organisateurs qui se produisent lorsque les pentes ont une valeur absolue plus petite que un ont déjàété décrits dans des travaux précédents, et nous nous concentrons donc sur la présentation des centres organisateurs qui se produisent si une ou plusieurs pentes ont des valeurs absolues plus grandes que un. Par là même, nous montronségalement que le comportement de chaque centre organisateur peutêtre expliqué en utilisant quatre scénariosélémentaires de bifurcation : les scénarios d'incrément de période et d'addition de période en domaine périodique, et les scénarios d'incrément de bandcount et d'addition de bandcount en domaine chaotique.Mots clefs...

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“…These two maps, characterized by a decreasing jump and an increasing jump, respectively, have been recently studied in [7] where an analysis of the socalled "principal tongues" ( [28], [29], [30], [12], [13], [4], [1], [2]) or "tongues of first degree" (Leonov,[26], [27], Mira [31], [32]) is provided. We recall that the principal tongues are regions, in the parameters' space, where a periodic cycle of period k exists, with one periodic point on a side of the discontinuity point and the remaining (k − 1) points on the other side (for any integer k > 1).…”

Section: A Family Of Piecewise Linear Maps With Two Discontinuitiesmentioning

confidence: 99%

“…In this section we are interested to analyze the behavior of the continuous map (1) for increasing values of λ up to the limiting case λ → +∞. map with one discontinuity.…”

Section: Lr(k − 1 Times)r ⊕ Lr(k Times)r = Lr(k − 1 Times)rlmentioning

confidence: 99%

See 1 more Smart Citation

Periodic Cycles and Bifurcation Curves for One-Dimensional Maps with Two Discontinuities

Bischi¹,

Gardini²,

Merlone³

2009

Journal of Dynamical Systems and Geometric Theories

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Abstract. Starting from a family of discontinuous piece-wise linear one-dimensional maps, recently introduced as a dynamic model in social sciences, we propose a geometric method for finding the analytic expression of the bifurcation curves, in the space of the parameters, that bound the regions characterized by the existence of stable periodic cycles of any period. The conditions for the creation and the destruction of periodic cycles, as well as the analytic expressions of the bifurcation conditions, are obtained by studying the occurrence of border-collision bifurcations. In this paper we consider the case of maps formed by three linear portions separated by two discontinuity points. After summarizing the bifurcation structure associated with one-dimensional maps with only one discontinuity point, we show how this is modified by the introduction of a second discontinuity point. Finally we show how the considered map can be obtained as the limit case of a family of continuous maps as a parameter is increased without bounds, and we show how the low period cycles, which are typical of the discontinuous map we consider, emerge from the more complex (i.e. chaotic) behaviors observed in the continuous maps when a parameter value is large enough. From the point of view of the social application the increasing values of the parameter can be interpreted as higher degrees of impulsivity of the agents involved in binary decisions.

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On multi-parametric bifurcations in a scalar piecewise-linear map

Cited by 85 publications

References 71 publications

The bandcount increment scenario. I. Basic structures

The bandcount increment scenario. I. Basic structures

Breaking the continuity of a piecewise linear map

Periodic Cycles and Bifurcation Curves for One-Dimensional Maps with Two Discontinuities

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