Codimension-three bifurcations: Explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps

Аврутин, Виктор; Schanz, Michael; Banerjee, Soumitro

doi:10.1103/physreve.75.066205

Cited by 38 publications

(57 citation statements)

References 19 publications

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“…Therefore, we consider in the current work the case of negative discontinuity, leaving the more complex case of positive discontinuity for future investigations. As shown in the study by Avrutin et al (2007b), in this case the region of periodic dynamics of system (2.1) is organized mainly by two codimension-3 big bang bifurcations occurring at the points ða; b; mÞZ ð0; 0; 0Þ and ða; b; mÞZ ð0; 0; 1Þ. Owing to the symmetry property of system (2.1) given by ðða; b; mÞ/ ðb; a;KmKlÞÞ 0 ðx /KxÞ; ð2:2Þ…”

Section: Investigated Systemmentioning

confidence: 64%

“…It follows directly from the results presented in Avrutin et al (2007b) that in the part of the parameter space we consider here, the only possible stable periodic orbits are O LR n . Hereby for each n the existence region P LR n of the orbit O LR n in the three-dimensional parameter space is bounded by two border collision bifurcation surfaces which can be straightforward calculated using the conditions that the first (respectively last) point of the orbit O LR n collides with the boundary xZ0 from the left (respectively right) side.…”

Section: Periodic Dynamics and Boundaries Of The Chaotic Domainmentioning

confidence: 79%

“…As shown for instance in the study by Avrutin et al (2007b) for the parameter l, only three characteristic cases have to be considered, namely l 2 fK1; 0; 1g. The case of the continuous map (lZ0) is already well investigated for both periodic and chaotic domains (Maistrenko et al 1993;Nusse & Yorke 1995;Di Bernardo et al 1999;Zhusubaliyev & Mosekilde 2003;Avrutin et al 2007a).…”

Section: Investigated Systemmentioning

confidence: 99%

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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: 64%

Section: Periodic Dynamics and Boundaries Of The Chaotic Domainmentioning

confidence: 79%

Section: Investigated Systemmentioning

confidence: 99%

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The bandcount increment scenario. I. Basic structures

2008

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“…Typically, at such a point in the parameter space an infinite number of existence regions of periodic orbits emerges. In the case that these orbits are stable, their stability regions may cover an extended part of the parameter space and hence explain several 1D bifurcation scenarios observed in parts of the parameter space located far away from the organizing center [9]. Organizing centers can also lead to the appearance of unstable periodic orbits.…”

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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“…Concerning the periodic solutions, we considered the characteristic case of a negative jump (l!0, whereby the investigated system can always be reduced to the case lZK1 by a suitable scaling) and the case a!0, where the periodic domain is organized by the period increment scenario (Avrutin et al 2007b) with coexisting attractors, sometimes also referred to as 'multi-stability'. This scenario is formed by a sequence of periodic orbits O s LR k with kZ1, 2, ., whose stability regions P s LR k overlap pairwise.…”

Section: Introductionmentioning

confidence: 99%

The bandcount increment scenario. III. Deformed structures

2008

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Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely. In a series of three papers, we investigated 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 Part I, the basic structures in the chaotic region are explained by the bandcount increment scenario. In Part II, fine self-similar substructures nested into the bandcount increment scenario are explained by the bandcount-adding and -doubling scenarios, nested into each other ad infinitum. Hereby, we fixed in both previous parts one of the parameters to a non-generic value, and studied the remaining two-dimensional parameter subspace. In this Part III, finally we investigated the structures under variation of this third parameter. Remarkably, this step is the most important with respect to practical applications, since it cannot be expected that these operate exactly at the previously investigated specific value.

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Codimension-three bifurcations: Explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps

Cited by 38 publications

References 19 publications

The bandcount increment scenario. I. Basic structures

The bandcount increment scenario. I. Basic structures

Breaking the continuity of a piecewise linear map

The bandcount increment scenario. III. Deformed structures

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