Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves

Sushko, Iryna; Agliari, Anna; Gardini, Laura

doi:10.1016/j.chaos.2005.08.107

Cited by 57 publications

(32 citation statements)

References 11 publications

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“…As a result, under variation of a parameter we may have a transition between attracting cycles of any period or transition from an attracting cycle to multi-band chaotic attractors. Investigation of dynamical systems with a piecewise smooth function, motivated from a theoretical point of view as well as by practical applications (Nusse and Yorke 1995;Nusse et al 1994;Avrutin and Schanz 2006;Sushko et al 2006), is a central topic of many scientific works published in recent years (see also Di Bernardo et al (2008) and Zhusubaliyev and Mosekilde (2003) for survey books and references therein).…”

Section: Properties Of the Central Mapmentioning

confidence: 99%

Border Collision Bifurcations in a Footloose Capital Model with First Nature Firms

et al. 2011

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Section: Properties Of the Central Mapmentioning

confidence: 99%

Border Collision Bifurcations in a Footloose Capital Model with First Nature Firms

et al. 2011

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“…There is an enormous literature on the study of PWS continuous maps, see, for example, [1,18,[27][28][29][30]32,[34][35][36][37][38][39][40]. It is well known from these works that the bifurcations in the one-dimensional unimodal PWL continuous maps have been completely understood.…”

Section: Introductionmentioning

confidence: 99%

Period-adding and the Farey tree structure in a class of one-dimensional discontinuous nonlinear maps

2016

Nonlinear Dyn

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In this paper, we consider the dynamics of a class of one-dimensional discontinuous nonlinear maps which is linear on one side of the phase space and nonlinear on the other side. The period-adding phenomena of the map are investigated by studying the period-m orbit which has m − 1 iterations on the linear side and one iteration on the nonlinear side, where m ≥ 2 is an integer. Analytical conditions for the existence and stability of such kind of periodic orbits are found. Our numerical simulations suggest that the stable period-m orbit of this type plays important roles in the dynamics of the map. As the bifurcation parameter varies, such kind of periodic orbits form finite or infinite period-adding sequences. Between those "primary" periodic orbits, there are alternative series of other periodic orbits organized by the Farey tree structure or chaotic orbits.

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“…As an example, he inserted the cautious mechanism in the simplest case of pure exchange economy introduced in the well-known works of Day (see [4]) and Day and Pianigiani ([5]). The analytical description of the model resulted expressed by a bimodal 1D continuous piecewise linear map which showed the possibility of erratic the system functions at the bifurcation value ( [24,25]). One more peculiarity of piecewise smooth maps is robustness of chaotic attractors [3], that means persistence of a chaotic attractor under parameter perturbations.…”

Section: Introductionmentioning

confidence: 99%

Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model

Foroni

Avellone

Panchuk

2015

Chaos, Solitons & Fractals

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a r t i c l e i n f o Article history: Available online xxx a b s t r a c tTâtonnement processes are usually interpreted as auctions, where a fictitious agent sets the prices until an equilibrium is reached and the trades are made. The main purpose of such processes is to explain how an economy comes to its equilibrium. It is well known that discrete time price adjustment processes may fail to converge and may exhibit periodic or even chaotic behavior. To avoid large price changes, a version of the discrete time tâtonnement process for reaching an equilibrium in a pure exchange economy based on a cautious updating of the prices has been proposed two decades ago. This modification leads to a one dimensional bimodal piecewise smooth map, for which we show analytically that degenerate bifurcations and border collision bifurcations play a fundamental role for the asymptotic behavior of the model.

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Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves

Cited by 57 publications

References 11 publications

Border Collision Bifurcations in a Footloose Capital Model with First Nature Firms

Border Collision Bifurcations in a Footloose Capital Model with First Nature Firms

Period-adding and the Farey tree structure in a class of one-dimensional discontinuous nonlinear maps

Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model

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