Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: unbounded chaotic sets

Makrooni, Roya; Khellat, Farhad; Gardini, Laura

doi:10.1080/10236198.2015.1045893

Cited by 12 publications

(12 citation statements)

References 53 publications

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“…0 is reached again. That is, the trajectories cannot be divergent, as it also occurs in the invertible case studied in [14] and in the non-invertible case of Range AI considered in [13].…”

Section: Preliminary Propertiesmentioning

confidence: 87%

“…As shown above, the branch f R becomes tangent to the main diagonal when (13) holds, and it intersects the main diagonal in two distinct points for b F R ð Þ , b , 0: Recall that in generic cases, a monotone increasing continuous map can only have fixed point whose stability is alternating. Since f 0 R ðxÞ .…”

Section: Preliminary Propertiesmentioning

confidence: 95%

“…In [13] we have considered the case of non-invertible map f ðxÞ in a particular range in which the trajectories cannot be divergent:…”

Section: Introductionmentioning

confidence: 99%

“…In particular, much attention has been devoted to the square-root singularities in impact oscillators, following the works of Nordmark ([17], [18]). In this paper we consider the PWS map proposed in [13] given by…”

Section: Introductionmentioning

confidence: 99%

“…As in [11] and [13] we consider system (1) for positive values of the parameter m, and for any m . 0 the transformation ðx; a; b; mÞ !…”

Section: Introductionmentioning

confidence: 99%

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Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: divergence and bounded dynamics

Makrooni

Khellat

Gardini

2015

Journal of Difference Equations and Applications

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In this work we continue the study of a family of 1D piecewise smooth maps, defined by a linear function and a power function with negative exponent, proposed in engineering studies. The range in which a point on the right side is necessarily mapped to the left side, and chaotic sets can only be unbounded, has been already considered. In this work we are characterizing the remaining ranges, in which more iterations of the right branch are allowed and in which divergent trajectories occur. We prove that in some regions a bounded chaotic repellor always exists, which may be the only nondivergent set, or it may coexist with an attracting cycle. In another range, in which divergence cannot occur, we prove that unbounded chaotic sets always exist. The role of particular codimension-two points is evidenced, associated with fold bifurcations and border collision bifurcations (BCBs), related to cycles having the same symbolic sequences. We prove that they exist related to the border collision of any admissible cycle. We show that each BCB, each fold bifurcation and each homoclinic bifurcation is a limit set of infinite families of other BCBs.

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“…0 is reached again. That is, the trajectories cannot be divergent, as it also occurs in the invertible case studied in [14] and in the non-invertible case of Range AI considered in [13].…”

Section: Preliminary Propertiesmentioning

confidence: 87%

Section: Preliminary Propertiesmentioning

confidence: 95%

“…In [13] we have considered the case of non-invertible map f ðxÞ in a particular range in which the trajectories cannot be divergent:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…As in [11] and [13] we consider system (1) for positive values of the parameter m, and for any m . 0 the transformation ðx; a; b; mÞ !…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: divergence and bounded dynamics

Makrooni

Khellat

Gardini

2015

Journal of Difference Equations and Applications

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Alternating Smooth and Nonsmooth Bifurcations in a Discontinuous Linear-Power Map

Gardini

Makrooni

Sushko

2017

Trends in Mathematics

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Nonlinear dynamics and game-theoretic modeling in economics and finance

Bischi,

Baiardi,

Lamantia

et al. 2024

Ann Oper Res

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In this foreword to the Special Issue “Nonlinear dynamics and game-theoretical modeling in economics and finance” we review the contributions in the issue highlighting the economic results and the connections with aspects of dynamic analysis. Indeed, the common theme of the contributions is the focus on system dynamics and the latest analytical techniques. This issue is devoted to celebrating the 70th birthday of Professor Laura Gardini, who inspired a generation of scholars in the study of discrete-time systems and global methods of analysis.

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Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: unbounded chaotic sets

Cited by 12 publications

References 53 publications

Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: divergence and bounded dynamics

Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: divergence and bounded dynamics

Alternating Smooth and Nonsmooth Bifurcations in a Discontinuous Linear-Power Map

Nonlinear dynamics and game-theoretic modeling in economics and finance

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