Complete periodic behaviours of real and complex bang bang dynamical systems

Chang, Yen Chih; Cheng, Sui Sun

doi:10.1080/10236198.2013.816694

Cited by 4 publications

(8 citation statements)

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“…A classical result of Montgomery establishes that any pointwise periodic homeomeorphism in an euclidian space is periodic ( [15]). In the series of papers [6,7] and [9], Chang, Cheng and Wang provide three explicit examples of pointwise periodic maps that are not periodic. The examples given by these authors in the above mentioned references belong to the family of discontinuous maps: Gpx, yq " py, ´x ´ρy `signpyqq , where signpyq "…”

Section: Introductionmentioning

confidence: 99%

Pointwise periodic maps with quantized first integrals

Cima,

Gasull,

Mañosa

et al. 2020

Preprint

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In a series of papers, Chang, Cheng and Wang studied the periodic behavior of some piecewise linear maps in the plane. These examples are valuable under the light of the classical result of Montgomery about periodic homeomorphisms, since they are pointwise periodic but not periodic. We revisit them from the point of view of their properties as integrable systems. We describe their global dynamics in terms of the dynamics induced by the maps on the level sets of certain first integrals that we also find. We believe that some of the features that the first integrals exhibit are interesting by themselves, for instance the set of values of the integrals are discrete. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of some prescribed tiles of certain regular tessellations. The existence of these quantized integrals is quite novel in the context of discrete dynamic systems theory.

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Section: Introductionmentioning

confidence: 99%

Pointwise periodic maps with quantized first integrals

Cima,

Gasull,

Mañosa

et al. 2020

Preprint

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“…By [1][2][3], we have observed the importance of nonnegative solutions of (7) with = 0, 1, and −1, respectively. However, we also realize the difficulties of analyzing the solution of (7) with ∉ {−1, 0, 1} since the breaking of into companion and error sequences is not applicable for all the solutions anymore.…”

mentioning

confidence: 99%

“…Some of these behaviors can be explained (see [1][2][3]) but some not (at least to the best of our knowledge). Similar models of piecewise constant dynamic systems which exhibit similar behaviors with parameters can be found in many recent investigations; see for examples [4][5][6][7][8][9][10] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…In [1][2][3], we are lucky to obtain complete information about the periodic solutions of (4) when = 1 and = 1, 0, or −1 by breaking the solutions into two sequences, a companion and an error (more specifically, for a solution = { } ∈N of (4) with = 1 and ∈ {−1, 0, 1}, the companion sequence is = { } ∈N which is an even integral sequence so that = 2[(1/2) ], where 2[ /2] is the greatest even integer that is less than or equal to ∈ R, and the error sequence = { } ∈N is defined by = − , ∈ Z).…”

Section: Introductionmentioning

confidence: 99%

“…In [1][2][3], we conducted complete analyses on the periodic solutions (7) with = 1 and ∈ {−1, 0, 1}. By means of breaking a solution into two subsequences, the companion sequence and the error sequence, we successfully show that both the companion sequence and error sequence are periodic so that is periodic.…”

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confidence: 99%

See 2 more Smart Citations

Nonnegative Periodic Solutions of a Three-Term Recurrence Relation Depending on Two Real Parameters

Chang

Cheng

Yeh

2017

Discrete Dynamics in Nature and Society

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Simple dynamic systems representing time varying states of interconnected neurons may exhibit extremely complex behaviors when bifurcation parameters are switched from one set of values to another. In this paper, motivated by simulation results, we examine the steady states of one such system with bang-bang control and two real parameters. We found that nonnegative and negative periodic states are of special interests since these states are solutions of linear nonhomogeneous three-term recurrence relations. Although the standard approach to analyse such recurrence relations is the method of finding the general solutions by means of variation of parameters, we find novel alternate geometric methods that offer the tracking of solution trajectories in the plane. By means of this geometric approach, we are then able, without much tedious computation, to completely characterize the nonnegative and negative periodic solutions in terms of the bifurcation parameters.

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