An elementary approach to dynamics and bifurcations of skew tent maps

Lindström, Torsten; Thunberg, Hans

doi:10.1080/10236190801927462

Cited by 12 publications

(21 citation statements)

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“…Khan and Mitra use this method to calculate an invariant measure for a piecewise linear system; as does Matsumoto (2005), who develops a similar method. Lindström and Thunberg (2008), by contrast, obtain a necessary and sufficient condition under which a piecewise linear system has a Liapounov exponent larger than unity. (2012), Grandmont (1985Grandmont ( , 2008, and Bhattacharya and Majumdar (2007).…”

Section: Preliminariesmentioning

confidence: 91%

“…These studies deal with models that have an upward-sloping lower boundary of state variables with a slope flatter than 1. Khan and Mitra (2012) is concerned with what they call the Robinson-Solow-Srinivasan model (Khan and Mitra 2005) and use the results of Boyarsky and Scarowsky (1979) and Lindström and Thunberg (2008) in order to prove the existence of observable chaos. 1 Yano and Furukawa (2012) show that their endogenous-exogenous growth model is subject to a lower boundary of state variables similar to our two-sector model and, like this study, demonstrate the possibility of ergodic chaos by using the result of Sato and Yano (2012).…”

Section: Introductionmentioning

confidence: 99%

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Optimal ergodic chaos under slow capital depreciation

Sato

Yano

2013

Int J of Economic Theory

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It has been known that, in a standard two-sector dynamic model, slow capital depreciation may result in a unimodal system with a kinked peak and that this system may be topological chaos, which is unobservable in Grandmont's sense. However, whether or not slow capital depreciation can cause observable chaos has not been examined in the existing literature. The present study demonstrates that slow capital depreciation may cause ergodic chaos, which is observable, by using a recent result of Sato and Yano (2012).

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Section: Preliminariesmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Optimal ergodic chaos under slow capital depreciation

Sato

Yano

2013

Int J of Economic Theory

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“…Related statistical methods are the Akaike information criterion AIC (Akaike (1973)) and later modifications. These methods possess cross-validation properties, too, but the idea with those methods is to find a model that is as close as possible to the correct model if such a model exists.Recently, Lindström and Thunberg (2008) stressed that the generic dynamical properties of typical maps fitted to data differ from those of maps usually accepted for mechanistic modeling. In fact, the simplest nonlinear models used in statistical data analysis (continuous TAR(1) models) do not possess period-doublings at all, so their build-up of complicated dynamics is entirely different from the generic bifurcation routes in the mechanistic modeling approach (Devaney (1989) and Lindström and Thunberg (2008)).…”

mentioning

confidence: 99%

“…Recently, Lindström and Thunberg (2008) stressed that the generic dynamical properties of typical maps fitted to data differ from those of maps usually accepted for mechanistic modeling. In fact, the simplest nonlinear models used in statistical data analysis (continuous TAR(1) models) do not possess period-doublings at all, so their build-up of complicated dynamics is entirely different from the generic bifurcation routes in the mechanistic modeling approach (Devaney (1989) and Lindström and Thunberg (2008)). Yet, in one-dimension they still possess at most one periodic attractor for each parameter value and this property agrees with the corresponding property for many mechanistic single species models.…”

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confidence: 99%

Fitting TAR(1) models to Sheperd data

Lindström¹

2007

Proc Appl Math and Mech

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A number of results that will be reported in detail elsewhere are outlined. The fundamental results are that typical models fitted to data of nonlinear mechanistic models possess non-standard bifurcation routes and they in essential cases fit repellers to attractors of the true mechanistic map.In this paper we explain briefly the ideas of a study where mechanistic and data-based models in population biology are examined side by side. The mechanistic modeling approach starts from mechanisms assumed to take part in the process under study. For instance, a predator-prey model in continuous time may be written as follows (cf. Kuang and Freedman (1988)),The first term in the first equation describes growth, natural death, and competition, the second term describes predation. In the second equation, the first term describes the conversion of prey biomass into predator biomass, whereas the last term describe natural death of the predator population. Each term in the equations can thus be interpreted as a model of each mechanism that is taken into account. The different terms have been justified with various success, and the second term in the first equation has that far the most compelling mechanistic explanation (Holling (1959)). The mechanistic justification of the first term of the first equation has, on the other hand been called into question (cf. Kooi, Boer, and Kooijman (1998)). We restrict the study to one dimensional discrete population models, since in this case there exist a theory of their generic dynamical properties (Devaney (1989)). In many cases, resources are used by individuals who do not obtain enough resources for reproduction, which causes a decrease of population size (Łomnicki (1988)) and therefore a large class of one-dimensional population models possess humped right hand sides.The mechanistic model under study is the Sheperd model, which describes the growth of a single species population under non-linear intra-specific competition corresponding to a humped right hand side. Such models undergo the perioddoubling route to chaos in many cases. It is no difference if we select another model to describe the nonlinear competition mechanisms, in most cases the same bifurcations will happen if the growth rate parameter is increased (Devaney (1989)). In the single species case, there also exists sufficiently general theory for how many periodic attractors such maps may have, cf. Guckenheimer, Oster and Ipaktchi (1977). It turns out that the considered cases do not possess more than at most one periodic attractor.When ecological data are collected, the map describing the nonlinear mechanisms is hardly available, and we may call into question whether such a correct model even exists (Burnham and Andersson (2002)). In order to select an appropriate model, different approaches have been developed and most attempts use least squares selection or likelihood methods combined with some procedure that penalizes the number of parameters used in the model. One well-known procedure is cross-validation (CV) that e...

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“…The study of invariant densities is continued in [14]. In [12] the authors classify the dynamics of skew tent maps in terms of two bifurcation parameters. In [4] equi-topological entropy regions are called isentropes and connectedness of isentropes is verified for real multimodal polynomial interval maps with only real critical points.…”

Section: Introductionmentioning

confidence: 99%

Equi-topological entropy curves for skew tent maps in the square

Buczolich

Keszthelyi

2017

Mathematica Slovaca

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We consider skew tent maps T α,β (x) such that (α, β) ∈ [0, 1] 2 is the turning point of T α,β , that is, T α,β = β α x for 0 ≤ x ≤ α and T α,β (x) = β 1−α (1 − x) for α < x ≤ 1. We denote by M = K(α, β) the kneading sequence of T α,β and by h(α, β) its topological entropy. For a given kneading squence M we consider equikneading, (or equi-topological entropy, or isentrope) curves (α, ϕ M (α)) such that K(α, ϕ M (α)) = M . To study the behavior of these curves an auxiliary function Θ M (α, β) is introduced. For this function Θ M (α, ϕ M (α)) = 0, but it may happen that for some kneading sequences Θ M (α, β) = 0 for some β < ϕ M (α) with (α, β) still in the dynamically interesting quarter of the unit square. Using Θ M we show that the curves (α, ϕ M (α)) hit the diagonal {(β, β) : 0.5 < β < 1} almost perpendicularly if (β, β) is close to (1, 1). Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for M = RLLRC the curve (α, ϕ M (α)) is not orthogonal to the diagonal. On the other hand, for M = RLC it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J.C. Marcuard, M. Misiurewicz and E. Visinescu.

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An elementary approach to dynamics and bifurcations of skew tent maps

Cited by 12 publications

References 12 publications

Optimal ergodic chaos under slow capital depreciation

Optimal ergodic chaos under slow capital depreciation

Fitting TAR(1) models to Sheperd data

Equi-topological entropy curves for skew tent maps in the square

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