From the Poincaré–Birkhoff Fixed Point Theorem to Linked Twist Maps: Some Applications to Planar Hamiltonian Systems

Pascoletti, Anna; Zanolin, Fabio

doi:10.1007/978-1-4614-7333-6_14

Cited by 3 publications

(5 citation statements)

References 15 publications

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“…Nonetheless, we show that it is still possible to prove the existence of chaotic dynamics for the associated Poincaré map via the LTMs technique dealing with a different geometrical configuration for orbits in the phase plane, in agreement with the results obtained in other contexts e.g. in [9,38]. Regarding the nonisochronicity of the centers, it was proven in [48,49] that the period of the orbits increases with the energy level in the settings that we are going to analyze.…”

Section: Introductionsupporting

confidence: 89%

“…Namely, as shown in different contexts e.g. in [9,38], a geometrical configuration connected with LTMs may be obtained even when the center position is not affected by a variation of a certain parameter, as long as the latter suitably influences the shape of the orbits. We show two such frameworks in Figures 6 and 7.…”

Section: A Biological Framework With Intraspecific Competitionmentioning

confidence: 99%

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Chaotic dynamics in the presence of medical malpractice litigation: a topological proof via linked twist maps for two evolutionary game theoretic contexts

Pireddu¹

2020

Preprint

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In the present work we reconsider the evolutionary game theoretic models by Antoci et al. (2016Antoci et al. ( , 2018 describing the dynamic outcomes arising from the interactions between patients and physicians, whose behavior is subject to clinical and legal risks. In particular, Antoci et al. (2016) analyzed the case of positive defensive medicine, while Antoci et al. (2018) dealt with the case of negative defensive medicine. We show that, when the models admit a nonisochronous center, it is possible to prove the existence of chaotic dynamics for the Poincaré map associated with those systems via the method of Linked Twist Maps (LTMs). To such aim we exploit in both frameworks, using a similar rationale, the periodic dependence on time of a model parameter influencing the position of the center and describing some risk associated with certain medical interventions, whose seasonal variation is empirically grounded. We also provide the ecological interpretation of the same model, proposed by Harvie et al. (2007) and connected with intraspecific competition and environmental carrying capacity in predator-prey settings. In this case, it is sensible to assume a seasonal variation both for the carrying capacities and for the intrinsic growth rates of the two populations. Although such parameters influence the shape of the orbits, but do not affect the center position, we show that it is still possible to prove the existence of chaotic dynamics for the associated Poincaré map via the LTMs technique dealing with a different geometrical configuration for orbits in the phase plane.

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Section: Introductionsupporting

confidence: 89%

Section: A Biological Framework With Intraspecific Competitionmentioning

confidence: 99%

Chaotic dynamics in the presence of medical malpractice litigation: a topological proof via linked twist maps for two evolutionary game theoretic contexts

Pireddu¹

2020

Preprint

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“…Example 1. We take α = 0.02, β = 0.01, χ = 0.7, ρ = 0.6, σ (I) = 3, σ (I I) = 2.5, c = 0.45, η = 0.3, δ = 1, µ = 1.2 and recall (15), System (MI) has a center in P (I) = (0.141, 0.167) while System (MI I) has a center in P (I I) = (0.170, 0.167).…”

Section: The Goodwin Model Modified Formulationmentioning

confidence: 99%

“…We stress, however, that the same assumption about a periodic variation between two different values made on any other model parameter would produce analogous results in terms of generated dynamics, since all parameters influence the center position (We remark that this is a sufficient, albeit not necessary, condition in order to apply the LTMs method whenever we deal with a nonisochronous center, as long as the switching times between the regimes described by the two different parameter values are large enough. As shown, for example, in [12,15], the LTMs technique can be used even when, in consequence of the periodic perturbation of one of the model parameters, the center position does not vary, but the shape of the orbits is modified in a suitable manner).…”

Section: Introductionmentioning

confidence: 99%

A Proof of Chaos for a Seasonally Perturbed Version of Goodwin Growth Cycle Model: Linear and Nonlinear Formulations

Pireddu

2023

Axioms

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We show the existence of complex dynamics for a seasonally perturbed version of the Goodwin growth cycle model, both in its original formulation and for a modified formulation, encompassing nonlinear expressions of the real wage bargaining function and of the investment function. The need to deal with a modified formulation of the Goodwin model is connected with the economically sensible position of orbits, which have to lie in the unit square, in contrast to what occurs in the model’s original formulation. In proving the existence of chaos, we follow the seminal idea by Goodwin of studying forced models in economics. Namely, the original and the modified formulations of Goodwin model are described by Hamiltonian systems, characterized by the presence of a nonisochronous center, and the seasonal variation of the parameter, representing the ratio between capital and output, which is common to both frameworks, is empirically grounded. Hence, exploiting the periodic dependence on time of that model parameter we enter the framework of Linked Twist Maps. The topological results valid in this context allow us to prove that the Poincaré map, associated with the considered systems, is chaotic, focusing on sets that lie in the unit square, and also when dealing with the original version of the Goodwin model. Accordingly, the trademark features of chaos follow, such as sensitive dependence on initial conditions and positive topological entropy.

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“…A broad but somewhat fuzzy distinction can be made between two groups of topological tools. One class consists of those devices that provide existence results directly on the grounds of how the involved functions interact with the topology of the space they operate upon; examples in this group are Brouwer or Schauder or Kakutani fixed point theorems [22,31,32], the Ważewski theorem [33,34] or the Birkhoff twist-map theorem [35][36][37][38]. A second group constitutes more complex constructions that attach some invariant to the map and then relate it to topological properties of the space.…”

mentioning

confidence: 99%

Introduction to: Topological degree and fixed point theories in differential and difference equations

Pera

Spadini

2021

Phil. Trans. R. Soc. A.

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From the Poincaré–Birkhoff Fixed Point Theorem to Linked Twist Maps: Some Applications to Planar Hamiltonian Systems

Cited by 3 publications

References 15 publications

Chaotic dynamics in the presence of medical malpractice litigation: a topological proof via linked twist maps for two evolutionary game theoretic contexts

Chaotic dynamics in the presence of medical malpractice litigation: a topological proof via linked twist maps for two evolutionary game theoretic contexts

A Proof of Chaos for a Seasonally Perturbed Version of Goodwin Growth Cycle Model: Linear and Nonlinear Formulations

Introduction to: Topological degree and fixed point theories in differential and difference equations

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