Bifurcations and Transitions to Chaos in the Three-Dimensional Lotka–Volterra Map

Gardini, Laura; Lupini, R.; Messia, M. G.

doi:10.1137/0147031

Cited by 28 publications

(26 citation statements)

References 15 publications

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“…The dynamic behaviors observed in this model are not yet well understood from a theoretical point of view; present knowledge cannot completely define the nature of the bifurcations involved (besides the well-known ((local)) bifurcations cited in the above example). Dynamics qualitatively similar to those observed here have been found in other maps on the plane from several authors (among others: Gumowski and Mira, 1980;Argoul and Arneodo, 1984;Grebogi, Ott and Yorke, 1983;Mira, 1987; and in particular, on Lotka-Volterra maps: Gardini, Lupini, Mammana and Messia, 1987;Gumowski and Mira, 1980).…”

Section: Remarksupporting

confidence: 91%

A Nonlinear Model of the Business Cycle With Money and Finance (*)

Gallegati¹,

Gardini²

1991

Metroeconomica

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In this paper we analyze the working of a capitalist system with sophisticated financial institutions, where the Modigliani-Miller theorem does not apply, so that the financial part of the economy affects the working of the real side. This model, in which the real and the financial sectors are connected throught a simple portfolio approach, and where the ratio between cash flow and debt commitment affects the pace of investment, explains the emergence of instability and, more generally, the cyclical behavior of the economy as an internally generated phenomenon of a capitalist system. As far as the dynamic behavior of the model is concerned, we can envisage the following process: when investment activity is low, the system evolves toward a stationary state; as soon as the accumulation rate starts growing, the model shows the existence of limited cycles and, when the elasticity of investment to internal finance increases further, a chaotic dynamic appears. Finally, we expect a situation of financial crisis to emerge when the elasticity of investment to internal financing constraint goes beyond a certain level, i.e. when the realized flow of profits is not sufficiently high to validate the debt commitments. The working of the model is referred to an economics with no technical change, while its dynamic properties will be investigated by the use of numerical simulation.

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Section: Remarksupporting

confidence: 91%

A Nonlinear Model of the Business Cycle With Money and Finance (*)

Gallegati¹,

Gardini²

1991

Metroeconomica

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“…In several applied models (see, e.g., [1,[3][4][5]) it has been observed that closed invariant curves in three dimensional (3D for short) maps can undergo a kind of doubling bifurcation sequence. The bifurcation mechanism leading to such a dynamic behavior was announced as an open problem at the European Conference on Iteration Theory in 2006 (ECIT-06) [6].…”

Section: Introductionmentioning

confidence: 99%

“…As an example, in Fig.1 we show the closed invariant attracting curve and a sequence of the bifurcations mentioned above in the Lotka-Volterra model represented by the 3D map x = x + Rx(1 − x − ay − bz) y = y + Ry(1 − bx − y − az) z = z + Rz(1 − ax − by − z) (1) where the parameters R and a are fixed as R = 1, a = 0.5, and b is varied (taken from [5] and [3]). Our aim is to give a qualitative description of the possible mechanism of such bifurcations.…”

Section: Introductionmentioning

confidence: 99%

Doubling bifurcation of a closed invariant curve in 3D maps

Gardini

Sushko

2012

ESAIM: Proc.

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Abstract. The object of the present paper is to give a qualitative description of the bifurcation mechanisms associated with a closed invariant curve in three-dimensional maps, leading to its doubling, not related to a standard doubling of tori. We propose an explanation on how a closed invariant attracting curve, born via Neimark-Sacker bifurcation, can be transformed into a repelling one giving birth to a new attracting closed invariant curve which has doubled loops.

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“…Our aim is to investigate how well Runge-Kutta methods do at modelling ordinary differential equations by looking at the resulting maps as dynamical systems. Chaos in numerical analysis has been investigated before: the midpoint method in the papers by Yamaguti & Ushiki [1981] and Ushiki [1982], the Euler method by Gardini et al [1987], the Euler method and the Heun method by Peitgen & Richter [1986], and the Adams-Bashforth-Moulton methods in a paper by Prüfer [1985]. These studies dealt with the chaotic dynamics of the maps produced in their own right, without relating them to the original differential equations.…”

Section: Introductionmentioning

confidence: 99%

The Dynamics of Runge–kutta Methods

Cartwright

Piro

1992

Int. J. Bifurcation Chaos

163

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The first step in investigating the dynamics of a continuous-time system described by an ordinary differential equation is to integrate to obtain trajectories. In this paper, we attempt to elucidate the dynamics of the most commonly used family of numerical integration schemes, Runge–Kutta methods, by the application of the techniques of dynamical systems theory to the maps produced in the numerical analysis.

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Bifurcations and Transitions to Chaos in the Three-Dimensional Lotka–Volterra Map

Cited by 28 publications

References 15 publications

A Nonlinear Model of the Business Cycle With Money and Finance (*)

A Nonlinear Model of the Business Cycle With Money and Finance (*)

Doubling bifurcation of a closed invariant curve in 3D maps

The Dynamics of Runge–kutta Methods

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