Hopf bifurcation and transition to chaos in Lotka-Volterra equation

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

doi:10.1007/bf00275811

Cited by 34 publications

(16 citation statements)

References 11 publications

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“…Ameodo et al (1980), (1982) use a mixture of numerical evidence and theoretical argument to support the hypothesis of a Silnikov-type strange attractors in three and higher dimensions. See also Takeuchi and Adachi (1984) and Gardini et al (1989). May and Leonard (1975) show that other kinds of wild behavior are possible in LotkaVolterra systems of three competitors.…”

Section: Relation To Lotka-volterra Models and To Other Lit-eraturementioning

confidence: 99%

Chaos in game dynamics

Skyrms

1992

J Logic Lang Inf

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Two examples demonstrate the possibility of extremely complicated non-convergent behavior in evolutionary game dynamics. For the Taylor-Jonker flow, the stable orbits for three strategies were investigated by Zeeman. Chaos does not occur with three strategies. This papers presents numerical evidence that chaotic dynamics on a "strange attractor" does occur with four strategies. Thus phenomenon is closely related to known examples of complicated behavior in Lotka-Volterra ecological models.

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Section: Relation To Lotka-volterra Models and To Other Lit-eraturementioning

confidence: 99%

Chaos in game dynamics

Skyrms

1992

J Logic Lang Inf

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“…Substituting i=1 : i (%) c i and i=1 ; i (%) c i for r and w respectively in Eqs. (10) yields that : i (%) and ; i (%) satisfies the scalar differential equations with initial value : 1 (0)=; 1 (0)=1 and : i (0)=; i (0)=0 for i 2. Solving the scalar differential equations, we find the singular point E (1, 1, 1) is an unstable focus with positive first focal value at curve 3= 2 +2= 1 =0 by Theorem 4.1 in [3].…”

Section: Bifurcations For a 3d Competitivementioning

confidence: 99%

Limit Cycles for the Competitive Three Dimensional Lotka–Volterra System

Xiao¹,

Li²

2000

Journal of Differential Equations

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In the first part of this paper, it is proved that the number of limit cycles of the competitive three-dimensional Lotka Volterra system in R 3 + is finite if this system has not any heteroclinic polycycles in R 3 + . In the second part of this paper, a 3D competitive Lotka Volterra system with two small parameters is discussed. This system always has a heteroclinic polycycle with three saddles. It is proved that there exists one parameter range in which the system is persistence and has at least two limit cycles, and there exists other parameter ranges in which the system is not persistence and has at least one limit cycle. Academic Press

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“…This feedback effect is discussed in greater detail in the following sections. 9 See, for instance, Koch (1974), Butler & Waltman (1981), Smith (1982), Cushing (1984), Gardini, Lupini & Messia (1989), Hofbauer & So (1994), Korobeinikov & Wake (1999), Hsu, Hwang & Kuang (2001), Loladze, Kuang, Elser & Fagan (2004), Feng & Hinson (2005; a collection of results for a class of such systems might be found in Zeeman (1993) and Zeeman & Zeeman (2002); for a more general discussion in the context of dynamical possibilities in three-dimensional differential equation systems, see Kuznetsov (1997).…”

Section: Final Modelmentioning

confidence: 99%

Macrodynamics of debt-financed investment-led growth with interest rate rules

Datta

2016

Journal of Post Keynesian Economics

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This paper demonstrates the diverse dynamical possibilities arising out of a simple macroeconomic model of debt-financed investment-led growth in the presence of interest rate rules. We show possibilities of convergence to steady state, growth cycles around it as well as various complex dynamics. We investigate whether, given this framework, the financial sector can provide endogenous bounds to an otherwise unstable system. The effectiveness of monetary policy in the form of a Taylor-type interest rate rule targeting capacity utilization is examined under this context.

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Hopf bifurcation and transition to chaos in Lotka-Volterra equation

Cited by 34 publications

References 11 publications

Chaos in game dynamics

Chaos in game dynamics

Limit Cycles for the Competitive Three Dimensional Lotka–Volterra System

Macrodynamics of debt-financed investment-led growth with interest rate rules

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