Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

Ghaziani, Reza Khoshsiar; Govaerts, W.; Sonck, Charlotte

doi:10.1016/j.nonrwa.2011.11.009

Cited by 61 publications

(3 citation statements)

References 23 publications

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“…These results were extended in the recent book [55] where bifurcations of the period-8 and period-16 orbits in this duopoly model were also studied. In reference [24] the same unfolding of the fold-flip bifurcation was found in a predator-prey map. Our studies extend these results to explain mechanisms of the appearance of hyperchaotic attractors.…”

Section: The Case B = 0 (Numerical Analysis Of the 2d Mirá Map)supporting

confidence: 53%

Scenarios for the creation of hyperchaotic attractors in 3D maps

et al. 2023

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We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map.

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Section: The Case B = 0 (Numerical Analysis Of the 2d Mirá Map)supporting

confidence: 53%

Scenarios for the creation of hyperchaotic attractors in 3D maps

et al. 2023

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“…Special windows are provided to help with maintaining systems, diagrams and curves when generating a large amount of data. In Reference [21] this software is extensively used to study the bifurcation behavior of fixed points and cycles in biological models. This software was also used in a study of resonance tongues in Reference [16] and in the analysis of an economic model in Reference [1].…”

Section: The Basics Of the Use Of Matcontmmentioning

confidence: 99%

Numerical Bifurcation Analysis of Homoclinic Orbits Embedded in One-Dimensional Manifolds of Maps

Neirynck

Govaerts

Kuznetsov

et al. 2018

ACM Trans. Math. Softw.

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We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing one-dimensional stable and unstable manifolds. The methods are implemented in M at C ont M, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits, and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows us to detect all known codimension 1 and 2 bifurcations in three-dimensional (3D) maps, including tangencies and generalized tangencies. M at C ont M provides a graphical user interface, enabling interactive control for all computations. As the prime new feature, we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing us to compute all homoclinic orbits to saddle points in 3D maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis, and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features.

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“…The discrete-time models sometimes have richer dynamical behaviors. For instance, the single-species discrete-time models have bifurcations, chaos and more complex dynamical behaviors (see [6,[12][13][14][15][16][18][19][20][21][22]). For the flip bifurcation and Hopf bifurcation of discrete models, see also [6,[12][13][14]24].…”

Section: Introductionmentioning

confidence: 99%

Complex dynamics of a discrete predator–prey system with a strong Allee effect on the prey and a ratio-dependent functional response

Fang

2018

Adv Differ Equ

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In this paper, we consider the complex dynamics of a discrete predator-prey system with a strong Allee effect on the prey and a ratio-dependent functional response, which is the discrete version of the continuous system in (Nonlinear Anal., Real World Appl. 16:235-249, 2014). First, by giving several examples to display the limitations and errors of the local stability of the equilibrium point p obtained in (Nonlinear Anal., Real World Appl. 16:235-249, 2014), we provide an easily verified and complete discrimination criterion for the local stability of this equilibrium. Then we study some properties of its discrete version, especially for the stability and bifurcation for the equilibrium point E 1 , which has not been considered in any literature to the best of our knowledge. By using the center manifold theorem and bifurcation theory, we consider the flip bifurcation of this system at E 1 and obtain the stability of the closed orbits bifurcated from E 1 . The numerical simulations not only show the correctness of our theoretical analysis, but also we find some new and interesting dynamics of this system.

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Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

Cited by 61 publications

References 23 publications

Scenarios for the creation of hyperchaotic attractors in 3D maps

Scenarios for the creation of hyperchaotic attractors in 3D maps

Numerical Bifurcation Analysis of Homoclinic Orbits Embedded in One-Dimensional Manifolds of Maps

Complex dynamics of a discrete predator–prey system with a strong Allee effect on the prey and a ratio-dependent functional response

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