It is well known that the locus of boundary crises in smooth systems contains gaps that give rise to periodic windows. We show that this phenomenon can also be observed in an impacting system, and that the mechanism by which these gaps are created is different. Namely, here gaps are created and disappear at points along the branches of boundary crises where they are intersected by branches of grazing bifurcations. We locate a novel type of double-crisis vertex which we call a grazing-crisis vertex. Additionally, we illustrate several types of basin-boundary metamorphosis that are intricately related with grazing bifurcations.
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Sci Rep
October 2024
Department of Mathematics, University of Kalyani, Kalyani, 741235, India.
Several experimental evidences and field data documented that zooplankton may alter its behavioral response in the presence of toxic phytoplankton, reducing its consumption to the point of starvation. This paper is devoted to the mathematical study of such interactions of toxic phytoplankton with grazer zooplankton. The non-toxic phytoplankton is assumed to adopt a density-dependent refuge strategy to avoid over-predation by zooplankton.
View Article and Find Full Text PDFChaos
October 2024
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China.
In this paper, we consider the grazing-sliding bifurcations in a dry-friction oscillator on a moving belt under periodic excitation. The system is a nonlinear piecewise smooth system defined in two zones whose analytical expressions of the solutions are not available. Thus, we obtain conditions of the existence of grazing-sliding orbits numerically by the shooting method.
View Article and Find Full Text PDFChaos
September 2024
Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan.
The neural oscillator model proposed by Matsuoka is a piecewise affine system that exhibits distinctive periodic solutions. Although such typical oscillation patterns have been widely studied, little is understood about the dynamics of convergence to certain fixed points and bifurcations between the periodic orbits and fixed points in this model. We performed fixed point analysis on a two-neuron version of the Matsuoka oscillator model, the result of which explains the mechanism of oscillation and the discontinuity-induced bifurcations such as subcritical/supercritical Hopf-like, homoclinic-like and grazing bifurcations.
View Article and Find Full Text PDFJ Math Biol
July 2024
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India.
Group defense in prey and hunting cooperation in predators are two important ecological phenomena and can occur concurrently. In this article, we consider cooperative hunting in generalist predators and group defense in prey under a mathematical framework to comprehend the enormous diversity the model could capture. To do so, we consider a modified Holling-Tanner model where we implement Holling type IV functional response to characterize grazing pattern of predators where prey species exhibit group defense.
View Article and Find Full Text PDFChaos
May 2024
Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville, Edwardsville, Illinois 62026-1805, USA.
The periodic motions of discontinuous nonlinear dynamical systems are very difficult problems to solve in engineering and physics. Until now, except for numerical studies, one cannot find a better way to solve such problems. In fact, one still has difficulty obtaining periodic motions in continuous nonlinear dynamical systems.
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