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Discontinuity, Nonlinearity, and Complexity 14(4) (2025) 669--686 | DOI:10.5890/DNC.2025.12.005

Satyabhan Singh, Prajjwal Gupta, Anupam Priyadarshi

Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India

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Abstract

Reducing the planar systems into their normal form, diffeomorphism of the plane with a smooth discriminant curve for neighborhoods of each critical point is used. In this study, using co-ordinate transformations, a generalization of the normal form is developed to understand the dynamics at infinity of dynamical systems. With the obtained normal form of dynamical planar systems, the equilibrium points and their stability can be studied quite easily, as normal form simplifies the involved equations of dynamical systems. We study behavior at infinity of a predator-prey model (Rosenzweig McArthur Model) through Poincar\'{e} compactification using the normal form of the model. The extensive analysis of dynamics at infinity of the model reveals that due to the presence of a saddle at the poles, the flow of the system solutions is trapped in the positive plane, and hence the system has an attracting region in $R^2_+$. The qualitative behavior of equilibrium points at infinity can be understood by using the Poincar\'{e} compactification through its generalized normal form. These methods are frequently used to analyze the behavior of the escapes to infinity in a family of Hamiltonian systems so-called Manev-type problems. Further, the classical results of the Kepler problem can be obtained using Poincare Compactification in Manev problems. A variation of the Newtonian gravitational n-body problem with a potential that depends on the distance and velocity was studied due to Poincare compactification.

Acknowledgments

The first two authors SS and PG acknowledged the financial support provided by UGC, India while the corresponding author AP expressed his thanks to IoE BHU for its financial support. We are also thankful for various online free software such as P4 (https://mat.uab.cat/$\mathrm{\sim}$artes/p4/p4.htm) and pplane8 \\ (https://aeb019.hosted.uark.edu/pplane.html).

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