The Geometry of Change: Qualitative Analysis and Bifurcation in Non-Linear Differential Equations

Author(s)	Dr. Deepak Gupta, Er Abhishek Chaurasiya, Aditi Mishra
Country	India
Abstract	Analytical solutions to non linear equations are notoriously difficult and often impossible to write down exactly. For systems that model real world dynamics, such as metabolic pathways in the body or turbulence in fluids exact algebraic formulas are known not to exist. This paper looks at the geometric approach started by Henri Poincaré. That approach moves the focus from finding numbers to mapping the overall structure of the system. We study the math of phase space how to check stability using Jacobian linearization and the shape changes that mark bifurcation thresholds. By evaluating shifts in predator prey dynamics and the boundary conditions governing deterministic chaos within atmospheric models this study provides a rigorous exposition of dynamical systems theory establishing dynamical systems theory as the foundation for algorithmic modeling and automated engineering frameworks.
Keywords	Jacobian Matrix, Non-Linear Differential Equations, Saddle-Node Bifurcation and Structural Instability
Field	Mathematics
Published In	Volume 8, Issue 6, November-December 2026
Published On	2026-11-03

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