DefinePK

Abstract

In physics, biology, engineering, and other scientific fields, partial differential equations (PDEs) are fundamental tools for modelling dynamic systems. Stability and bifurcation are two basic ideas that frequently control how their solutions behave, particularly in different situations. The conditions under which solutions to nonlinear PDEs either stay stable or experience qualitative changes as a result of parameter variation are examined in this research paper. To investigate the dynamics of representative systems like reaction-diffusion equations and the Navier-Stokes equations, numerical simulations are used in conjunction with analytical methods such as Lyapunov stability theory, linearisation techniques, and bifurcation theory, including pitchfork and Hopf bifurcations. In-depth analysis is done on how boundary conditions, eigenvalue spectra, and nonlinear feedback affect the behaviour of the solution. In addition to offering useful insights into the critical transitions seen in real-world systems like fluid flow, chemical reactions, and biological pattern formation, our goal is to advance the theoretical understanding of solution trajectories in PDEs.