Bifurcation and Stability Analysis of Nonlinear Systems with Computational Validation

Abstract

This paper explores the dynamical aspects of a family of nonlinear systems using a combination of analytical and computational approaches. The main aim is to examine the stability features and bifurcation effects, which are brought about by changes in important system parameters. A nonlinear dynamical model is developed and studied in order to determine the equilibrium point of a nonlinear dynamical model and the structural features of the model. System stability is checked with the help of analytical techniques, such as Jacobian-based eigenvalue analysis and Lyapunov criteria of stability. With these methods, equilibrium points can be categorized as stable, unstable and marginally stable. In addition, bifurcation analysis is performed to calculate the important parameter values where a qualitative change in system behavior takes place, e.g. saddle-node and Hopf bifurcations. Computational simulations are carried out to confirm the theoretical results based on the use of numerical integration. To explain the dynamic responses of the system in different conditions, time series analysis, phase portraits, and bifurcation diagrams are produced. The largest Lyapunov exponent is calculated as well to indicate the occurrence of chaotic behavior. The findings not only identify specific regions of stability but also show the influence of changes in the parameter on the transitions between the stable equilibria and periodic or chaotic dynamics. There is a good consistency between analytical predictions and the outcomes of the computed results. This combination method improves the accuracy of stability measurements as well as offers more profound understanding about complex nonlinear systems, and it can apply to engineering, physics, and applied mathematics.