A mathematical framework for analyzing stability and bifurcation in nonlinear dynamical systems
Amanveer Singhamanveer.singh.orp@chitkara.edu.inCentre of Research Impact and Outcome (CRIO)Chitkara UniversityRajpura, Punjab, 140417, IndiaView full profile → , Chirag R. Trivedichiragkumar.trivedi270034@paruluniversity.ac.inDepartment of Applied Sciences and HumanitiesFaculty of Engineering and TechnologyParul UniversityVadodara, Gujarat, 391760, IndiaView full profile → , *Samir N. AjaniCorresponding authorsamir.ajani@gmail.comSchool of Computer Science and EngineeringRamdeobaba University (RBU)Nagpur, Maharashtra, 440013, IndiaView full profile → , V. Selva Sharmilasharmila-maths@dayanandasagar.eduDepartment of MathematicsDayanand Sagar College of EngineeringBangalore, Karnataka, 560111, IndiaView full profile → , Vaishali Sunilsingh Bayasvaishali.bayas1@vit.eduDepartment of Electronics and Telecommunication EngineeringVishwakarma Institute of TechnologyPune, Maharashtra, 411037, IndiaView full profile → , Shweta Bandhekarshweta.bandhekar@rungta.orgDepartment of Computer Science and EngineeringRungta College of Engineering and TechnologyRungta International Skills UniversityBhilai, Chhattisgarh, 490024, IndiaView full profile → , Delsy Jyothidelsy.jyothi@presidencyuniversity.inPresidency School of Computer Science and EngineeringPresidency UniversityBangalore, Karnataka, 560119, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Apr 2026
- Published Online:
- 27 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2833
- Pages:
- 3231–3243
Abstract
The behavior of nonlinear dynamical systems is complex and demands systematic mathematical tools to understand their stability, transitions and the dependence of the behavior on the parameters of the system. This study proposes a mathematical model of analyzing the phenomena of equilibrium stability and bifurcation in the nonlinear dynamics system. The framework develops nonlinear state equations, finds equilibrium points and uses Jacobian linearization of the system to describe the local behavior of the system. Eigenvalue analysis is used to identify stability conditions and changes in the control parameters are explored to look for critical bifurcation points. The proposed method combines the stability criteria, the parameter sensitivity and bifurcation detection in a unique analytical approach. To systematically investigate changes in the dynamics of a system near the critical conditions, mathematical algorithms are developed. The framework offers an efficient way to forecast qualitative changes, to determine stability regions and to perform robust analysis of nonlinear systems in various scientific and engineering contexts.
Keywords
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References
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