Oscillatory Behavior and Stability Switching in Fourth-Order Nonlinear Delay Differential Equations
Umar Ado *
Mathematics Department, Modibbo Adama University, Yola, Nigeria.
Bashir Inuwa
Mathematics Department, Modibbo Adama University, Yola, Nigeria.
Alhaji Tahir
Mathematics Department, Modibbo Adama University, Yola, Nigeria.
Saidu Garba
Mathematics Department, Modibbo Adama University, Yola, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
This Paper investigates the oscillatory behavior and stability transitions in three-dimensional fourth-order delay differential equations (DDEs), which are crucial for modeling complex systems with memory or timedelay effects. The primary aim is to establish sufficient conditions under which solutions exhibit oscillations and to identify criteria for stability switches driven by varying delay parameters. The methodology employs analytical techniques including characteristic equation analysis, normal form reduction, and integrability conditions to derive precise criteria for oscillation and bifurcation. The results reveal that under specific growth and delay conditions, the system exhibits sustained oscillatory behavior and transitions in stability can occur at critical parameter values. These findings deepen the understanding of delay-induced dynamics in higher-order systems and provide a rigorous theoretical foundation for future analysis and application in fields such as engineering, biology, and physics.
Keywords: Oscillatory behavior, stability switches, delay differential equations, fourth-order systems, analytical methods