Studying highly nonlinear oscillators using the non-perturbative methodology

Area of Science:

• Applied Mathematics
• Nonlinear Dynamics
• Engineering Mathematics

Background:

• Nonlinear oscillators (NOSs) are prevalent in various scientific and engineering fields.
• Traditional methods for analyzing strong NOSs often rely on perturbation techniques, which have limitations.

Purpose of the Study:

• To introduce and examine a novel non-perturbative methodology (NPM) for analyzing strong nonlinear ordinary differential equations (ODEs).
• To demonstrate the simplicity, efficiency, and accuracy of NPM compared to existing perturbation methods.

Main Methods:

• The study employs the general He's frequency formula (HFF) within the NPM framework.
• The NPM transforms nonlinear ODEs into equivalent linear ODEs, yielding new frequency and damping terms.
• Theoretical results are verified using numerical comparisons with Mathematical Software (MS).

Main Results:

• The NPM provides analytical representations for strong NOSs with reduced computational effort.
• Numerical comparisons show excellent consistency between theoretical and precise numerical solutions.
• The NPM overcomes limitations of Taylor expansion used in traditional perturbation methods.

Conclusions:

• The non-perturbative solution (NPS) derived from NPM is a more reliable tool for analyzing strong NOSs.
• NPM enables stability analysis, a capability lacking in older conventional approaches.
• The NPS is versatile and applicable to a wide range of nonlinear problems in applied science and engineering, particularly dynamical systems.