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<p>Nonlinear oscillators, including the Duffing, Van der Pol and Duffing-van der Pol systems, play a&#xa0;crucial role in modeling complex dynamical phenomena across various scientific and engineering disciplines. This study introduces the Hermite wavelet method (HWM) as an efficient and accurate computational framework for solving these nonlinear oscillators under diverse parameter configurations. Utilizing the operational matrix of integration, HWM transforms nonlinear equations into systems of algebraic equations. The proposed method demonstrates superior accuracy, rapid convergence, and computational efficiency, achieving high precision even with a&#xa0;small number of collocation points. Comparative analysis with existing numerical and techniques including RK4, validates the robustness of HWM through numerical experiments, visual comparisons, and close agreement with exact solutions. Additionally, phase portraits are generated and analyzed to characterize the complex behavior of the system and to visualize its stability patterns. HWM’s simplicity and versatility make it a&#xa0;powerful tool for addressing nonlinear oscillatory problems, with potential extensions to higher-dimensional systems, fractional-order equations, and chaotic dynamics. This study establishes HWM as a&#xa0;reliable and practical approach for advancing the computational methods used in nonlinear dynamics.</p>

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Study of nonlinear oscillators using hermite wavelet operational matrix approach

  • 期刊论文

  • 发表日期：

    2026年7月31日

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  • Drishti Rajput,
  • Vinod Mishra

摘要

Nonlinear oscillators, including the Duffing, Van der Pol and Duffing-van der Pol systems, play a crucial role in modeling complex dynamical phenomena across various scientific and engineering disciplines. This study introduces the Hermite wavelet method (HWM) as an efficient and accurate computational framework for solving these nonlinear oscillators under diverse parameter configurations. Utilizing the operational matrix of integration, HWM transforms nonlinear equations into systems of algebraic equations. The proposed method demonstrates superior accuracy, rapid convergence, and computational efficiency, achieving high precision even with a small number of collocation points. Comparative analysis with existing numerical and techniques including RK4, validates the robustness of HWM through numerical experiments, visual comparisons, and close agreement with exact solutions. Additionally, phase portraits are generated and analyzed to characterize the complex behavior of the system and to visualize its stability patterns. HWM’s simplicity and versatility make it a powerful tool for addressing nonlinear oscillatory problems, with potential extensions to higher-dimensional systems, fractional-order equations, and chaotic dynamics. This study establishes HWM as a reliable and practical approach for advancing the computational methods used in nonlinear dynamics.