<p>This study presents two main contributions concerning the oscillatory behavior of Duffing-type nonlinear RLC circuits with a nonlinear voltage-charge capacitor. First, using the well-known Riccati-type technique, we establish a comparison theorem linking the oscillation of nonlinear differential equations to that of corresponding linear equations, showing that classical linear oscillation results can be extended to a wide class of nonlinear systems under suitable structural conditions. Second, we apply this theorem to Duffing-type nonlinear RLC electrical circuit models, deriving explicit conditions under which all solutions oscillate. This approach demonstrates how linear oscillation theory can be rigorously applied to nonlinear circuits. Moreover, the proposed approach is not restricted to circuit models and is applicable to differential equations with periodic coefficients, including special cases of nonlinear Mathieu-type differential equations. Such equations are known as models describing coefficient excitation and parametric resonance phenomena. Overall, these results indicate that linear oscillation theory provides a unified and effective analytical framework for the study of nonlinear differential equations arising in applied sciences and engineering.</p>

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A Note on the Linearized Oscillation Theorem for Nonlinear RLC Circuits of Duffing Oscillator Type

  • Kazuki Ishibashi,
  • Seiya Shirakusa

摘要

This study presents two main contributions concerning the oscillatory behavior of Duffing-type nonlinear RLC circuits with a nonlinear voltage-charge capacitor. First, using the well-known Riccati-type technique, we establish a comparison theorem linking the oscillation of nonlinear differential equations to that of corresponding linear equations, showing that classical linear oscillation results can be extended to a wide class of nonlinear systems under suitable structural conditions. Second, we apply this theorem to Duffing-type nonlinear RLC electrical circuit models, deriving explicit conditions under which all solutions oscillate. This approach demonstrates how linear oscillation theory can be rigorously applied to nonlinear circuits. Moreover, the proposed approach is not restricted to circuit models and is applicable to differential equations with periodic coefficients, including special cases of nonlinear Mathieu-type differential equations. Such equations are known as models describing coefficient excitation and parametric resonance phenomena. Overall, these results indicate that linear oscillation theory provides a unified and effective analytical framework for the study of nonlinear differential equations arising in applied sciences and engineering.