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<p>We analytically describe the oscillations of fractional-derivative versions of the Duffing oscillator and Brusselator. We consider low-level fractionality in the highest derivative such that the leading-order problem is the solvable harmonic oscillator. The method of multiple scales is used to determine the slow evolution of the oscillator and, in particular, the interplay between the effects of the fractional derivatives, damping, nonlinearities and forcing. For the Duffing oscillator, we study resonance due to forcing. For the Brusselator, we consider the bifurcation to limit cycles. A key result is that for both oscillators the evolution of the amplitude depends on the sum of the fractional derivatives.</p>

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Dynamics of nonlinear oscillators with low-level fractional derivatives

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  • 发表日期：

    2026年7月24日

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  • Thomas W. Carr

摘要

We analytically describe the oscillations of fractional-derivative versions of the Duffing oscillator and Brusselator. We consider low-level fractionality in the highest derivative such that the leading-order problem is the solvable harmonic oscillator. The method of multiple scales is used to determine the slow evolution of the oscillator and, in particular, the interplay between the effects of the fractional derivatives, damping, nonlinearities and forcing. For the Duffing oscillator, we study resonance due to forcing. For the Brusselator, we consider the bifurcation to limit cycles. A key result is that for both oscillators the evolution of the amplitude depends on the sum of the fractional derivatives.