Nonlinear dynamics and stability of a strongly nonlinear damped cubic-quintic oscillator

Abstract This paper presents a stability analysis of a strongly nonlinear damped cubic-quintic oscillator using three approaches: the classical multiple scales (CMS) method, an enriched multiple scales (EMS) method based on homotopy perturbation, and numerical continuation via MatCont. Comparisons across different nonlinearity regimes reveal that CMS accuracy degrades substantially when the perturbation parameter $$\varepsilon$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> is not small. In the cubic-dominant case ( $$\alpha _3 = 10$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:math> , $$\alpha _5 = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , $$\varepsilon = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ), CMS overestimates peak amplitudes by approximately 60% and mislocates bifurcation points, whereas EMS predictions remain within 1–2% of numerical results. Even under strong cubic nonlinearity ( $$\alpha _3 = 100$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>100</mml:mn> </mml:mrow> </mml:math> ), EMS maintains agreement within 3% while CMS errors reach 25%. The EMS method also accurately captures both stable and unstable solution branches, with stability boundaries matching Floquet-based numerical detection to within 0.1%. These results suggest that EMS may serve as a useful analytical tool for strongly nonlinear oscillators where traditional perturbation methods lose accuracy.