Basin Geometry, Period-Doubling Windows and Chaotic Dynamics in an Asymmetric Quintic van der Pol–Duffing–Helmholtz Oscillator

An asymmetric quintic van der Pol–Duffing–Helmholtz oscillator under harmonic forcing is analyzed using mean-corrected harmonic balance, basin computation, Poincaré shooting, Floquet multipliers and complementary chaos diagnostics. Because quadratic stiffness couples the zero Fourier mode to the fundamental, a zero-mean first-harmonic approximation is structurally inconsistent. The coupled mean–amplitude reduction is therefore benchmarked at several forcing levels against direct integration and against the conventional zero-mean truncation; higher-harmonic content is also reported to delimit its useful range. Here [Formula: see text] denotes the quadratic-stiffness asymmetry coefficient, [Formula: see text] the forcing frequency and [Formula: see text] the forcing amplitude. A symmetric [Formula: see text] reference isolates the equilibrium shift and stability asymmetry generated by the quadratic term. Basin fractions for coexisting period-1 and period-2 responses are checked under [Formula: see text]–[Formula: see text] grid refinement. At [Formula: see text], period-3 and period-6 branches undergo multiplier-confirmed flips near [Formula: see text] and [Formula: see text]. Their two-parameter continuation is extended substantially in frequency, with the period-3 curve developing a near-vertical turning region. Chaos is supported by converged positive largest Lyapunov exponents and an independent 0–1 test. Two Lyapunov atlases from distinct initial conditions quantify the basin dependence of the parameter-plane picture.