In this paper, a numerical study on the forced Duffing oscillator is performed, where its deterministic, stochastic, and fractional order dynamics are systematically investigated. Starting from the derivation of a planar dynamical system from a space-time fractional nonlinear model, it extends the formulation to include periodic forcing and Gaussian white noise to treat nonlinearity, external excitation, and uncertainty in a unified way. With the aid of the equilibrium manifold analysis, we find a rich structure of bifurcation points involving pitchfork and saddle-node bifurcations, which give rise to the global stability landscape from which we proceed to study the dynamics of the model. Three-dimensional phase space representations initial-condition dependence plots, and Poincaré section plots are used to show a hierarchy of regimes, including periodic, quasi-periodic, and deterministic chaos, with fractal sensitivity boundaries verifying sensitive dependence on initial conditions. Testing with surrogate data chaotic dynamics), positive Lyapunov exponents ($\lambda_1 \approx 0.1018$), and recurrence quantification analysis (DET $> 0.87$) further verifies the authenticity of the chaotic dynamics observed. The Fokker-Planck formalism offers a statistical view of the dynamics of noisy systems, showing how noise intensity affects stationary probability distributions and effective potential landscapes and how noise intensity affects the nature of the dynamics can be captured by power spectral analysis in the frequency domain. The results of the stochastic ensemble analysis show that moderate noise preserves chaotic signatures and gives rise to predictable statistical broadening, leading to final-state distributions that tend to normality following the central limit theorem. The conformable fractional-order extension ($\beta = 0.75$) shows that memory-dependent damping deforms the toroidal invariant sets in a systematic way over time, as it is able to model a history-dependent behavior that cannot be modeled using a classical integer-order model. A systematic bifurcation analysis is also presented to validate the choice of representative parameter values for the whole study, thus giving a parameter-dependent perspective of the transition between dynamical regimes.
Citation: Samina Samina, Maham Munawar, Adil Jhangeer, Mudassar Imran. Integrated diagnostic and visualization framework for characterizing nonlinear and chaotic oscillator dynamics[J]. AIMS Mathematics, 2026, 11(10): 33860-33910. doi: 10.3934/math.20261323
In this paper, a numerical study on the forced Duffing oscillator is performed, where its deterministic, stochastic, and fractional order dynamics are systematically investigated. Starting from the derivation of a planar dynamical system from a space-time fractional nonlinear model, it extends the formulation to include periodic forcing and Gaussian white noise to treat nonlinearity, external excitation, and uncertainty in a unified way. With the aid of the equilibrium manifold analysis, we find a rich structure of bifurcation points involving pitchfork and saddle-node bifurcations, which give rise to the global stability landscape from which we proceed to study the dynamics of the model. Three-dimensional phase space representations initial-condition dependence plots, and Poincaré section plots are used to show a hierarchy of regimes, including periodic, quasi-periodic, and deterministic chaos, with fractal sensitivity boundaries verifying sensitive dependence on initial conditions. Testing with surrogate data chaotic dynamics), positive Lyapunov exponents ($\lambda_1 \approx 0.1018$), and recurrence quantification analysis (DET $> 0.87$) further verifies the authenticity of the chaotic dynamics observed. The Fokker-Planck formalism offers a statistical view of the dynamics of noisy systems, showing how noise intensity affects stationary probability distributions and effective potential landscapes and how noise intensity affects the nature of the dynamics can be captured by power spectral analysis in the frequency domain. The results of the stochastic ensemble analysis show that moderate noise preserves chaotic signatures and gives rise to predictable statistical broadening, leading to final-state distributions that tend to normality following the central limit theorem. The conformable fractional-order extension ($\beta = 0.75$) shows that memory-dependent damping deforms the toroidal invariant sets in a systematic way over time, as it is able to model a history-dependent behavior that cannot be modeled using a classical integer-order model. A systematic bifurcation analysis is also presented to validate the choice of representative parameter values for the whole study, thus giving a parameter-dependent perspective of the transition between dynamical regimes.
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Samina Samina, Maham Munawar, Adil Jhangeer, Mudassar Imran. Integrated diagnostic and visualization framework for characterizing nonlinear and chaotic oscillator dynamics[J]. AIMS Mathematics, 2026, 11(10): 33860-33910. doi: 10.3934/math.20261323
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