Chaotic and complex dynamical systems exhibit extreme sensitivity to initial conditions: small perturbations can lead to rapidly diverging trajectories, making deterministic prediction infeasible. However, their statistical properties responses to external perturbations can remain robust and quantifiable. Understanding how these statistical responses change from linear to nonlinear regimes is therefore essential for characterizing the predictability of chaotic systems. Here we investigate this contrast using the Lorenz-63 system, a classical low-dimensional model for atmospheric convection.
We study its response to weak sinusoidal forcing, which provides an idealized representation of periodic environmental influences such as seasonal or oscillatory atmospheric variability. Rather than focusing only on the linear frequency response, we aim to quantify the leading second-order nonlinear response generated by single-frequency and multi-frequency perturbations harmonics and combination frequencies in the output spectrum will be used to identify how quadratic interactions grow with forcing amplitude and frequency.
By comparing second-order nonlinear responses with their corresponding linear responses, we aim to establish a practical criterion for the validity range of linear response theory. This frequency-domain framework provides a systematic way to identify when linear approximations remain effective and when nonlinear effects become significant in chaotic systems.Such understanding can improve the interpretation of response-based approaches in more realistic climate and environmental systems subjected to periodic external variability.

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