There are several critical points to consider here:
Numerical Noise. Higher tolerances of the ODE solvers may take larger steps, and in regions with rapidly changing dynamics; however, tighter tolerances may not capture the whole system behavior accurately. This can result in numerical noise and the appearance of spurious frequencies in the FFT.
To address the accuracy issues of numerical simulations:
(1) Refine a time step: Instead of drastically increasing tolerances, you might consider refining the time stepping in regions of interest. Consider a variable step solver that will take variable step algorithms which can automatically adjust the step size based on the local behavior of the system.
(2) Test Higher-Order Solvers, e.g. ODE113 and also, stiff solvers (e.g., ODE15s) which may be more suitable for stiff systems.
(3) Double-check Initial Conditions: In some unstable or very dynamic systems, small changes in initial conditions can lead to significant behavior changes.
All the best.
