Calculation the maximum Lyapunov exponent for the oscillatory system of Duffing with a degree memory

In the study of nonlinear systems, one of the important problems is the determination of the type of oscillations-periodic, quasi-periodic, random, chaotic. It is especially difficult to distinguish between quasi-periodic oscillations from chaotic and random oscillations, since quasi-periodic oscillations often have a very complex shape, visually weakly distinguishable from «random». A feature of chaotic oscillations is their high sensitivity to small changes in the initial conditions. Therefore, one of the most reliable ways of detecting chaos is to determine the rate of run-off of trajectories, which is estimated using the Lyapunov exponent spectrum. Using the construction of the spectrum of Max Lyapunov exponents, depending on the values of the control parameters, chaotic regimes of the Duffing fractal oscillator with variable power memory were found, and its phase trajectories.