Abstract
We train an artificial neural network which distinguishes chaotic and regular dynamics of the two-dimensional Chirikov standard map. We use finite length trajectories and compare the performance with traditional numerical methods which need to evaluate the Lyapunov exponent (LE). The neural network has superior performance for short periods with length down to 10 Lyapunov times on which the traditional LE computation is far from converging. We show the robustness of the neural network to varying control parameters, in particular we train with one set of control parameters, and successfully test in a complementary set. Furthermore, we use the neural network to successfully test the dynamics of discrete maps in different dimensions, e.g. the one-dimensional logistic map and a three-dimensional discrete version of the Lorenz system. Our results demonstrate that a convolutional neural network can be used as an excellent chaos indicator.
Highlights
The main cause of errors is due to fractal phase space structures at the boundaries between chaotic and regular dynamics
Trajectories launched in these regions yield sticky trajectories which can mimick regular ones for long times, only to escape at even larger times into the chaotic sea
We used a network trained with twodimensional standard map data to classify chaotic and regular dynamics in one- and three-dimensional maps
Summary
Chaotic dynamics exists in many natural systems, such as heartbeat irregularities, weather and climate. Chaotic dynamics exists in many natural systems, such as heartbeat irregularities, weather and climate1,2 Such dynamics can be studied through the analysis of proper mathematical models which generate nonlinear dynamics and determenistic chaos. In practice one integrates the tangent dynamics along a given trajectory and averages a finite time Lyapunov exponent λ (t). For a regular trajectory λN ∼ 1/N and λ = 0, at variance to a chaotic trajectory for which λN saturates at λ at a time N ≈ Tλ This saturation, and the value of λ can be safely confirmed and read off only on time scales N ≈ 102..103Tλ , without becoming a quantifiable distinguisher of the two types of trajectories, see Fig.. For small values of K e.g. K = 0.5 (Fig.1(a)) most of these orbits persist, with tiny regions of chaotic dynamics appearing which are not visible on the presented plotting scales. Further increase of K leads to a flooding of the regular islands by the chaotic sea
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