Transient Dynamics of the Lorenz System with a Parameter Drift

Abstract

Nonautonomous dynamical systems help us to understand the implications of real systems which are in contact with their environment as it actually occurs in nature. Here, we focus on systems where a parameter changes with time at small but non-negligible rates before settling at a stable value, by using the Lorenz system for illustration. This kind of systems commonly show a long-term transient dynamics previous to a sudden transition to a steady state. This can be explained by the crossing of a bifurcation in the associated frozen-in system. We surprisingly uncover a scaling law relating the duration of the transient to the rate of change of the parameter for a case where a chaotic attractor is involved. Additionally, we analyze the viability of recovering the transient dynamics by reversing the parameter to its original value, as an alternative to the control theory for systems with parameter drifts. We obtain the relationship between the paramater change rate and the number of trajectories that tip back to the initial attractor corresponding to the transient state.

On the approximation of basins of attraction using deep neural networks

“…We note here that this is in agreement with the findings in Ref. 19. The reconstruction of the basin of attraction with the proposed method is shown in Fig.…”

“…2 (c) in the form of a heat colormap. For the two well defined regions of the basins of attraction 19 , classification probabilities are close to their extreme values. In contrast, within the mixing area, the neural network is unable to decide the class and results in a probability value close to 0.5.…”

“…In particular, for r = 24.06 there is the birth of a chaotic attractor. For 24.06 < r < 24.74 there is a multistable regime where the chaotic attractors C + and C − coexist 19 and finally, a pair of unstable limit cycles-called homoclinic orbits-for 13.926 < r < 24.06. 28 .…”

“…Transient chaos occurs at 13.926 < r < 24.06, where a homoclinic bifurcation takes place and a chaotic saddle is born 30 . It is related to the stable and unstable manifolds of the chaotic saddle where points exactly on the stable manifold necessarily reach the chaotic saddle and never leave it 19 . This phenomenon gives rise to fractal regions at the boundaries of the basin of attraction between C + and C − .…”

“…In this context, the multiple coexisting attractors constitute the various classes and subsequently, the decision regions (the regions where the data have the same predicted class labels) have the role of the attractor basins. We illustrate the efficiency of the proposed method using the well studied Lorenz system 18 in a bistable regime 19 and moreover provide evidence that the complexity of the attractor basins has a significant effect on the quality of the obtained reconstruction, using the concept of basin entropy 20 .…”