Data driven approach to study the transition from dispersive to dissipative systems through dimensionality reduction techniques

Abstract

Complexity is often exhibited in dynamical systems, where certain parameters evolve with time in a strange and chaotic nature. These systems lack predictability and are common in the physical world. Dissipative systems are one of such systems where the volume of the phase space contracts with time. On the other hand, we employ dimensionality reduction techniques to study complicated and complex data, which are tough to analyse. The Principal Component Analysis (PCA) is a dimensionality reduction technique used as a means to study complex data. Through PCA, we studied the reduced dimensional features of the numerical data generated by a nonlinear partial differential equation called the Korteweg de Vries (KdV) equation, which is a nonlinear dispersive system, where solitary waves travel along a specific direction with finite amplitude. Dissipative nature, specific to that of the Lorenz system, were observed in the dimensionally reduced data, which implies a transition from a dispersive system to a dissipative system.

Figures (9)

• Figure 1: (A)-(C): The phase diagrams of the three variables of Lorenz system, taken two at a time. We see that the trajectories confine to a bounded region. The points oscillate in an irregular and non-periodic manner. (D)-(F): These are the time series plots of each variable.
• Figure 2: Normalised singular values of the covariance matrix of the trajectory matrix of the time series data $x(t)$ of the Lorenz system. For each $\kappa$, there are two distinct parts, the deterministic part represented by the declining line and the noise floor represented by the horizontal and nearly flat part. As we decrease the value of $\kappa$, the number of singular values in the deterministic part decreases. The noise floor show the small differences between the smaller but finite singular values.
• Figure 3: Reconstructed phase space of the Lorenz system using three components of the reduced matrix $X^\prime$, namely, $c_1$, $c_2$ and $c_3$, plotted pairwise.
• Figure 4: Evolution of solitons with time and space for different values of $N$. We can observe the periodic behavior due to the periodic boundary conditions and the appearance of inherent noise indicated by the small spikes.
• Figure 5: Normalised singular values of the covariance matrix of the KdV trajectory matrix. For smaller values of $N$, we can see the larger covariance between the larger singular values. The noise floor is almost non-existent for larger values of $N$.