Data based identification and prediction of nonlinear and complex dynamical systems

We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin

Layered Chaos in Mean-field and Quantum Many-body Dynamics

We investigate the dimension of the phase space attractor of a quantum chaotic many-body ratchet in the mean-field limit. Specifically, we explore a driven Bose-Einstein condensate in three distinct dynamical regimes - Rabi oscillations, chaos, and self-trapping regime, and for each of them we calculate the correlation dimension. For the ground state of the ratchet formed by a system of field-free non-interacting particles, we find four distinct pockets of chaotic dynamics throughout these regimes. We show that a measurement of a local density in each of the dynamical regimes, has an attractor characterized with a higher fractal dimension, $D_{R}=2.59\pm0.01$, $D_{C}=3.93\pm0.04$, and $D_{S}=3.05\pm0.05$, as compared to the global measure of current, $D_{R}=2.07\pm0.02$, $D_{C}=2.96\pm0.05$, and $D_{S}=2.30\pm0.02$. We find that the many-body case converges to mean-field limit with strong sub-unity power laws in particle number $N$, namely $N^{\alpha}$ with $\alpha_{R}={0.28\pm0.01}$, $\alpha_{C}={0.34\pm0.067}$ and $\alpha_{S}={0.90\pm0.24}$ for each of the dynamical regimes mentioned above. The deviation between local and global measurement of the attractor's dimension corresponds to an increase towards high condensate depletion which remains constant for long time scales in both Rabi and chaotic regimes. The depletion is found to scale polynomially with particle number as $N^{\beta}$ with $\beta_{R}={0.51\pm0.004}$ and $\beta_{C}={0.18\pm0.004}$ for the two regimes. Thus, we find a strong deviation from the mean-field results, especially in the chaotic regime of the quantum ratchet. The ratchet also reveals quantum revivals in the Rabi and self-trapped regimes but not in the chaotic regime. Based on the obtained results we outline pathways for the identification and characterization of the emergent phenomena in driven many-body systems

Transition from phase to generalized synchronization in time-delay systems

The notion of phase synchronization in time-delay systems, exhibiting highly non-phase-coherent attractors, has not been realized yet even though it has been well studied in chaotic dynamical systems without delay. We report the identification of phase synchronization in coupled nonidentical piece-wise linear and in coupled Mackey-Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from non-synchronized behavior to phase and then to generalized synchronization as a function of coupling strength. We have introduced a transformation to capture the phase of the non-phase coherent attractors, which works equally well for both the time-delay systems. The instantaneous phases of the above coupled systems calculated from the transformed attractors satisfy both the phase and mean frequency locking conditions. These transitions are also characterized in terms of recurrence based indices, namely generalized autocorrelation function $P(t)$, correlation of probability of recurrence (CPR), joint probability of recurrence (JPR) and similarity of probability of recurrence (SPR). We have quantified the different synchronization regimes in terms of these indices. The existence of phase synchronization is also characterized by typical transitions in the Lyapunov exponents of the coupled time-delay systems.Comment: Accepted for publication in CHAO

Analyzing Multiple Nonlinear Time Series with Extended Granger Causality

Identifying causal relations among simultaneously acquired signals is an important problem in multivariate time series analysis. For linear stochastic systems Granger proposed a simple procedure called the Granger causality to detect such relations. In this work we consider nonlinear extensions of Granger's idea and refer to the result as Extended Granger Causality. A simple approach implementing the Extended Granger Causality is presented and applied to multiple chaotic time series and other types of nonlinear signals. In addition, for situations with three or more time series we propose a conditional Extended Granger Causality measure that enables us to determine whether the causal relation between two signals is direct or mediated by another process.Comment: 16 pages, 6 figure