Detecting unstable periodic orbits in high-dimensional chaotic systems from time series: Reconstruction meeting with adaptation

Detecting unstable periodic orbits (UPOs) in chaotic systems based solely on time series is a fundamental but extremely challenging problem in nonlinear dynamics. Previous approaches were applicable but mostly for low-dimensional chaotic systems. We develop a framework, integrating approximation theory of neural networks and adaptive synchronization, to address the problem of time-series-based detection of UPOs in high-dimensional chaotic systems. An example of finding UPOs from the classic Mackey-Glass equation is presented.

Figure 1

(a) Three UPOs of periods $T_1=1.48$, $T_2=3.17$, and $T_3=4.53$, computed directly from the original Lorenz system. (b) The corresponding UPOs of approximately the same periods ($T_1^{*}=1.557$, $T_2^{*}=3.084$, and $T_3^{*}=4.588$, respectively), detected adaptively from the measured data based on system (6). (c) Spectra of the UPOs in panels (a) and (b). (d) Three detected UPOs of longer periods: $T_4^{*}=6.3$, $T_5^{*}=6.9$, and $T_6^{*}=9.6$.

Figure 2

(a) Time series $x_2\left(t\right)$ from the Lorenz system, (b) reconstructed attractor, (c) an UPO detected from the reconstructed attractor, and (d) time evolution of UPO trajectories, where the upper panel exhibits the detected UPO in panel (c) and the lower panel is the directly computed period-$T_3$ UPO shown in Fig. 1a.

Figure 3

For the Hindmarsh-Rose neuron model, (a) a chaotic attractor, (b) noise-perturbed and sparsely sampled time series, (c) a detected UPO from the time series data in panel (b) with the period $T=91.47$, (d) the corresponding UPO computed directly from the model with the period $T=91.22$, and (e) MSE vs the noise level.

Figure 4

(a) A chaotic attractor of the Mackey-Glass system and an UPO of period about $T=7.61$ computed directly from system equations. (b) UPO in panel (a) and the corresponding UPO (of period $T^{*}=7.60$) detected from our reconstructed system of RBF neural networks.