Chronotaxic Systems: A New Class of Self-Sustained Nonautonomous Oscillators

Nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies have often been treated as stochastic, inappropriately. We therefore formulate them as a new class and discuss how they generate complex behavior. We show how to extract the underlying dynamics, and we demonstrate that it is simple and deterministic, thus paving the way for a diversity of new systems to be recognized as deterministic. They include complex and nonautonomous oscillatory systems in nature, both individually and in ensembles and networks.

Figure 1

(a) Time evolution of chronotaxic system (4, 5) in the plane with polar coordinates (${\alpha }_{\mathbf{x}}$, $r_{\mathbf{x}}$) using $\epsilon <-1$ and ${\omega }_0(t)>0$. Four phase portraits separated by 2.5 s time intervals are shown. The time-dependent point attractor or steady state $\mathbf{(}{\alpha }_{\mathbf{x}}^A(t),r_0\mathbf{)}$ (large black disk in red circle) moves along a limit cycle (black circle). Trajectories (orange) from different initial conditions (white dots) approach the point attractor and then move together with it. This occurs because ${\alpha }_{\mathbf{x}}$ is coupled to a moving point $\mathbf{(}{\alpha }_{\mathbf{p}}(t),r_0\mathbf{)}$ (small black disk) which results from a nonautonomous contribution. The coupling determines instantaneous velocities (gray arrows) of the system and the angular distance ${\alpha }_{\mathbf{p}}(t)-{\alpha }_{\mathbf{x}}^A(t)=\arcsin (1/|\epsilon |)$. (b) The position (phase) ${\alpha }_{\mathbf{x}}$ of the chronotaxic system on the limit cycle when unperturbed (straight gray line) and perturbed (noisy black line). (c) A moving point attractor does not exist on a limit cycle, the perturbed phase strongly deviates from the unperturbed phase.

Figure 2

Different domains of analysis of a signal $x(t)=\cos \mathbf{(}{\alpha }_{\mathbf{x}}(t)\mathbf{)}$ from system (5). Change in frequency, ${\dot{\alpha }}_{\mathbf{x}}(t)$, and phase slips as seen in (a) the time domain, (b) Fourier power spectrum and (c) phase space obtained by time-delay embedding [28, 29] for $\epsilon =1.1$. (d) The embedding of the signal from the unperturbed $\mathbf{(}\xi (t)=0\mathbf{)}$ system (5) with $\tau =1.9\mathrm{s}$, which corresponds to the first minimum of the mutual information [30]. Two small parts of the embedded trajectory during time intervals $\mathbf{(}t,t+2\pi /{\omega }_0(t)\mathbf{)}$ reveal a time-dependent cycle. (e) Wavelet transform of the signal from the unperturbed system (5) with $\epsilon \ge 1$. (f) When a point attractor does not exist ($\epsilon =0.9$) continuous phase slips occur for the perturbed system. (g) Phase slips also occur for the perturbed system at the bifurcation point ($\epsilon =1$) but the attractor exists and can be reconstructed. When $\epsilon$ increases in time, phase slips become less frequent and almost disappear around $t=1400\mathrm{s}$ where $\epsilon (t)=1+5{\omega }_0^{-1}(t)\exp [-(t-1400{)}^2/5000]$.

Figure 3

Analysis of (a) respiration and (b) heart rate variability (HRV) recorded while respiration was paced. The top plots in (a) and (b) show the raw time series. Wavelet transforms of the detrended time series are shown at the bottom. The phase space plots (c) and (d) show the time-delay embedded time series of (c) detrended HRV, $h(t)$, with time-delay $\tau =1.9\mathrm{s}$, and (d) $x_h(t)=\cos \mathbf{(}{\alpha }_h(t)\mathbf{)}$ with time-delay $\tau =1.5\mathrm{s}$, where ${\alpha }_h(t)$ is a phase of $h(t)$ which was reconstructed using a method based on the synchrosqueezed wavelet transform [14].