Terminal Transient Phase of Chaotic Transients

Thomas Lilienkamp and Ulrich Parlitz

Phys. Rev. Lett. 120, 094101 – Published 28 February 2018

Abstract

Transient chaos in spatially extended systems can be characterized by the length of the transient phase, which typically grows quickly with the system size (supertransients). For a large class of these systems, the chaotic phase terminates abruptly, without any obvious precursors in commonly used observables. Here we investigate transient spatiotemporal chaos in two different models of this class. By probing the state space using perturbed trajectories we show the existence of a “terminal transient phase,” which occurs prior to the abrupt collapse of chaotic dynamics. During this phase the impact of perturbations is significantly different from the earlier transient and particular patterns of (non)susceptible regions in state space occur close to the chaotic trajectories. We therefore hypothesize that even without perturbations proper precursors for the collapse of chaotic transients exist, which might be highly relevant for coping with spatiotemporal chaos in cardiac arrhythmias or brain functionality, for example.

Received 13 June 2017
Revised 16 December 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.094101

Physics Subject Headings (PhySH)

Cardiac dynamics

Chaotic systems Dynamical systems High dimensional systems

Physics of Living SystemsNonlinear Dynamics

Authors & Affiliations

Thomas Lilienkamp^1,2,* and Ulrich Parlitz^1,2,3

¹Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
²Institute for Nonlinear Dynamics, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
³German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Robert-Koch-Straße 42a, 37075 Göttingen, Germany

^*thomas.lilienkamp@ds.mpg.de

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Vol. 120, Iss. 9 — 2 March 2018

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Images

Figure 1
Spatiotemporal chaos in the Fenton-Karma model (exemplary snapshot of the membrane potential $V_{m}$ of a two-dimensional simulation) (a) and the Morris-Lecar neuron model [exemplary (transient) episode of a one-dimensional ring, including self-termination at $t = 172.38 s$ ] (b), respectively. The white bar depicts the length of the terminal transient phase.
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Figure 2
Schematic sketch of the procedure of the application of small perturbations: The initial condition (solid black line) reaches the desired attractor (red cross) without any perturbations. At some point in time, small perturbations (small blue arrows) are applied to the initial condition. This leads to new perturbed trajectories, which also reach the attractor after their specific transient lifetime.
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Figure 3
The terminal transient phase in the Fenton-Karma model. In subplot (a), a single trajectory was chosen, which terminates at around $t = 242 T_{sp}$ (black arrow). $⟨ T ⟩_{pert}$ is plotted for various perturbation strengths (in color: [0.001, 0.01, 0.1, 1.00]). It drops (differently for each strength) already at a period of time before the actual collapse of the system, indicating that fewer perturbations can actually change the evolution with respect to the unperturbed trajectory, significantly. However, perturbations with a larger amplitude can shift the initial trajectory over a larger distance in the state space, which results in a higher chance of preventing the collapse. The self-termination is not visible in the pseudo electrocardiogram (ECG, dashed gray line) or the number of phase singularities ( $N_{PS}$ , solid black line) only instantly before the collapse, shown in the lower part of (a). For a perturbation strength of $Δ = 0.1$ , $⟨ T ⟩_{pert}$ was averaged over twenty trajectories [(b), time axis is normalized such that self-termination occurs at $t = 0$ ], indicating a ${TTP}_{FK} \approx 20 - 25 T_{sp}$ .
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Figure 4
Spatial structure of the lifetime of perturbed trajectories. The subplots (a), (b), and (c) show snapshots of the membrane potential $V_{m}$ during the TTP, at points in time which correspond to the black lines in Fig. 3 marked by $t_{1}$ , $t_{2}$ , and $t_{3}$ , respectively. The subplots (d),(e), and (f) show the map of transient times relative to the lifetime of the reference trajectory $T_{ref}$ , corresponding to the trajectories which were perturbed at the specific position. The solid black curves represent the contour of $V_{m} = 0.5$ of the corresponding snapshots of the upper plot. The locations of perturbations which can significantly change the transient time (with respect to the original, unperturbed trajectory) are clustered, and their number decreases as the system gets closer to the collapse of the (unperturbed) reference trajectory.
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Figure 5
The terminal transient phase in the tent map. Subplot (a) shows a cobweb plot for a representative initial condition $x_{0} = 0.13$ and $a = 2.2$ . In (b), the domains of the state space are shown, which will lead to self-termination in $n$ steps ( $n = 0$ means self-termination). For $n = - 1$ the domain corresponds to the exitlike window in subplot (a) [between the two intersections of the green dashed line ( $x_{n + 1} = 1$ ) and the black line (tent map)]. The gray arrows indicate a representative perturbation. The ratio between the perturbation strength (length of the arrow) and the domain sizes increases when departing from the collapse ( $n = 0$ ) backward in time. In subplot (c), $⟨ T ⟩_{pert}$ is shown for various perturbation strengths (colored lines), and also the theoretical approximation [Eq. (6)] is plotted for each perturbation strength (black dotted lines).
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