Conclusion

In numerical simulations of six systems we have shown the transient nature of chaotic dynamics in spatially extended systems and low-dimensional maps. We investigated two models which are used to simulate action potential propagation of cardiac tissue (Fenton-Karma model and the Bueno-Orovio-Cherry-Fenton model), a ring network of Morris-Lecar elements which models neuron activity and a one dimensional system of the Gray-Scott model which simulates a chemical reaction of two species. Furthermore, we extended our simulations to the tent map (one-dimensional) and the H´enon map (two-dimensional), which provide, due to their low dimensionality, a more direct access.

We determined the average transient lifetime hTi_IC of the dynamics by a large number of initial conditions for each model. Since this is a quantity based on the whole chaotic regime of the state space, we investigated (the vicinity of) single trajectories by small but finite perturbations in order to probe the local structure of the state space. In each model we could identify a Terminal Transient Phase (TTP) characterized by a decreasing transient lifetime averaged over all perturbed trajectories hTi_Pert. This indicates, that trajectories propagate before self-termination through a qualitatively different (concerning the preced-ing dynamics) region of the state space, although this is not visible in “common” observables (e.g. the pseudo ECG or the number of phase singularities). The typical duration for trajec-tories to travel through this “transition zone” is significant with respect to the intrinsic time scale governing the dynamics (e.g the length of a spiral rotation) (TTP_FK ≈ 20−25 T_Sp (Fenton-Karma model), TTPBOCF ≈ 15 −20 TSp (Bueno-Orovio-Cherry-Fenton model), TTP_ML≈1.5 s (Morris-Lecar network) and TTP_GS ≈60 a.u. (Gray-Scott model)). Fur-thermore we showed, that the drop ofhTi_Pert is related to spatial clusters of perturbations

which do not change the initial trajectory anymore, indicating that if the system is close to the collapse, only perturbations into specific directions can prevent the system from self-termination. Simulations of the tent map and the H´enon map provide an intuitive insight into the underlying mechanism in low-dimensional systems. We could determine the drop ofhTi_Pert analytically for the tent map and found a geometrical representation in the case of the H´enon map. This provides an idea of what the dynamics and the state space may look like during the TTP in high-dimensional systems.

With this study we investigated the temporal and spatial structure (concerning the lifetime) of the transition from a chaotic regime to the (non-chaotic) attractor of the system. It is noteworthy to emphasize that the transition from the chaotic regime to the attractor can not be understood as the entering of the basin of attraction of the attractor. Typical trajectories are the whole time located inside the basin of attraction, since they will definitely reach the attractor at some point. However, since the information of the collapse is already present in the system some time before upcoming the collapse (TTP), it should be possible, in principle, to identify an observable precursor for predicting temporally close self-termination (despite the fact that type-II-supertransients are characterized by an abrupt termination). In the low-dimensional systems this can already be done (e.g. determining the (former) basin of attraction in the case of the H´enon map). To devise suitable precursors of the collapse in high-dimensional systems one might employ data assimilation or machine learning methods, to exploit the fact that during the TTP trajectories are (partially) robust with respect to small perturbations.

The Terminal Transient Phase occurs in each model investigated here, low-dimensional maps, spatially extended one- and two-dimensional systems, with no-flux or periodic bound-ary conditions and one or two diffusive variables. In the future, it may be of interest whether a finite TTP and corresponding clustering effects in the perturbation space also exist in other systems (e.g. fluid dynamics [36]).

The existence of a TTP extends the general understanding of type-II-supertransients and also may open up new possibilities for applications. For example, in cardiac dynamics where occurring arrhythmias can be life-threatening and the standard defibrillation tech-nique comes along with severe side effects like additional tissue damage and considerable pain [18, 108, 19]. Since self-termination of such arrhythmias like ventricular fibrillation has been observed frequently [109], an observable precursor could improve the control of such arrhythmias. Possible applications could be heading in two directions: (i) a prediction of a close self-termination of e.g. ventricular fibrillation could prevent the submission of a defibrillation shock at all and reduce in this way the side-effects, and (ii) the energy of electrical pulses applied for defibrillation could be reduced by exploiting the state space structure. Either by detecting states where successful defibrillation can be achieved using a lower energy, or using a two step protocol and first reaching these states before defibrillating.

Last but not least, transient dynamics also play a role in the functionality of the brain (e.g.

information processing) [110], and thus studying these processes could benefit from taking the Terminal Transient Phase into account.

3.2. Terminal Transient Phase of Chaotic Transients

Summary of Results

In this study, we reveal the existence of a transition zone in the state space between the chaotic dynamics and the non-chaotic attractor, that means before the self-termination of the dynamics, called the Terminal Transient Phase (TTP). We show in each of the six investigated models, that the region of the state space which corresponds to the TTP has a qualitatively different structure than the state space region which is governed by the previous chaotic dynamics. That means, the state space before the collapse of the chaotic dynamics is different then before, (Fig. 3.17), and it should therefore in principle be possible to develop proper observables which can predict the upcoming self-termination.

"Terminal Transient Phase"

Figure 3.17: The (schematic) state space, depicting the conclusion of the study “Terminal Transient Phase of Chaotic Transients”. The difference of the state space structure between the Terminal Transient Phase (TTP) and the chaotic dynamics before the TTP is illustrated here by the red and blue color of an exemplary trajectory.

Since the TTP occurs in systems with very distinct dynamics, we believe that it is a general feature of transient dynamics. Hence, the existence of the transition zone significantly extends the general understanding of chaotic transients in the field of nonlinear dynamics.

Furthermore, the obtained results can also be relevant for possible applications, e.g. in the field of cardiac dynamics. An extended discussion of the results, including possible implications for applications can be found in the section “Discussion and Outlook” 4.1 on page 125.