Structures of the State Space

In the following, typical topological objects of the state space are discussed. In many cases, the dynamics of a system is determined by the coexistence of more than one of these objects.

Attractor

An attractor of a system is a subset of the state spaceX, which tends to have an attracting effect on trajectories. The subsetAof the state space is called an attractor of the system if it fulfills three properties:

• When x(t₁) is an element ofA, then it will remain inA ∀t(Ais an invariant set).

• For each attractor A there exists the basin of attraction, a neighborhood of A, called B(A). The basin is defined by the set of all points, which will asymptotically converge to the attractor in the limit t→ ∞.

• No non-empty subset of A exists, which fulfills the first two properties.

2.2. Mathematical Embedding

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Figure 2.12: The sinus rhythm in state space. The rhythmical contraction of the heart (sinus rhythm) can be interpreted as a trajectory in state space and is here shown schemat-ically. The resting state (fixed point), where the membrane potential everywhere in the heart is equal to the resting potential (section 2.1.1 on page 11) is in the figure marked by (1). The beginning of the contraction cycle initiated by an excitation of the av node can be interpreted as a perturbation from the resting state (2). As the plane wave propagates through the tissue, the state of the system travels through the state space ((3), (4) and (5)), until the plane wave has passed through the whole heart, and the system returns again to the resting state (1). (Simulation by Sebastian Stein).

Fixed Point

A stable fixed point is a special case of an attractor, where the attractor itself is given by a single point in the state space. By definition, the fixed point maps to itself. In many cases, the stability of the fixed point, thus the robustness against infinitesimal perturbations, and its basin are of interest. Actually, the principle of the basin of a fixed point can be found in cardiac dynamics as a practical example, in particular during the cycle of a usual (sinus rhythm) contraction. Here, the resting state between two contractions, defined by the absence of electrical waves, can be interpreted as a stable fixed point (marked by (1) in Fig. 2.12).

The fixed point is stable against small local perturbations of the membrane potential, which do not exceed the excitation threshold. However, if the perturbation is big enough (green arrow in Fig. 2.12), an action potential is induced. The external stimulus (which starts the cycle) is initiated by the sinus node and is forwarded by the electrical conduction system (section 2.1.2 on page 17). While the electrical wave propagates through the tissue ((3),(4) and (5)) the state of the system forms a trajectory in the state space. Since the system returns to its fixed point (1), the state in state space which corresponds to the initial perturbation (2) is thus also part of the basin of the fixed point.

As a remark, the aim of the above interpretation is to illustrate the general concept of a basin of attraction and should not provide an exact mathematical example. Actually, it is questionable if the resting state can be interpreted as a fixed point (with measure of zero) or should rather be modeled as an extended attractor⁴. Furthermore, strictly speaking the discussed resting state is not stable under biological considerations, since cardiac muscle cells will initiate excitation waves by themselves if the external stimulation by the sinus rhythm is missing due to some reason. However, the state space sketched in Fig. 2.12 provides a basic picture, which will be used to illustrate the scientific objective, the main concepts, and the obtained results of each study of this thesis.

Chaotic Attractor

Similar to the fixed point, the chaotic attractor (also called strange attractor) is a special case of a general attractor, too. The dynamics of a trajectory inside the attractor is chaotic, which manifests in the sensitivity of initial conditions. This behavior can be quantified by the calculation of the leading Lyapunov exponents (section 2.2.4). A chaotic attractor is the topological object which is responsible for a permanent chaotic dynamics (in comparison to transient chaos, see below). It has a fractal structure, which can be estimated by e.g.

the Kaplan-Yorke dimension (section 2.2.4). A specific trajectory comes arbitrarily close to each point of the attractor. As an example, a chaotic attractor can be found in the two-dimensional H´enon map [33] in a specific parameter regime. The H´enon map is discussed extensively in the study of section 3.2 on page 76.

Repeller

A repeller is similar to an attractor, whereas it has a repelling effect on neighboring tra-jectories. In some cases, a chaotic repeller can be responsible for chaotic transients, thus chaotic dynamics with a finite duration, followed by a non-chaotic behavior, governed by e.g. another attractor.

Chaotic Saddle

Chaotic saddles are invariant sets of the state space, which are besides chaotic repellers responsible for the occurrence of chaotic transients. They can be characterized by their influence on neighboring trajectories. In contrast to chaotic repellers which repel all close trajectories, chaotic saddles do have both, attracting and repelling directions. These direc-tions are actually hypersurfaces in the state space and are denoted as the stable manifold and the unstable manifold. Thestable manifold of a fixed point x^∗ is defined as all points x which will approach x^∗ ast→ ∞, whereas theunstable manifold is defined as the points x which will approach x^∗ as t→ −∞. They are exemplary denoted in Fig. 2.13 in orange (stable manifold) and green (unstable manifold), respectively.

4The main question here, is how the state of the system should be parametrized by the model, since this parametrization defines the state space at the end.

2.2. Mathematical Embedding

Figure 2.13: A saddle point in the state space. The saddle point is defined by the intersec-tion of thestable manifold (orange) and theunstable manifold (green) of the fixed point. In this sketch, the saddle point coexists with a sink (stable fixed point) and a source (unstable fixed point), as well as a stable limit cycle. Republished with permission of Royal Society, from [34]; permission conveyed through Copyright Clearance Center, Inc.

A chaotic saddle is defined by the intersections of the stable and unstable manifold [35].

This behavior is depicted in Fig. 2.14. If the stable manifold and the unstable manifold intersect at one point, they must do so infinitely many times, since if one point is part of both manifolds, also the image and the preimage of the point have to be part of both manifolds. This results in a complex intertwined structure of both manifolds.

Figure 2.14: The intersection of the stable and the unstable manifold of a fixed point.

The stable (orange) and unstable manifold (green) of a fixed point x^∗ are shown. The intersections of both manifolds (black circles) define the chaotic saddle.

In contrast to the transient chaotic dynamics governed by a repeller, long-living trajectories can in the case of a chaotic saddle initiate far away from the saddle. If they start in the proximity of the stable manifold, they are first attracted to the saddle. Since the repeller only has nonattracting directions, long-living episodes can only originate from the vicinity of the repeller. However, although a chaotic saddle has also attracting directions, the repeller

and the chaotic saddle are both globally non-attracting sets, since all typical trajectories will depart after a finite transient time. Mathematically, the chaotic saddle itself is an invariant set with a corresponding infinite lifetime, but since its measure is zero, one will in practice, for example in numerical simulations, never reach such a trajectory with sustained chaos.

Thus, although the chaotic saddle is the set which governs the dynamics, typical trajectories will in practice never reach the set itself, and the influence on the dynamics is mainly due to the attracting and non-attracting effect on neighboring trajectories.

Beside chaotic repellers, chaotic saddles are the sets in state space which are responsible for transient chaotic dynamics. They typically coexist with other topological objects in the state space, as for example an attractor or a limit cycle (as sketched in Fig. 2.13). Trajectories starting in the proximity of the saddle typically show chaotic dynamics for a transient time, until they reach these objects and as a consequence finish the chaotic dynamics. The length of the chaotic episode also depends sensitively on the initial condition. Chaotic transients play a major role in studies of this thesis (e.g. section 3.1 on page 68 or section 3.2 on page 76). That is why the phenomenon of transient chaos will be discussed in more detail in the next section.