Chaotic Dynamics in Planetary Systems by Sylvio Ferraz-Mello

Author:Sylvio Ferraz-Mello
Language: eng
Format: epub
ISBN: 9783031458163
Publisher: Springer Nature Switzerland

2.9 Lyapunov Characteristic Exponents (LCE)

We will deepen the issue of exponential sensitivity to initial conditions discussed in the introduction to the definition of chaos and see how it can provide another method to diagnose chaoticity. As we have already observed more than once, we will be more concerned with the qualitative and operational aspects than with demonstrations. Exponential sensitivity indicators appear in the literature under several names: Lyapunov numbers, Lyapunov characteristic numbers, Lyapunov exponents, or simply LCE (Lyapunov characteristic exponents). The inverse of the largest LCE is usually called “Lyapunov time”. These indicators are conceptually based on a series of theorems demonstrated by Russian mathematicians (Valery Oseledec, Yakov Pesin, etc.) and widely disseminated in the West by Giancarlo Benettin, Luigi Galgani and Jean-Marie Strelcyn (1976). We begin this section by presenting a method that is not the best, it even presents an often mediocre performance, but it makes use only of intuitive concepts.

Since chaos manifests itself as an asymptotically exponential divergence of two neighboring trajectories, to study it, we can construct solutions of the differential equations of motion starting at two nearby points, and and follow them. When we are in a chaotic region, the tendency of these solutions is to move exponentially away from one another. The distance between the two trajectories in phase space tends to grow exponentially (see Fig. 1.​3). But the exponential divergence itself conspires against such a simple method: if the two trajectories move apart exponentially, after a certain time, they will be so far away from one another that the distance between them no longer has any meaning. A simple example demonstrates this: Suppose that the trajectory in phase space is inside a sphere. Then, two paths on this sphere cannot move apart indefinitely because the maximum distance that two points can move apart is the diameter of the sphere. Therefore, if we follow two chaotic trajectories on a sphere, the distance between them can grow exponentially, but only until its value approaches the diameter of the sphere. If this fact is not taken into account, when we compare one trajectory with another, we will have meaningless results. This extreme example shows that there is a need to refine the idea, as discussed in Chap. 1 (Sect. 1.​2).

We are trying to detect the chaos behavior along a trajectory; the neighboring trajectory has the sole purpose of exploring space in the vicinity of the investigated trajectory. When it moves too far away from the investigated trajectory, it ceases to do it.

Let us observe Fig. 1.​4. There we have two solutions whose conditions differ by a tiny amount, and progressively they move away from one another. In the beginning, the two almost overlap, but gradually they move apart until after a certain point they are very far apart. After some time, the distance between the two solutions becomes too large and no longer serves to inform how these solutions diverge in the vicinity of one of them. Therefore, if the distance between the two solutions is

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