Lyapunov indices with two nearby trajectories in a curved spacetime

We compare three methods for computing invariant Lyapunov exponents (LEs) in general relativity. These methods involve the geodesic deviation vector technique (M1), the two-nearby-orbits method with projection operations and with coordinate time as the independant variable (M2), and the two-nearby-orbits method without projection operations and with proper time as the independent variable (M3). An analysis indicates that M1 and M3 do not need any projection operation. In general, the values of LEs from the three methods are almost the same. However, M2 fails for some specific cases. As a result, M3 is the most preferable to calculate LEs in most cases. In addition, we propose to construct the invariant fast Lyapunov indictor (FLI) with two-nearby-trajectories and give its algorithm in order to quickly distinguish chaos from order. Taking a static axisymmetric spacetime as a physical model, we apply different algorithms of the FLI to explore the global dynamics of phase space of the system where regions of chaos and order are clearly identified.

Figure 1

Poincaré surface of section at the plane $\theta =\frac{\pi }{2}$ ($\dot{\theta }<0$) for the full relativistic core-shell configuration consisting of a Schwarzschild black hole and a purely octopolar shell with parameters $E=0.9679$, $L=3.8$, and $\mathcal{O}=7.012\times {10}^{-7}$.

Figure 2

LEs for a chaotic orbit with the initial conditions $r=9$ and $\dot{r}=0.025$. (a) (M1) coincides nearly with (c) (M3), and (b) describes the failure of M2 for computing LEs. (d) denotes the domain admissible for the motion of $\theta$, as an interpretation for an obstacle to the application of M2. For more details, see the text.

Figure 3

The relation of FLIs with proper time by use of the index (23) for the regular orbit $M$ in the left (colored light gray). The black dots correspond to the result obtained from Eq. (22). The right is the same as the left but for the chaotic orbit $N$ in Fig. 1.

Figure 4

Left: the evolution of the FLIs with the initial values of $r$ ranging from the interval [5.77, 22.34] when the indicator (23) is used and $\dot{r}=0$ is fixed at the initial time. Each orbit is integrated numerically till $\tau$ reaches ${10}^5$. Two types of orbits consisting of regular and chaotic orbits are obtained according to distinct values of FLIs. Right: the same as the left, but $r=9$ is fixed first, and $\dot{r}$ takes values from the interval $[-0.1428,0.1428]$.

Figure 5

Regions of different values of the FLIs on the plane $\theta =\frac{\pi }{2}$. Initial conditions are colored white if their $\mathrm{FLIs}\le 6$, and black if $\mathrm{FLIs}>6$, which are corresponded to the regular region I, and chaotic region II, respectively. Here, all intersection points of an orbit with the plane should be regarded to have the same final value of the FLI.