Distinguishing quasiperiodic dynamics from chaos in short-time series

We propose a procedure to distinguish quasiperiodic from chaotic orbits in short-time series, which is based on the recurrence properties in phase space. The histogram of the return times in a recurrence plot is introduced to disclose the recurrence property consisting of only three peaks imposed by Slater’s theorem. Noise effects on the statistics are studied. Our approach is demonstrated to be efficient in recognizing regular and chaotic trajectories of a Hamiltonian system with mixed phase space.

Figure 1

RPs of four different cases: (a) periodic, (b) quasiperiodic, (c) chaotic, and (d) Gaussian white noise. The line with double arrows in (a) is used to denote a white vertical line.

Figure 2

RPs and the corresponding histograms of the white vertical lines. (a),(b) $ϵ=0.2$ and (c),(d) $ϵ=0.3$. The maximal distance is denoted as $d_3$ in (a).

Figure 3

(a) The maximum white vertical line versus $ϵ$ for the golden mean. The numbers on the plateaus of the curve satisfy the Fibonacci sequence. (b) The $ϵ$ values against the maximal white vertical lines in a logarithmic scale plot. The straight line is with slope $-1$ as expected by Eq. (7).

Figure 4

(a) Three white vertical lines in dependence of $ϵ$ values for the golden mean. (b) The $ϵ$ values against the white vertical lines in a logarithmic plot.

Figure 5

(a) The crossings of trajectories with the Poincaré section defined by $x=0$, $\dot{x}<0$ for ten random initial conditions for the case with energy $H=0.125$. (b) About ten oscillations of a quasiperiodic orbit with the initial values $(\mathrm{0.004\hspace{0.17em}793},\mathrm{0.479\hspace{0.17em}291},\mathrm{0.149\hspace{0.17em}994},-\mathrm{0.001\hspace{0.17em}275})$. (c) A chaotic orbit with the initial conditions $(\mathrm{0.004\hspace{0.17em}876},\mathrm{0.487\hspace{0.17em}553},-\mathrm{0.001\hspace{0.17em}108},-\mathrm{0.110\hspace{0.17em}757})$.

Figure 6

RPs of the Poincaré section points of two typical trajectories: (a) quasiperiodic and (b) chaotic.

Figure 7

(a),(d) The projection of the original trajectory in the $(x,\dot{x})$ plane. (b),(e) Recurrence plots. (c),(f) The corresponding histogram of white vertical lines in RPs. The first row (a)–(c) is for the quasiperiodic orbit and the second row (d)–(e) is for the chaotic case.

Figure 8

(a),(d) The projections of the embedded trajectory in the $(x,x_{t+\tau })$ plane. (b),(e) Recurrence plots. (c),(f) The corresponding histogram of white vertical lines in RPs. The first row (a)–(c) is for the quasiperiodic case and the second row (d)–(e) is for the chaotic case.

Figure 9

Power spectra for two cases: (a) quasiperiodic and (b) chaotic. The $y$ axes are in logarithmic scale.

Figure 10

The effective recurrence interval is denoted by thick arcs. (a) The Euclidean norm yields the same arc length independent of the position due to the symmetry property. (b) The maximum norm yields recurrence intervals of different length due to the invariance of the orientation $\mathrm{\Delta }X$ in the horizontal ($\mathrm{\Delta }Y$ in the vertical, respectively) direction to the reference point.

Figure 11

White vertical lines versus the position on the Poincaré section. (a) The Euclidean norm and (b) the maximum norm.

Figure 12

(a) Schematic description of the relationship between the flow and the recurrence regime on the Poincaré section. The lines with arrows indicate the direction of the flow showing different ways to pass the area. The central circle is the reference point. $({\mathrm{a}}_4$) shows the effect of the sampling denoted by black points. (b) RP of quasiperiodic trajectory obtained from Eq. (2). The maximum norm and a sampling time $\mathrm{\Delta }t=0.01$ are used.

Figure 13

Histograms of the white vertical lines of the RPs of Eqs. (2) with $R=4$ but for two different values of the radius: (a) $r=2.5$ and (b) $r=3.5$ ($ϵ=0.25$ in both cases).

Figure 14

(Color online) (a) Poincaré section of Eq. (2), $x_3=\mathrm{const}$, corrupted by 10% of independent Gaussian noise. The original circle is marked by red color. The phase variable $\Phi$ is defined as $\arctan x_2∕(x_1-R)$. (b) RP with $ϵ=0.2$. Compared to Fig. 2a, the continuous lines are broken, leading to subdistances.

Figure 15

(a) Histogram of white vertical lines in the RP based on the Poincaré section points of Eq. (2) corrupted by 10% of noise. (b) The return times as a function of the position on the Poincaré section.

Figure 16

(Color online) Quasiperiodic trajectory with observational noise for six different levels. (a) The number of white vertical lines versus $ϵ∕\sigma$. The noise/signal level is given in the legend value. A close look at smaller values is shown in the inset. (b) The same plot of (a) with the $y$ axis in logarithmic scale.

Figure 17

(Color online) The number of white vertical lines versus $ϵ∕\sigma$ for two chaotic systems with observational noise for six different levels. The $y$ axes are in logarithmic scale and the noise/signal ratio is given in the legend. (a) The Bernoulli map and (b) the Rössler system.

Figure 18

(a) A short piece of a trajectory on the torus $({\varphi }_1,{\varphi }_2)$. (b) Schematic illustration for the relationship between two errors.

Figure 19

(a) The maximal vertical distances versus $ϵ$ for the silver mean. The Pell numbers are denoted by $▵$, while $\circ$ for the Farey neighbors of the Pell numbers. The inset shows the sequence when the $ϵ$ is less than $0.04$. (b) Double-logarithmic plot of (a).