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Geometric and dynamic perspectives on phase-coherent and noncoherent chaos
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10.1063/1.3677367
/content/aip/journal/chaos/22/1/10.1063/1.3677367
http://aip.metastore.ingenta.com/content/aip/journal/chaos/22/1/10.1063/1.3677367

Figures

Image of FIG. 1.
FIG. 1.

Two-dimensional projection of a part of the trajectory of the Rössler system [Eq. (1) ] in the (a) PC (a = 0.165) and (b) NPC (funnel) regimes (a = 0.265).

Image of FIG. 2.
FIG. 2.

RT distribution p(τ) with τ = lΔt (zoom for short times) for one realization of the Rössler system with (a) PC and (b) NPC chaos. The threshold ɛ has been chosen to yield a recurrence rate RR = 0.03.

Image of FIG. 3.
FIG. 3.

(Color online) Color-coded representations of local RN properties ((a) and (b) local clustering coefficient , (c) and (d) logarithm of betweenness centrality log bv ) for the Rössler system with [(a) and (c)] PC and [(b) and (d)] NPC chaos (RR = 0.03). In (c) and (d), black circles indicate vertices in poorly populated regions of phase space with bv  < 1.

Image of FIG. 4.
FIG. 4.

Scatter plot between the RN measures and log bv for the Rössler system with (a) PC and (b) NPC chaos (RR = 0.03). ρs gives the values of the rank-order correlation coefficient (Spearman’s Rho) between both quantities.

Image of FIG. 5.
FIG. 5.

Probability density function of the RN measures [(a) and (b)] and [(c) and (d)] log bv for the Rössler system with [(a) and (c)] PC and [(b) and (d)] NPC chaos. The different symbols represent the results obtained for the same trajectory with different choices of the recurrence rate (RR = 0.02 (), 0.03 , and 0.04 (•)).

Image of FIG. 6.
FIG. 6.

Behavior of different measures for phase coherence for the Rössler system in dependence on the parameter a (error bars indicate standard deviations obtained from 100 independent realizations of the system for each value of a): (a) Largest Lyapunov exponents λ 1 (solid line, ) and λ 2 (dashed line) calculated from the dynamical equations, indicating the location of periodic windows, (b) phase diffusion coefficient D, (c) coherence index CI, and (d) generalized coherence factor GCF (RR = 0.03). Shaded areas indicate the presence of periodic windows evaluated by means of the largest Lyapunov exponents.

Image of FIG. 7.
FIG. 7.

Behavior of RN-based characteristics for the Rössler system in dependence on the parameter a (RR = 0.03, error bars indicate standard deviations obtained from 100 independent realizations of the system for each value of a): (a) global clustering coefficient , (b) network transitivity , (c) average path length , (d) assortativity coefficient , and [(e) and (f)] standard deviation and [(g) and (h)] skewness of the local clustering coefficient and logarithmic betweenness centrality ( , σ log  b , , and γ log  b , respectively).

Image of FIG. 8.
FIG. 8.

Mean values k (a) and standard deviations σk (b) of the distribution of degree centrality kv for the RNs obtained from 100 independent realizations (error bars indicating ensemble means and standard deviations) of the Rössler system (N = 10 000). The desired recurrence rate has been approximated by selecting the threshold ɛ based on a Monte Carlo sampling of inter-point distances from the trajectory in order to enhance computational efficiency.

Image of FIG. 9.
FIG. 9.

As in Fig. 7 for a fixed recurrence threshold, ɛ = 0.2776 (corresponding to RR ≈ 0.03 at a = 0.15).

Image of FIG. 10.
FIG. 10.

Phase portraits of the Mackey-Glass system (14) for (a) τ = 13, (b) τ = 13.5 (after the period-doubling bifurcation), (c) τ = 15.5, and (d) τ = 17.

Image of FIG. 11.
FIG. 11.

Behavior of different statistical characteristics for individual realizations of the Mackey-Glass system in dependence on the parameter τ: (a) Largest Lyapunov exponent λ 1 estimated from the variational equations of a discretized version of the system with 10 000 variables representing (x(t), x(t − τ/9999), …, x(t − τ)), (b) phase diffusion coefficient D and (c) coherence index (CI) obtained via Hilbert transform of x(t). In addition, (d) shows the mean generalized coherence factor GCF obtained from 100 realizations for each value of τ (RR = 0.03, embedding dimension 3 and delay τ/2, i.e., x i  = (x(ti ), x(ti  − τ/2), x(ti  − τ))).

Image of FIG. 12.
FIG. 12.

Behavior of RN-based characteristics for the Mackey-Glass system in dependence on the parameter τ (RR = 0.03, embedding parameters as in Fig. 11(d) , error bars indicate standard deviations obtained from 100 independent realizations of the system for each value of a): (a) global clustering coefficient , (b) network transitivity , (c) average path length , (d) assortativity coefficient , and [(e) and (f)] standard deviation and [(g) and (h)] skewness of the local clustering coefficient and logarithmic betweenness centrality ( , σ log  b , , and γ log  b , respectively).

Image of FIG. 13.
FIG. 13.

Estimates of the marginal density p(x) of the Mackey-Glass system (14) for (a) τ = 13, (b) τ = 13.5, (c) τ = 15.5, and (d) τ = 17. One clearly observes the period-doubling bifurcation (b) and the emergence of a cusp point and, subsequently, the secondary loop structure (c) in terms of the present local maxima.

Tables

Generic image for table
Table I.

Mean values and standard deviations (in brackets) of the different measures for phase coherence for the considered realizations of the Rössler system (averages over 100 independent realizations for every value of a, fixed RR = 0.03) taken over all parameter values in the PC and NPC regimes, and P-values of the associated U-test: generalized coherence factor GCF, global RN measures , , , and , and standard deviation and skewness of the distributions of the local RN measures and log bv (from top to bottom). Symbols indicate the significance of the different parameters as discriminatory statistics (—: insignificant, *: significant at 5% level, **: significant at 1% level, ***: significant at 0.1% level).

Generic image for table
Table II.

As in Table I , results obtained with a fixed recurrence threshold ɛ = 0.2776.

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/content/aip/journal/chaos/22/1/10.1063/1.3677367
2012-02-02
2014-04-21
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Geometric and dynamic perspectives on phase-coherent and noncoherent chaos
http://aip.metastore.ingenta.com/content/aip/journal/chaos/22/1/10.1063/1.3677367
10.1063/1.3677367
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