^{1,2,3}, Reik V. Donner

^{1}and Jürgen Kurths

^{1,4,5}

### Abstract

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic Rössler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.

Oscillatory processes can be frequently observed in natural and technological systems. Often, the corresponding dynamics is not strictly periodic, but shows more complex temporal variability patterns characterized by a fast divergence of trajectories with arbitrarily close initial conditions.

^{1–3}There are numerous examples of such chaotic oscillators for which long-term predictions of amplitudes and phases are not possible. Therefore, studying their phase dynamics has recently attracted particular interest, e.g., regarding the process of phase synchronization between different coupled systems.

^{4,5}However, most existing methods suitable for this purpose require the explicit definition of an appropriate phase variable, which can become a non-trivial problem in the case of noncoherent chaotic oscillations. Therefore, studying the phase coherenceproperties of chaotic systems has become an important problem in both theoretical and experimental studies.

^{6}In this work, we propose some methods based on the concept of recurrences in phase space, which allow studying complementary aspects of chaotic oscillators relating to the geometric structure of, and the dynamics on the attractor. Specifically, we derive a detailed characterization of changes of the geometric structure of complex systems in phase space with varying control parameters, which accompany transitions from phase-coherent to noncoherent dynamics.

This work has been partially funded by the Leibniz society (project ECONS) and the Federal Ministry for Education and Research (BMBF) via the Potsdam Research Cluster for Georisk Analysis, Environmental Change and Sustainability (PROGRESS). The authors thank Wei Zou for providing a code for estimating the largest Lyapunov exponent of the Mackey-Glass system, and Istvan Kiss for fruitful discussions. Complex network measures have been calculated on the IBM iDataPlex Cluster at the Potsdam Institute for Climate Impact Research using the software package igraph.^{88}

I. INTRODUCTION

II. METHODS

A. Recurrence time statistics

B. Recurrence network analysis

C. Recurrence properties of phase-coherent and noncoherent Rössler systems

III. QUANTIFYING PHASE COHERENCE OF CHAOTIC OSCILLATORS

A. Phase and frequency of chaotic oscillators

B. Traditional measures of phase coherence

C. RP-based indicators of phase coherence

IV. EXAMPLE I: BIFURCATION SCENARIO OF THE RÖSSLER SYSTEM

A. Traditional and recurrence times-based measures

B. Recurrence network measures

C. Discriminatory skills of RP-based phase coherence indicators

D. Impact of the homoclinic point on RN measures

V. EXAMPLE II: BIFURCATION SCENARIO OF THE MACKEY-GLASS SYSTEM

VI. CONCLUSIONS

### Key Topics

- Chaos
- 39.0
- Attractors
- 31.0
- Coherence
- 28.0
- Statistical properties
- 23.0
- Thermodynamic properties
- 22.0

## Figures

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).

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).

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.

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.

(Color online) Color-coded representations of local RN properties ((a) and (b) local clustering coefficient , (c) and (d) logarithm of betweenness centrality log *b _{v} *) 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

*b*< 1.

_{v}(Color online) Color-coded representations of local RN properties ((a) and (b) local clustering coefficient , (c) and (d) logarithm of betweenness centrality log *b _{v} *) 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

*b*< 1.

_{v}Scatter plot between the RN measures and log *b _{v} * for the Rössler system with (a) PC and (b) NPC chaos (

*RR*= 0.03).

*ρ*gives the values of the rank-order correlation coefficient (Spearman’s Rho) between both quantities.

_{s}Scatter plot between the RN measures and log *b _{v} * for the Rössler system with (a) PC and (b) NPC chaos (

*RR*= 0.03).

*ρ*gives the values of the rank-order correlation coefficient (Spearman’s Rho) between both quantities.

_{s}Probability density function of the RN measures [(a) and (b)] and [(c) and (d)] log *b _{v} * 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 (•)).

Probability density function of the RN measures [(a) and (b)] and [(c) and (d)] log *b _{v} * 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 (•)).

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.

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.

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).

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).

Mean values *k* (a) and standard deviations *σ _{k} * (b) of the distribution of degree centrality

*k*for the RNs obtained from 100 independent realizations (error bars indicating ensemble means and standard deviations) of the Rössler system (

_{v}*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.

Mean values *k* (a) and standard deviations *σ _{k} * (b) of the distribution of degree centrality

*k*for the RNs obtained from 100 independent realizations (error bars indicating ensemble means and standard deviations) of the Rössler system (

_{v}*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.

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

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

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.

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.

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*(*t _{i} *),

*x*(

*t*−

_{i}*τ*/2),

*x*(

*t*−

_{i}*τ*))).

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*(*t _{i} *),

*x*(

*t*−

_{i}*τ*/2),

*x*(

*t*−

_{i}*τ*))).

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).

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).

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.

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

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 *b _{v} * (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).

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 *b _{v} * (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).

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