scholarly journals A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 103
Author(s):  
Bulcsú Sándor ◽  
Bence Schneider ◽  
Zsolt I. Lázár ◽  
Mária Ercsey-Ravasz
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
State Transition ◽  
Transition Probabilities ◽  
Coarse Graining ◽  
Special Focus ◽  
Entropy Of Transition ◽  
Transition Networks

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.

Sparse Recovery and Dictionary Learning to Identify the Nonlinear Dynamical Systems: One Step Toward Finding Bifurcation Points in Real Systems

2019 ◽  
Vol 29 (03) ◽  
pp. 1950030 ◽  
Author(s):  
Fahimeh Nazarimehr ◽  
Aboozar Ghaffari ◽  
Sajad Jafari ◽  
Seyed Mohammad Reza Hashemi Golpayegani
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dictionary Learning ◽  
Nonlinear Dynamical Systems ◽  
Sparse Recovery ◽  
Governing Equations ◽  
Nonlinear Dynamical ◽  
Bifurcation Points ◽  
Long Time

Modeling real dynamical systems is an important challenge in many areas of science. Extracting governing equations of systems from their time-series is a possible solution for such a challenge. In this paper, we use the sparse recovery and dictionary learning to extract governing equations of a system with parametric basis functions. In this algorithm, the assumption of sparsity in the functions of dynamical equations is used. The proposed algorithm is applied to different types of discrete and continuous nonlinear dynamical systems to show the generalization ability of this method. On the other hand, transition from one dynamical regime to another is an important concept in studying real world complex systems like biological and climate systems. Lyapunov exponent is an early warning index. It can predict bifurcation points in dynamical systems. Computation of Lyapunov exponent is a major challenge in its application in real systems, since it needs long time data to be accurate. In this paper, we use the predicted governing equation to generate long time-series, which is needed for Lyapunov exponent calculation. So the proposed method can help us to predict bifurcation points by accurate calculation of Lyapunov exponents.

CHARACTERIZATION OF CHAOTICITY IN THE TRANSIENT CURRENT THROUGH PMMA THIN FILMS

Fractals ◽  
2006 ◽  
Vol 14 (02) ◽  
pp. 125-131 ◽  
Author(s):  
A. HACINLIYAN ◽  
Y. SKARLATOS ◽  
H. A. YILDIRIM ◽  
G. SAHIN
Keyword(s):  
Thin Films ◽  
Time Series ◽  
Lyapunov Exponent ◽  
Power Law ◽  
Chaotic Behavior ◽  
Transient Current ◽  
Positive Lyapunov Exponent ◽  
Aluminum Films ◽  
Thin Aluminum

Chaotic behavior in the transient current through thin Aluminum-PMMA-Aluminum films has been analyzed for times ranging up to 30,000s, in the temperature range 293–363K for applied voltages in the range 10–80V. Time series analysis reveals a positive Lyapunov exponent consistently and reproducibly throughout this range. Power law relaxation as reflected by the autocorrelation function and the positive Lyapunov exponent show parallel behaviors as a function of applied electric field.

scholarly journals Insight into earthquake sequencing: analysis and interpretation of time-series constructed from the directed graph of the Markov chain model

2015 ◽  
Vol 2 (1) ◽  
pp. 399-424
Author(s):  
M. S. Cavers ◽  
K. Vasudevan
Keyword(s):  
Time Series ◽  
Markov Chain ◽  
Directed Graph ◽  
State Transition ◽  
Transition Probabilities ◽  
Markov Chain Model ◽  
Fano Factor ◽  
Chain Model ◽  
Intrinsic Mode Functions ◽  
Allan Factor

Abstract. Directed graph representation of a Markov chain model to study global earthquake sequencing leads to a time-series of state-to-state transition probabilities that includes the spatio-temporally linked recurrent events in the record-breaking sense. A state refers to a configuration comprised of zones with either the occurrence or non-occurrence of an earthquake in each zone in a pre-determined time interval. Since the time-series is derived from non-linear and non-stationary earthquake sequencing, we use known analysis methods to glean new information. We apply decomposition procedures such as ensemble empirical mode decomposition (EEMD) to study the state-to-state fluctuations in each of the intrinsic mode functions. We subject the intrinsic mode functions, the orthogonal basis set derived from the time-series using the EEMD, to a detailed analysis to draw information-content of the time-series. Also, we investigate the influence of random-noise on the data-driven state-to-state transition probabilities. We consider a second aspect of earthquake sequencing that is closely tied to its time-correlative behavior. Here, we extend the Fano factor and Allan factor analysis to the time-series of state-to state transition frequencies of a Markov chain. Our results support not only the usefulness the intrinsic mode functions in understanding the time-series but also the presence of power-law behaviour exemplified by the Fano factor and the Allan factor.

scholarly journals Brief Communication: Earthquake sequencing: analysis of time series constructed from the Markov chain model

2015 ◽  
Vol 22 (5) ◽  
pp. 589-599 ◽  
Author(s):  
M. S. Cavers ◽  
K. Vasudevan
Keyword(s):  
Time Series ◽  
Markov Chain ◽  
State Transition ◽  
Transition Probabilities ◽  
Markov Chain Model ◽  
Fano Factor ◽  
Chain Model ◽  
Sequencing Analysis ◽  
Intrinsic Mode Functions ◽  
Allan Factor

Abstract. Directed graph representation of a Markov chain model to study global earthquake sequencing leads to a time series of state-to-state transition probabilities that includes the spatio-temporally linked recurrent events in the record-breaking sense. A state refers to a configuration comprised of zones with either the occurrence or non-occurrence of an earthquake in each zone in a pre-determined time interval. Since the time series is derived from non-linear and non-stationary earthquake sequencing, we use known analysis methods to glean new information. We apply decomposition procedures such as ensemble empirical mode decomposition (EEMD) to study the state-to-state fluctuations in each of the intrinsic mode functions. We subject the intrinsic mode functions, derived from the time series using the EEMD, to a detailed analysis to draw information content of the time series. Also, we investigate the influence of random noise on the data-driven state-to-state transition probabilities. We consider a second aspect of earthquake sequencing that is closely tied to its time-correlative behaviour. Here, we extend the Fano factor and Allan factor analysis to the time series of state-to-state transition frequencies of a Markov chain. Our results support not only the usefulness of the intrinsic mode functions in understanding the time series but also the presence of power-law behaviour exemplified by the Fano factor and the Allan factor.

Detecting and Predicting Tipping Points

2019 ◽  
Vol 29 (08) ◽  
pp. 1930022 ◽  
Author(s):  
Xiaoyi Peng ◽  
Michael Small ◽  
Yi Zhao ◽  
Jack Murdoch Moore
Keyword(s):  
Time Series ◽  
Logistic Map ◽  
Pitchfork Bifurcation ◽  
Permutation Entropy ◽  
Dynamical Regime ◽  
Tipping Points ◽  
Edge Betweenness ◽  
Network Methods ◽  
Entropy Of Transition ◽  
Transition Networks

Tipping points are sudden, and sometimes irreversible and catastrophic, changes in a system’s dynamical regime. Complex networks are now widely used in the analysis of time series from a complex system. In this paper, we investigate the scope of network methods to indicate tipping points. In particular, we verify that the permutation entropy of transition networks constructed from time series observations of the logistic map can distinguish periodic and chaotic regimes and indicate bifurcations. The permutation entropy of transition networks, the mean edge betweenness of visibility graphs and the number of code words in compression networks, are each shown to indicate the onset of transition of a pitchfork bifurcation system. Our study shows that network methods are effective in detecting transitions. Network-based forecasts can be applied to models of real systems, as we illustrate by considering a lake eutrophication model.

Complex Networks Approach for Dynamical Characterization of Nonlinear Systems

2019 ◽  
Vol 29 (13) ◽  
pp. 1950188 ◽  
Author(s):  
Vander L. S. Freitas ◽  
Juliana C. Lacerda ◽  
Elbert E. N. Macau
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Complex Networks ◽  
Discrete Time ◽  
Betweenness Centrality ◽  
Bifurcation Diagrams ◽  
Computationally Expensive ◽  
Nonlinear Maps

Bifurcation diagrams and Lyapunov exponents are the main tools for dynamical systems characterization. However, they are often computationally expensive and complex to calculate. We present two approaches for dynamical characterization of nonlinear systems via the generation of an undirected complex network that is built from their time series. Periodic windows and chaos can be detected by analyzing network statistics like average degree, density and betweenness centrality. Results are assessed in two discrete time nonlinear maps.

scholarly journals Recovering Bistable Systems from Psychological Time Series

2019 ◽  
Author(s):  
Jonas M B Haslbeck ◽  
Oisín Ryan
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Mental Disorders ◽  
Complex Systems ◽  
Transition Probabilities ◽  
Sampling Frequency ◽  
Bistable System ◽  
Series Data ◽  
Global Dynamics ◽  
Systems Model

Conceptualizing mental disorders as complex dynamical systems has become a popular framework to study mental disorders. Especially bistable dynamical systems have received much attention, because their properties map well onto many characteristics of mental disorders. While these models were so far mostly used as stylized toy models, the recent surge in psychological time series data promises the ability to recover such models from data. In this paper we investigate how well popular (e.g., the Vector Autoregressive model) and more advanced (e.g., differential equation estimation) data analytic tools are suited to recover bistable dynamical systems from time series. Using a simulated high-frequency time series (measurement every six seconds) as an ideal case we show that while it is possible to recover global dynamics (e.g., position of fixed points, transition probabilities) it is difficult to recover the microdynamics (i.e., moment to moment interactions) of a bistable system. Repeating all analyses with a sampling frequency typical for Experience Sampling Method studies (measurement every 90 minutes) showed that the recovery of the global dynamics was still successful, but no microdynamics could be recovered. These results raise two fundamental issues involved in studying mental disorders from a complex systems perspective: first, it is generally unclear what to conclude from a statistical model about an underlying complex systems model; and second, if the sampling frequency is too low, it is impossible to recover microdynamics. In response to these results we propose a new modeling strategy based on substantively plausible dynamical systems models.

scholarly journals An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1511
Author(s):  
Kei Inoue
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Chaotic Maps ◽  
Chaotic Map ◽  
Calculation Formula ◽  
Two Dimensional ◽  
Information Dynamics ◽  
The Difference ◽  
Chaos Degree

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.

THE ONSET OF CHAOS IN NONLINEAR DYNAMICAL SYSTEMS DETERMINED WITH A NEW FRACTAL TECHNIQUE

Fractals ◽  
2005 ◽  
Vol 13 (01) ◽  
pp. 19-31 ◽  
Author(s):  
DORA E. MUSIELAK ◽  
ZDZISLAW E. MUSIELAK ◽  
KENNY S. KENNAMER
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Lyapunov Method ◽  
Nonlinear Dynamical Systems ◽  
Computational Time ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
Experimental Time Series ◽  
Method Agreement

A new fractal technique is used to investigate the onset of chaos in nonlinear dynamical systems. A comparison is made between this fractal technique and the commonly used Lyapunov exponent method. Agreement between the results obtained by both methods indicates that this technique may be used in a manner analogous to the Lyapunov exponents to predict onset of chaos. It is found that the fractal technique is much easier to implement than the Lyapunov method and it requires much less computational time. This fractal technique can easily be adopted to investigate the onset of chaos in many nonlinear dynamical systems and can be used to analyze theoretical and experimental time series.

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