Novel gradient calculation method for the largest Lyapunov exponent of chaotic systems

2016 ◽  
Vol 85 (3) ◽  
pp. 1377-1392 ◽  
Author(s):  
Haitao Liao
Keyword(s):  
Lyapunov Exponent ◽  
Calculation Method ◽  
Chaotic Systems ◽  
Largest Lyapunov Exponent ◽  
Gradient Calculation ◽  
The Largest Lyapunov Exponent

Numerical Analysis and Improved Algorithms for Lyapunov-Exponent Calculation of Discrete-Time Chaotic Systems

2016 ◽  
Vol 26 (13) ◽  
pp. 1650219 ◽  
Author(s):  
Jianbin He ◽  
Simin Yu ◽  
Jianping Cai
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Discrete Time ◽  
Chaotic Systems ◽  
Orthogonal Decomposition ◽  
Largest Lyapunov Exponent ◽  
Error Message ◽  
Eigenvalue Method ◽  
Calculation Results ◽  
The Largest Lyapunov Exponent

Lyapunov exponent is an important index for describing chaotic systems behavior, and the largest Lyapunov exponent can be used to determine whether a system is chaotic or not. For discrete-time dynamical systems, the Lyapunov exponents are calculated by an eigenvalue method. In theory, according to eigenvalue method, the more accurate calculations of Lyapunov exponent can be obtained with the increment of iterations, and the limits also exist. However, due to the finite precision of computer and other reasons, the results will be numeric overflow, unrecognized, or inaccurate, which can be stated as follows: (1) The iterations cannot be too large, otherwise, the simulation result will appear as an error message of NaN or Inf; (2) If the error message of NaN or Inf does not appear, then with the increment of iterations, all Lyapunov exponents will get close to the largest Lyapunov exponent, which leads to inaccurate calculation results; (3) From the viewpoint of numerical calculation, obviously, if the iterations are too small, then the results are also inaccurate. Based on the analysis of Lyapunov-exponent calculation in discrete-time systems, this paper investigates two improved algorithms via QR orthogonal decomposition and SVD orthogonal decomposition approaches so as to solve the above-mentioned problems. Finally, some examples are given to illustrate the feasibility and effectiveness of the improved algorithms.

scholarly journals The futility of long-term predictions in bipolar disorder: mood fluctuations are the result of deterministic chaotic processes

2021 ◽  
Vol 9 (1) ◽  
Author(s):  
Abigail Ortiz ◽  
Kamil Bradler ◽  
Maxine Mowete ◽  
Stephane MacLean ◽  
Julie Garnham ◽  
...  
Keyword(s):  
Bipolar Disorder ◽  
Fractal Dimension ◽  
Lyapunov Exponent ◽  
Detrended Fluctuation Analysis ◽  
Largest Lyapunov Exponent ◽  
Fluctuation Analysis ◽  
Mood Regulation ◽  
The Largest Lyapunov Exponent ◽  
Depressive Episodes ◽  
Detrended Fluctuation

Abstract Background Understanding the underlying architecture of mood regulation in bipolar disorder (BD) is important, as we are starting to conceptualize BD as a more complex disorder than one of recurring manic or depressive episodes. Nonlinear techniques are employed to understand and model the behavior of complex systems. Our aim was to assess the underlying nonlinear properties that account for mood and energy fluctuations in patients with BD; and to compare whether these processes were different in healthy controls (HC) and unaffected first-degree relatives (FDR). We used three different nonlinear techniques: Lyapunov exponent, detrended fluctuation analysis and fractal dimension to assess the underlying behavior of mood and energy fluctuations in all groups; and subsequently to assess whether these arise from different processes in each of these groups. Results There was a positive, short-term autocorrelation for both mood and energy series in all three groups. In the mood series, the largest Lyapunov exponent was found in HC (1.84), compared to BD (1.63) and FDR (1.71) groups [F (2, 87) = 8.42, p < 0.005]. A post-hoc Tukey test showed that Lyapunov exponent in HC was significantly higher than both the BD (p = 0.003) and FDR groups (p = 0.03). Similarly, in the energy series, the largest Lyapunov exponent was found in HC (1.85), compared to BD (1.76) and FDR (1.67) [F (2, 87) = 11.02; p < 0.005]. There were no significant differences between groups for the detrended fluctuation analysis or fractal dimension. Conclusions The underlying nature of mood variability is in keeping with that of a chaotic system, which means that fluctuations are generated by deterministic nonlinear process(es) in HC, BD, and FDR. The value of this complex modeling lies in analyzing the nature of the processes involved in mood regulation. It also suggests that the window for episode prediction in BD will be inevitably short.

Estimation of the Largest Lyapunov Exponent of Discontinuous Systems Using Chaos Synchronization

1999 ◽  
Author(s):  
Andrzej Stefanski ◽  
Jerzy Wojewoda ◽  
Tomasz Kapitaniak ◽  
John Brindley
Keyword(s):  
Lyapunov Exponent ◽  
Mechanical System ◽  
Dry Friction ◽  
Chaos Synchronization ◽  
Largest Lyapunov Exponent ◽  
Discontinuous Systems ◽  
System A ◽  
The Largest Lyapunov Exponent

Abstract Properties of chaos synchronization have been used for estimation of the largest Lyapunov exponent of a discontinuous mechanical system. A method for such estimation is proposed and an example is shown, based on coupling of two identical systems with dry friction which is modelled according to the Popp-Stelter formula.

Automatic identification of eye movements using the largest lyapunov exponent

2018 ◽  
Vol 41 ◽  
pp. 10-20 ◽  
Author(s):  
Alexandra I. Korda ◽  
Pantelis A. Asvestas ◽  
George K. Matsopoulos ◽  
Errikos M. Ventouras ◽  
Nikolaos Smyrnis
Keyword(s):  
Eye Movements ◽  
Lyapunov Exponent ◽  
Largest Lyapunov Exponent ◽  
Automatic Identification ◽  
The Largest Lyapunov Exponent

Improvement the Largest Lyapunov Exponent for Measurement the Northeast Monsoon Prediction

2018 ◽  
Vol 879 ◽  
pp. 217-221
Author(s):  
Sunisa Saiuparad
Keyword(s):  
Lyapunov Exponent ◽  
Measurement Method ◽  
Limit Theorems ◽  
Climate Model ◽  
Global Climate ◽  
Global Climate Model ◽  
Largest Lyapunov Exponent ◽  
Northeast Monsoon ◽  
Medium Range Weather Forecast ◽  
The Largest Lyapunov Exponent

Thailand is an agricultural country. So that, the water resources are important. The water management is very important for keep the water used in necessary time. The monsoon is causes a heavy rain. So that, the monsoon prediction by the global climate model is important. The accuracy of the forecasts by the predictability measurement method is very important. In this research, the northeast monsoon prediction in Thailand by the global climate model. The data from The Bjerknes Centre for Climate Research (BCCR), University of Bergen, Norway. The global climate model is Bergen Climate Model (BCM) Version 2.0 (BCCR-BCM2.0) of the Intergovernmental Panel on Climate Change (IPCC) and the European Centre for Medium Range Weather Forecast (ECMWF) are used. The largest Lyapunov exponent (LLE) is the predictability measurement method for verify the efficiency of the global climate model and improvement the LLE by limit theorems. The result to show that the improvement the LLE by limit theorems can be measure the accuracy of the northeast monsoon prediction in Thailand by the global climate model are suitable.

Stochastic Stability of Gyroscopic Systems Under Bounded Noise Excitation

2018 ◽  
Vol 18 (02) ◽  
pp. 1850022 ◽  
Author(s):  
Jian Deng
Keyword(s):  
Lyapunov Exponent ◽  
Parametric Resonance ◽  
Stochastic Stability ◽  
Largest Lyapunov Exponent ◽  
Gyroscopic System ◽  
Bounded Noise ◽  
Rotating Shaft ◽  
Dynamic Stochastic ◽  
The Largest Lyapunov Exponent ◽  
Gyroscopic Systems

Dynamic stochastic stability of a two-degree-of-freedom gyroscopic system under bounded noise parametric excitation is studied in this paper through moment Lyapunov exponent and the largest Lyapunov exponent. A rotating shaft subject to stochastically fluctuating thrust is taken as a typical example. To obtain these two exponents, the gyroscopic differential equation of motion is first decoupled into Itô stochastic differential equations by using the method of stochastic averaging. Then mathematical transformations are used in these Itô equation to obtain a partial differential eigenvalue problem governing moment Lyapunov exponents, the slope of which at the origin is equal to the largest Lyapunov exponent. Depending upon the numerical relationship between the natural frequency and the excitation frequencies, the gyroscopic system may fall into four types of parametric resonance, i.e. no resonance, subharmonic resonance, combination additive resonance, and combination differential resonance. The effects of noise and frequency detuning parameters on the parametric resonance are investigated. The results pave the way to utilize or control the vibration of gyroscopic systems under stochastic excitation.

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