scholarly journals Largest Lyapunov Exponents and Bifurcations of Stochastic Nonlinear Systems

1996 ◽  
Vol 3 (4) ◽  
pp. 313-320 ◽  
Author(s):  
C.W.S. To ◽  
D.M. Li
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Systems ◽  
Random Excitation ◽  
Van Der Pol Oscillator ◽  
Stochastic Nonlinear Systems ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Points ◽  
Largest Lyapunov Exponents ◽  
The Largest Lyapunov Exponent

Two commonly adopted expressions for the largest Lyapunov exponents of linearized stochastic systems are reviewed. Their features are discussed in light of bifurcation analysis and one expression is selected for evaluating the largest Lyapunov exponent of a linearized system. An independent method, developed earlier by the authors, is also applied to determine the bifurcation points of a van der Pol oscillator under parametric random excitation. It is shown that the bifurcation points obtained by the independent technique agree qualitatively and quantitatively with those evaluated by using the largest Lyapunov exponent of the linearized oscillator.

Numerical Simulation of Dynamic Stability of Fractional Stochastic Systems

2018 ◽  
Vol 18 (10) ◽  
pp. 1850128 ◽  
Author(s):  
Jian Deng
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponent ◽  
Dynamic Stability ◽  
Stochastic Systems ◽  
Simulation Method ◽  
Largest Lyapunov Exponent ◽  
Stochastic Dynamic ◽  
Data Normalization ◽  
Moment Lyapunov Exponent ◽  
The Largest Lyapunov Exponent

The modern theory of stochastic dynamic stability is founded on two main exponents: the largest Lyapunov exponent and moment Lyapunov exponent. Since any fractional viscoelastic system is indeed a system with memory, data normalization during iterations will disregard past values of the response and therefore the use of data normalization seems not appropriate in numerical simulation of such systems. A new numerical simulation method is proposed for determining the [Formula: see text]th moment Lyapunov exponent, which governs the [Formula: see text]th moment stability of the fractional stochastic systems. The largest Lyapunov exponent can also be obtained from moment Lyapunov exponents. Examples of the two-dimensional fractional systems under wideband noise and bounded noise excitations are presented to illustrate the simulation method.

scholarly journals Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Jian Deng
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Stability ◽  
Random Fluctuation ◽  
Largest Lyapunov Exponent ◽  
Real Noise ◽  
The Largest Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment stochastic stability and almost-sure stochastic stability of two-degree-of-freedom coupled viscoelastic systems, under the parametric excitation of a real noise, are investigated through the moment Lyapunov exponents and the largest Lyapunov exponent, respectively. The real noise is also called the Ornstein-Uhlenbeck stochastic process. For small damping and weak random fluctuation, the moment Lyapunov exponents are determined approximately by using the method of stochastic averaging and a formulated eigenvalue problem. The largest Lyapunov exponent is calculated through its relation with moment Lyapunov exponents. The stability index, the stability boundaries, and the critical excitation are obtained analytically. The effects of various parameters on the stochastic stability of the system are then discussed in detail. Monte Carlo simulation is carried out to verify the approximate results of moment Lyapunov exponents. As an application example, the stochastic stability of a flexural-torsional viscoelastic beam is studied.

Chaotic Characteristics of Heart Sound Signals Based on the Largest Lyapunov Exponent

2013 ◽  
Vol 411-414 ◽  
pp. 1117-1124 ◽  
Author(s):  
Li Sha Sun ◽  
Li Li Sun
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Heart Sound ◽  
Mean Value ◽  
Sound Signal ◽  
Largest Lyapunov Exponent ◽  
Normal Heart ◽  
The Mean ◽  
Largest Lyapunov Exponents

Seeking a way non-invasive and adaptive to differentiate the normal and abnormal heart sound signals to provide more valuable reference method to the clinical diagnosis. This paper made the largest Lyapunov exponent as the mainline. According to the unity of the whole signal in different stages, a method to study the characteristic in stage was proposed. First of all, we made phase space reconstitution to the typical seven normal and abnormal heart sound signals. Then, we calculated the largest Lyapunov exponents according to the phase space reconstitution parameters. At last, we compared and analyzed the mean values of the largest Lyapunov exponents. The mean value of the normal heart sound signal in S1 was 0.145, which was much larger than that of the abnormal signals and the mean value of the normal heart sound signal in S2 was larger than that of the abnormal ones, too. This conclusion means that there are chaotic characteristic in the heart sound signals and the degree of chaos in normal heart sounds is higher than that in the abnormal heart sound signals.

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.

Lyapunov Exponents and Stochastic Stability of Two-Dimensional Parametrically Excited Random Systems

1993 ◽  
Vol 60 (3) ◽  
pp. 677-682 ◽  
Author(s):  
S. T. Ariaratnam ◽  
Wei-Chau Xie
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Random Systems ◽  
Random Fluctuation ◽  
Largest Lyapunov Exponent ◽  
Random Disturbance ◽  
Two Dimensional ◽  
Parametrically Excited ◽  
The Largest Lyapunov Exponent

The variation of the largest Lyapunov exponent for two-dimensional parametrically excited stochastic systems is studied by a method of linear transformation. The sensitivity to random disturbance of systems undergoing bifurcation is investigated. Two commonly occurring examples in structural dynamics are considered, where the random fluctuation appears in the stiffness term or the damping term. The boundaries of almost-sure stochastic stability are determined by the vanishing of the largest Lyapunov exponent of the linearized system. The validity of the approximate results is checked by numerical simulation.

scholarly journals Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions

Photonics ◽  
2021 ◽  
Vol 8 (12) ◽  
pp. 554
Author(s):  
Gary McCormack ◽  
Rejish Nath ◽  
Weibin Li
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Long Range ◽  
Optical Lattice ◽  
Mean Field ◽  
Interaction Strength ◽  
Soft Core ◽  
Largest Lyapunov Exponent ◽  
Localized State ◽  
The Largest Lyapunov Exponent

We study the chaos and hyperchaos of Rydberg-dressed Bose–Einstein condensates (BECs) in a one-dimensional optical lattice. Due to the long-range, soft-core interaction between the dressed atoms, the dynamics of the BECs are described by the extended Bose-Hubbard model. In the mean-field regime, we analyze the dynamical stability of the BEC by focusing on the ground state and localized state configurations. Lyapunov exponents of the two configurations are calculated by varying the soft-core interaction strength, potential bias, and length of the lattice. Both configurations can have multiple positive Lyapunov exponents, exhibiting hyperchaotic dynamics. We show the dependence of the number of the positive Lyapunov exponents and the largest Lyapunov exponent on the length of the optical lattice. The largest Lyapunov exponent is directly proportional to areas of phase space encompassed by the associated Poincaré sections. We demonstrate that linear and hysteresis quenches of the lattice potential and the dressed interaction lead to distinct dynamics due to the chaos and hyperchaos. Our work is relevant to current research on chaos as well as collective and emergent nonlinear dynamics of BECs with long-range interactions.

EFFECTS OF TREND AND PERIODICITY ON THE CORRELATION DIMENSION AND THE LYAPUNOV EXPONENTS

2008 ◽  
Vol 18 (12) ◽  
pp. 3679-3687 ◽  
Author(s):  
AYDIN A. CECEN ◽  
CAHIT ERKAL
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Time Series Data ◽  
Logistic Map ◽  
Model Identification ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
The Impact

We present a critical remark on the pitfalls of calculating the correlation dimension and the largest Lyapunov exponent from time series data when trend and periodicity exist. We consider a special case where a time series Zi can be expressed as the sum of two subsystems so that Zi = Xi + Yi and at least one of the subsystems is deterministic. We show that if the trend and periodicity are not properly removed, correlation dimension and Lyapunov exponent estimations yield misleading results, which can severely compromise the results of diagnostic tests and model identification. We also establish an analytic relationship between the largest Lyapunov exponents of the subsystems and that of the whole system. In addition, the impact of a periodic parameter perturbation on the Lyapunov exponent for the logistic map and the Lorenz system is discussed.

Experimental and Simulation of a Cam and Translated Roller Follower Over a Range of Speeds

2016 ◽  
Author(s):  
Louay S. Yousuf ◽  
Dan B. Marghitu
Keyword(s):  
Experimental Data ◽  
Lyapunov Exponent ◽  
High Resolution ◽  
Lyapunov Exponents ◽  
Planar Motion ◽  
Largest Lyapunov Exponents ◽  
Set Up ◽  
The Impact ◽  
Impact With Friction

In this study a cam and follower mechanism is analyzed. There is a clearance between the follower and the guide. The mechanism is analyzed using SolidWorks simulations taking into account the impact and the friction between the roller follower and the guide. Four different follower guide’s clearances have been used in the simulations like 0.5, 1, 1.5, and 2 mm. An experimental set up is developed to capture the general planar motion of the cam and follower. The measures of the cam and the follower positions are obtained through high-resolution optical encoders (markers). The effect of follower guide’s clearance is investigated for different cam rotational speeds such as 100, 200, 300, 400, 500, 600, 700 and 800 R.P.M. Impact with friction is considered in our study to calculate the Lyapunov exponent. The largest Lyapunov exponents for the simulated and experimental data are analyzed and selected.

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.

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