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

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.

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Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

Mathematical Problems in Engineering ◽

10.1155/2018/4725148 ◽

2018 ◽

Vol 2018 ◽

pp. 1-14 ◽

Cited By ~ 1

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.

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Stochastic Stability of Gyroscopic Systems Under Bounded Noise Excitation

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455418500220 ◽

2018 ◽

Vol 18 (02) ◽

pp. 1850022 ◽

Cited By ~ 5

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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ON THE LOCAL STOCHASTIC STABILITY OF NONLINEAR COMPLEX NETWORKS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127410025491 ◽

2010 ◽

Vol 20 (01) ◽

pp. 177-184 ◽

Cited By ~ 1

Author(s):

ZHI-LONG HUANG ◽

ZHOU YAN ◽

XIAO-LING JIN ◽

GUANRONG CHEN

Keyword(s):

Lyapunov Exponent ◽

Theoretical Analysis ◽

Complex Networks ◽

Stochastic Stability ◽

Largest Lyapunov Exponent ◽

Stochastic Perturbations ◽

Approximate Analytical Solutions ◽

Coupling Strengths ◽

The Largest Lyapunov Exponent ◽

The One

The local stochastic stability of nonlinear complex networks is studied, subject to stochastic perturbations to the coupling strengths and stochastic parametric excitations to the nodes. The complex network is first linearized at its trivial solution and then the linearized network is reduced to N independent subsystems by using a suitable linear transformation, where N is the size of the network. The largest Lyapunov exponent for each subsystem is then calculated and all the approximate analytical solutions are evaluated for some specific cases. It is found that the largest Lyapunov exponent among all subsystems is the one associated with the subsystem that has the largest or the smallest eigenvalue of the configuration matrix of the network. Finally, an example is given to demonstrate the validity and accuracy of the theoretical analysis.

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Numerical Simulation of Dynamic Stability of Fractional Stochastic Systems

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455418501286 ◽

2018 ◽

Vol 18 (10) ◽

pp. 1850128 ◽

Cited By ~ 2

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.

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Largest Lyapunov Exponents and Bifurcations of Stochastic Nonlinear Systems

Shock and Vibration ◽

10.1155/1996/316740 ◽

1996 ◽

Vol 3 (4) ◽

pp. 313-320 ◽

Cited By ~ 2

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.

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ON THE COMPUTATION OF LYAPUNOV EXPONENTS FOR DISCRETE TIME SERIES: APPLICATIONS TO TWO-DIMENSIONAL SYMPLECTIC AND DISSIPATIVE MAPPINGS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127400001857 ◽

2000 ◽

Vol 10 (12) ◽

pp. 2791-2805 ◽

Cited By ~ 3

Author(s):

ELENA LEGA ◽

GABRIELLA DELLA PENNA ◽

CLAUDE FROESCHLÉ ◽

ALESSANDRA CELLETTI

Keyword(s):

Time Series ◽

Lyapunov Exponent ◽

Discrete Series ◽

Dissipative Systems ◽

Data Series ◽

Largest Lyapunov Exponent ◽

Two Dimensional ◽

Chaotic Motions ◽

Conservative Systems ◽

The Largest Lyapunov Exponent

Many techniques have been developed for the measure of the largest Lyapunov exponent of experimental short data series. The main idea, underlying the most common algorithms, is to mimic the method of computation proposed by Benettin and Galgani [1979]. The aim of the present paper is to provide an explanation for the reliability of some algorithms developed for short time series. To this end, we consider two-dimensional mappings as model problems and we compare the results obtained using the Benettin and Galgani method to those obtained using some algorithms for the computation of the largest Lyapunov exponent when dealing with short data series. In particular we focus our attention on conservative systems, which are not widely investigated in the literature. We show that using standard techniques the results obtained for discrete series related to area-preserving mappings are often unreliable, while dissipative systems are easier to analyze. In order to overcome the problem arising with conservative systems, we develop an alternative method, which takes advantage of the existing techniques. In particular, all algorithms provide a good approximation of the largest Lyapunov exponent in the strong chaotic symplectic case and in the dissipative one. However, the application of standard algorithms provides results which are not in agreement with the theoretical expectation for weak chaotic motions, and sometimes also for regular orbits. On the contrary, the method that we propose in this paper seems to work well for the weak chaotic case. Because of the speed of computation, we suggest to use all algorithms to cross-check the results.

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

International Journal of Bipolar Disorders ◽

10.1186/s40345-021-00235-3 ◽

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.

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

Volume 7A: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8041 ◽

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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