Some Properties of Chaotic Modified of Bogdanov Map

In this paper we have proposed a novel amplitude suppression algorithm for EEG signals collected during epileptic seizure. Then we have proposed a measure of chaoticity for a chaotic signal, which is somewhat similar to measuring sensitive dependence on initial conditions by measuring Lyapunov exponent in a chaotic dynamical system. We have shown that with respect to this measure the amplitude suppression algorithm reduces chaoticity in a chaotic signal (EEG signal is chaotic). We have compared our measure with the estimated largest Lyapunov exponent measure by the largelyap function, which is similar to Wolf's algorithm. They fit closely for all but one of the cases. How the algorithm can help to improve patient specific dosage titration during vagus nerve stimulation therapy has been outlined.

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SCALE-INVARIANCE OF NORMALIZED YEARLY MEAN GRAIN YIELD ANOMALY SERIES

Advances in Complex Systems ◽

10.1142/s0219525907001161 ◽

2007 ◽

Vol 10 (03) ◽

pp. 395-412 ◽

Cited By ~ 1

Author(s):

RICARDO DAVID VALDEZ-CEPEDA ◽

OLIVIA DELGADILLO-RUIZ ◽

RAFAEL MAGALLANES-QUINTANAR ◽

GERARDO MIRAMONTES-de LEÓN ◽

JOSÉ LUIS GARCÍA-HERNÁNDEZ ◽

...

Keyword(s):

Lyapunov Exponent ◽

Grain Yield ◽

Scaling Laws ◽

Crop Yields ◽

Initial Conditions ◽

Southern Oscillation ◽

Triticum Aestivum L ◽

Time Frequency ◽

Complex Processes ◽

External Forcings

In crop science, tools of non-linear dynamics, fractals, chaos, intermittency and self-organized criticality may be employed and applied to the analysis of spatial variability and temporal behavior of agro-meteorological variables, soil properties, plant attributes, commercial yields, and prices of the agricultural products in order to gain knowledge about underlying complex processes. A search on the occurrence of particular scaling laws in Mexico's normalized yearly mean grain yield anomaly series of maize (Zea mays L.), beans (Phaseolus vulgaris L.), wheat (Triticum aestivum L.) and rice (Oriza sativa L.), using a variography approach is reported in this work. Additionally, power spectrum determination, time-frequency analysis, and estimation of Lyapunov exponent were performed for each profile in order to obtain useful information on the frequency contents and signs at which important frequencies occur as well as to determine their sensitivity to initial conditions. Fractal analysis gives us the order maize < wheat < rice < beans of sensitivity to external forcings, which was the same as that obtained through the Lyapunov exponent values. Results confirm that the final outputs (crop yields) in agricultural systems are affected by the magnetic and sunspot cycles of the Sun, the El Niño Southern Oscillation (ENSO), and the quasi-biannual oscillation, and possibly by the so-called heliospheric mid-term-quasi-periodicities, which act on different time (or spatial and spatial-time) scales. In particular, the maize normalized year anomaly series is clearly correlated with March, April, May and June series of the ENSO index.

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CHAOTIC ANALYSIS OF A TIME SERIES COMPOSED OF SEISMIC EVENTS RECORDED IN JAPAN

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000071 ◽

1994 ◽

Vol 04 (01) ◽

pp. 87-98 ◽

Cited By ~ 14

Author(s):

G.P. PAVLOS ◽

L. KARAKATSANIS ◽

J.B. LATOUSSAKIS ◽

D. DIALETIS ◽

G. PAPAIOANNOU

Keyword(s):

Time Series ◽

Lyapunov Exponent ◽

Initial Conditions ◽

Low Frequency ◽

Underlying Mechanism ◽

Largest Lyapunov Exponent ◽

Seismic Events ◽

Earthquake Dynamics ◽

The Largest Lyapunov Exponent ◽

Low Dimensional

A chaotic analysis approach was applied to an earthquake time series recorded in the Japanese area in order to test the assumption that the earthquake process could be the manifestation of a chaotic low dimensional process. For the study of the seismicity we have used a time series consisting of time differences between two consecutive seismic events with magnitudes greater than 2.6. The results of our study show that the underlying mechanism, as expressed by the time series, can be described by low dimensional chaotic dynamics. The power spectrum of the time series shows a power law profile with two slopes, α=1.4 in the low frequency and α=0.05 in the high frequency regions, while the slopes of the correlation integrals show an apparent plateau at the scaling region, which saturates at the value D≈3.2. The largest Lyapunov exponent was found to be ≈0.9. The positive value of the largest Lyapunov exponent reveals strong sensitivity to initial conditions of the supposed earthquake dynamics.

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The Chaotic Motion of the Solar System

International Astronomical Union Colloquium ◽

10.1017/s025292110006588x ◽

1993 ◽

Vol 132 ◽

pp. 21-21

Author(s):

J. Laskar

Keyword(s):

Lyapunov Exponent ◽

Solar System ◽

Initial Conditions ◽

Maximum Lyapunov Exponent ◽

Secular Resonance ◽

Critical Argument ◽

Outer Planets ◽

Fundamental Frequencies ◽

Chaotic Nature ◽

Evolution With Time

AbstractIn a previous paper (Laskar, Nature, 338, 237-238), the chaotic nature of the solar system excluding Pluto was established by the numerical computation of the maximum Lyapunov exponent of its secular system over 200 Myr. In the present an explanation is given for the exponential divergence of the orbits: it is due to the transition from libration to circulation of the critical argument of the secular resonance 2(g4−g3)−(s4−s3) related to the motions of perihelions and nodes of the Birth and Mars. An other important secular resonance is identified: (g1−g5)−(s1−s2). Its critical argument stays in libration over 200 Myr with a period of about 10 Myr and amplitude from 85° to 135°. The main features of the solutions of the inner planets are now identified when taking these resonances into account. Estimates of the size of the chaotic regions are determined by a new numerical method using the evolution with time of the fundamental frequencies. The size of the chaotic regions in the inner solar system are large and correspond to variations of about 0.2 arcsec/year in the fundamental frequencies. The chaotic nature of the inner solar system can thus be considered as robust against small variations of the initial conditions or of the model. The chaotic regions related to the outer planets frequencies are very thin except for g6 which present variations sufficiently large to be significant over the age of the solar system.

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Dynamics and Stability of a Nonlinear Brake Model

Volume 6C: 18th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc2001/vib-21579 ◽

2001 ◽

Author(s):

Jörg Wauer ◽

Jürgen Heilig

Keyword(s):

Stability Analysis ◽

Lyapunov Exponent ◽

Nonlinear Model ◽

Numerical Evaluation ◽

Critical Speed ◽

Initial Conditions ◽

Brake Disc ◽

Disc Brake ◽

Brake Pad ◽

The Stability

Abstract The dynamics of a nonlinear car disc brake model is investigated and compared with a simplified linear model. The rotating brake disc is approximated by a rotating ring. The brake pad is modeled as a point mass which is in contact with the rotating ring and visco-elastically suspended in axial and circumferential direction. The stability analysis for the nonlinear model is performed by a numerical evaluation of the top Lyapunov-exponent. Several parameter studies for the nonlinear model are discussed. It is shown that dynamic instabilities of the nonlinear model are estimated at subcritical rotating speeds lower than 10% of the critical speed. Further, the sensitivity of the nonlinear model to the initial conditions and the stiffness ratios is demonstrated.

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Numerical Experiments on the Motion of the Outer Planets

Symposium - International Astronomical Union ◽

10.1017/s0074180900090896 ◽

1992 ◽

Vol 152 ◽

pp. 25-32 ◽

Cited By ~ 2

Author(s):

Gerald D. Quinlan

Keyword(s):

Lyapunov Exponent ◽

Solar System ◽

Probability Distribution ◽

Numerical Experiments ◽

Initial Conditions ◽

Major Axis ◽

Solar Systems ◽

Outer Planets ◽

System A ◽

Real Value

We have integrated the motion of the four Jovian planets on Myr timescales in fictitious solar systems in which the orbits differ from those of the real solar system. A change of ≲1% in the major axis of any one of the planets from its real value can lead to chaotic motion with a Lyapunov exponent larger than 10-5 yr−1. A survey of fifty solar systems with initial conditions chosen at random from a reasonable probability distribution shows the majority of them to be chaotic.

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LOW-DIMENSIONAL BEHAVIOR OF NONLINEAR TEARING MODE DYNAMICS IN A ROTATING PLASMA

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127493000957 ◽

1993 ◽

Vol 03 (05) ◽

pp. 1155-1168 ◽

Cited By ~ 1

Author(s):

M. PERSSON ◽

C. R. LAING

Keyword(s):

Lyapunov Exponent ◽

Initial Conditions ◽

Largest Lyapunov Exponent ◽

Tearing Mode ◽

Oscillatory State ◽

The Largest Lyapunov Exponent ◽

Low Dimensional ◽

Sensitivity To Initial Conditions

Simulations of resistive magnetohydrodynamics in a rotating plasma are analyzed by calculating the correlation dimensions and the local intrinsic dimensions. A rotating plasma with a nonlinear oscillatory state is found to be associated with a low-dimensional attractor. The solutions are also checked for sensitivity to initial conditions, characterized by the sign of the largest Lyapunov exponent.

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Chaos Detection and Optimal Control in a Cannibalistic Prey–Predator System with Harvesting

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420501710 ◽

2020 ◽

Vol 30 (12) ◽

pp. 2050171

Author(s):

Harsha Kharbanda ◽

Sachin Kumar

Keyword(s):

Lyapunov Exponent ◽

Equilibrium Point ◽

Chaotic Behavior ◽

Initial Conditions ◽

Optimal Level ◽

Optimal Harvesting ◽

Predator Population ◽

Maximum Lyapunov Exponent ◽

Catch Per Unit Effort ◽

Chaos Detection

This paper deals with a stage-structured predator–prey system which incorporates cannibalism in the predator population and harvesting in both population. The predator population is categorized into two divisions; adult predator and juvenile predator. The adult predator and prey species are harvested via hypothesis of catch-per-unit-effort, whereas juveniles are safe from being harvested. Mathematically, the dynamic behavior of the system such as existing conditions of equilibria with their stability is studied. The global asymptotic stability of prey-free equilibrium point and nonzero equilibrium point, if they exist, is proved by considering respective Lyapunov functions. The system undergoes transcritical and Hopf–Andronov bifurcations. The impacts of predator harvesting rate and prey harvesting rate on the system are analyzed by taking them as bifurcation parameters. The route to chaos is discussed by showing maximum Lyapunov exponent to be positive with sensitivity dependence on the initial conditions. The chaotic behavior of the system is confirmed by positive maximum Lyapunov exponent and non-integer Kaplan–Yorke dimension. Numerical simulations are executed to probe our theoretic findings. Also, the optimal harvesting policy is studied by applying Pontryagin’s maximum principle. Harvesting effort being an emphatic control instrument is considered to protect prey–predator population, and preserve them also through an optimal level.

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Domains of Attraction for an Autoparametric Mass-Pendulum System: A Lyapunov Exponent Map Approach

Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2003/vib-48594 ◽

2003 ◽

Author(s):

Khalid El-Rifai ◽

George Haller ◽

Anil K. Bajaj

Keyword(s):

Lyapunov Exponent ◽

Finite Time ◽

Initial Conditions ◽

Transient Behavior ◽

System Response ◽

Pendulum System ◽

Domains Of Attraction ◽

Nonlinear Dynamical ◽

Finite Time Lyapunov Exponents ◽

Finite Time Lyapunov Exponent

Many recent studies have been performed on resonantly excited mass-pendulum systems with autoparametric (internal) resonance capturing interesting local steady state phenomena. The objective of this work is to explore the transient behavior in such systems. The domains of attraction of the time-periodic system provide some help in understanding the transient dynamics, and these are sought using a recently developed algorithm that solves for the finite-time Lyapunov exponent over a grid of initial conditions. Though the use of finite-time Lyapunov exponents in nonlinear dynamical analyses is not novel, its application to multi-degree-offreedom forced nonlinear systems has not been reported in the literature. In addition to identifying regions of different final states, the technique used captures different levels of attraction within a domain. This sheds some light on the role played by other modes present in a multi-degree-of-freedom system in shaping the overall system response.

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