scholarly journals Time Recurrence Analysis of a Near Singular Billiard

2019 ◽  
Vol 24 (2) ◽  
pp. 50 ◽  
Author(s):  
Rodrigo Simile Baroni ◽  
Ricardo Egydio de Carvalho ◽  
Bruno Castaldi ◽  
Bruno Furlanetto
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Singular Limit ◽  
Largest Lyapunov Exponent ◽  
Classical Dynamics ◽  
Sharp Drop ◽  
Nonlinear Part ◽  
Return Times

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.

The Chaotic Attractor of the Sediment Movement

Fractals ◽  
1998 ◽  
Vol 06 (02) ◽  
pp. 191-196
Author(s):  
Fengsu Chen ◽  
Kongxian Xue ◽  
Wenkang Cai
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Attractor ◽  
Yangtze River ◽  
Correlation Dimension ◽  
Chaotic Behavior ◽  
Largest Lyapunov Exponent ◽  
The Yangtze River ◽  
Sediment Movement ◽  
Reconstructed Phase Space

We consider the chaotic behavior of the sediment movement with the observed data of the Yangtze River in China and the method of the reconstructed phase space and we find that in the sediment movement there is an attractor. As far as the real example mentioned in this paper is concerned, the correlation dimension and the largest Lyapunov exponent are around 6.6 and 0.013 respectively. These results are crucially referential for estimating the mode of the sediment movement, designing the scheme of the sediment observation, and studying the predictability problem of the sediment.

Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors

1994 ◽  
Vol 263 ◽  
pp. 93-132 ◽  
Author(s):  
George Broze ◽  
Fazle Hussain
Keyword(s):  
Phase Diagram ◽  
Phase Space ◽  
Lyapunov Exponent ◽  
Conceptual Model ◽  
Chaotic Attractor ◽  
Excitation Frequency ◽  
Chaotic Attractors ◽  
Largest Lyapunov Exponent ◽  
Open Flow ◽  
Vortex Pairing

Conclusive experimental evidence is presented for the existence of a low-dimensional temporal dynamical system in an open flow, namely the near field of an axisymmetric, subsonic free jet. An initially laminar jet (4 cm air jet in the Reynolds number range 1.1 × 104 [Lt ] ReD × 9.1 × 104) with a top-hat profile was studied using single-frequency, longitudinal, bulk excitation. Two non-dimensional control parameters – forcing frequency StD (≡fexD/Ue, where fez is the excitation frequency, D is the jet exit diameter and Ue is the exit velocity) and forcing amplitude af (≡ u’f/Ue, where u’f is the jet exit r.m.s. longitudinal velocity fluctuation at the excitation frequency) – were varied over the ranges 10-4 < af < 0.3 and 0.3 < StD < 3.0 in order to construct a phase diagram. Periodic and chaotic states were found over large domains of the parameter space. The periodic attractors correspond to stable pairing (SP) and stable double pairing (SDP) of rolled-up vortices. One chaotic attractor, near SP in the parameter space, results from nearly periodic modulations of pairing (NPMP) of vortices. At large scales (i.e. approximately the size of the attractor) in phase space, NPMP exhibits approximately quasi-periodic behaviour, including modulation sidebands around ½fex in u-spectra, large closed loops in its Poincaré sections, correlation dimension v ∼ 2 and largest Lyapunov exponent λ1 ∼ 0. But investigations at smaller scales (i.e. distances greater than, but of the order of, trajectory separation) in phase space reveal chaos, as shown by v > 2 and λ1 > 0. The other chaotic attractor, near SDP, results from nearly periodic modulations of the first vortex pairing but chaotic modulations of the second pairing and has a broadband spectrum, a dimension 2.5 [Lt ] v [Lt ] 3 and the largest Lyapunov exponent 0.2 [Lt ] λ1 [Lt ] 0.7 bits per orbit (depending on measurement locations in physical and parameter spaces).A definition that distinguishes between physically and dynamically open flows is proposed and justified by our experimental results. The most important conclusion of this study is that a physically open flow, even one that is apparently dynamically open due to convective instability, can exhibit dynamically closed behaviour as a result of feedback. A conceptual model for transitional jets is proposed based on twodimensional instabilities, subharmonic resonance and feedback from downstream vortical structures to the nozzle lip. Feedback was quantified and shown to affect the exit fundamental–subharmonic phase difference ϕ – a crucial variable in subharmonic resonance and, hence, vortex pairing. The effect of feedback, the sensitivity of pairings to ϕ, the phase diagram, and the documented periodic and chaotic attractors demonstrate the validity of the proposed conceptual model.

scholarly journals The Dynamics of Philippine Foreign Exchange Rates (2013-2017): A Test for Chaos

2019 ◽  
Vol 8 (11) ◽  
pp. 486-489
Keyword(s):  
Time Series ◽  
Exchange Rate ◽  
Lyapunov Exponent ◽  
Exchange Rates ◽  
Foreign Exchange ◽  
Chaotic Behavior ◽  
The Philippines ◽  
Series Data ◽  
Largest Lyapunov Exponent ◽  
Foreign Exchange Rates

In most studies on dynamics of time series financial data, the absence of chaotic behavior is generally observed. However, this theory is not yet established in the dynamics of foreign exchange rates. Conflicting claims of presence and absence of chaos in foreign exchange rates open door for further investigation considering various deterministic factors. This work examines the dynamics of exchange rate of the Philippine Peso against selected foreign currencies. Time series data were collected for eight (8) of Philippine’s top trading partners as categorized according to economic condition. The data obtained with permission from the Central Bank of the Philippines covered the years 2013 to 2017. Data sets were plotted revealing non-linear movement of Philippine exchange rates against time. The foreign exchange rate time series obtained per currency were examined for chaotic behavior by computing the Largest Lyapunov Exponents (LLE). A positive Lyapunov exponent is an indication of sensitivity dependence, i.e, a chaotic dynamics; whereas, a negative Lyapunov exponent indicates otherwise. Computed LLE’s varied per currency but all were found to be negative. Therefore, using the Largest Lyapunov Exponent Test (LLE), analysis of the time series of Philippine foreign exchange rates shows little evidence of chaotic patterns.

COLOR MAP OF LYAPUNOV EXPONENTS OF INVARIANT SETS

1999 ◽  
Vol 09 (07) ◽  
pp. 1459-1463 ◽  
Author(s):  
MARCO MONTI ◽  
WILLIAM B. PARDO ◽  
JONATHAN A. WALKENSTEIN ◽  
EPAMINONDAS ROSA ◽  
CELSO GREBOGI
Keyword(s):  
Lyapunov Exponent ◽  
Parameter Space ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Invariant Sets ◽  
Largest Lyapunov Exponent ◽  
Specific Behavior ◽  
Practical Applications ◽  
The Lorenz System ◽  
Different Levels

The largest Lyapunov exponent of the Lorenz system is used as a measure of chaotic behavior to construct parameter space color maps. Each color in these maps corresponds to different values of the Lyapunov exponent and indicates, in parameter space, the locations of different levels of chaos for the Lorenz system. Practical applications of these maps include moving in parameter space from place to place without leaving a region of specific behavior of the system.

DYNAMICS OF ELECTROSTATICALLY ACTUATED MICRO-ELECTRO-MECHANICAL SYSTEMS: SINGLE DEVICE AND ARRAYS OF DEVICES

2009 ◽  
Vol 19 (03) ◽  
pp. 1007-1022 ◽  
Author(s):  
V. Y. TAFFOTI YOLONG ◽  
P. WOAFO
Keyword(s):  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Mechanical Systems ◽  
Largest Lyapunov Exponent ◽  
Mems Devices ◽  
Micro Electro Mechanical Systems ◽  
Electrostatically Actuated ◽  
In Series ◽  
Electrostatic Coupling

The dynamical behavior of micro-electro-mechanical systems (MEMS) with electrostatic coupling is studied. A nonlinear modal analysis approach is applied to decompose the partial differential equation into a set of ordinary differential equations. The stability analysis of the equilibrium points is investigated. The amplitudes of the harmonic oscillatory states in the triple resonant states are obtained and discussed. Chaotic behavior is investigated using bifurcations diagram and the largest Lyapunov exponent. The dynamics of the MEMS with multiple functions in series is also investigated as well as the transitions boundaries for the complete synchronization state in a shift-invariant set of coupled MEMS devices.

CHAOTIC ANALYSES FOR SPACE SERIES OF GOLD GRADE

2004 ◽  
Vol 18 (17n19) ◽  
pp. 2730-2733 ◽  
Author(s):  
YAN-SHI XIE ◽  
GUANG-HAO CHEN ◽  
KAI-XUAN TAN
Keyword(s):  
Fractal Dimension ◽  
Phase Space ◽  
Lyapunov Exponent ◽  
Dynamic Process ◽  
Chaotic Attractor ◽  
Space Dimension ◽  
Chaotic Dynamic ◽  
Largest Lyapunov Exponent ◽  
Gold Grade ◽  
Chaotic Theory

A new powerful tool, chaotic theory, has been used to study mineralization through chaotic analysis for space series of gold grade in this paper. Both of the most important chaotic measures, Largest Lyapunov exponent (LLE) and fractal dimensional, for space series of gold grade in one gold deposit are computed. The positive LLE suggests that the space series of gold grade are chaotic series. When the phase space dimension approach 8~10, a chaotic attractor appears and their fractal dimension values vary from 1.94 to 3.99. It indicates that the evolution of ore-forming fluid and the enrichment and deposition of gold element are chaotic dynamic process.

scholarly journals A Fractional-Order Chaotic System with an Infinite Number of Equilibrium Points

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Ping Zhou ◽  
Kun Huang ◽  
Chun-de Yang
Keyword(s):  
Lyapunov Exponent ◽  
Fractional Order ◽  
Chaotic System ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Chaotic Synchronization ◽  
Order System ◽  
Largest Lyapunov Exponent ◽  
Synchronization Scheme

A new 4D fractional-order chaotic system, which has an infinite number of equilibrium points, is introduced. There is no-chaotic behavior for its corresponded integer-order system. We obtain that the largest Lyapunov exponent of this 4D fractional-order chaotic system is 0.8939 and yield the chaotic attractor. A chaotic synchronization scheme is presented for this 4D fractional-order chaotic system. Numerical simulations is verified the effectiveness of the proposed scheme.

DETECTING THE STATE OF THE DUFFING OSCILLATOR BY PHASE SPACE TRAJECTORY AUTOCORRELATION

2013 ◽  
Vol 23 (04) ◽  
pp. 1350065 ◽  
Author(s):  
VAHID RASHTCHI ◽  
MOHSEN NOURAZAR
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Duffing Oscillator ◽  
Simple Calculation ◽  
The State ◽  
Detection Methods ◽  
Largest Lyapunov Exponent ◽  
Chaotic Oscillator ◽  
The Largest Lyapunov Exponent ◽  
Phase Space Trajectory

Detecting the state of the Duffing oscillator, a type of well-known chaotic oscillator, deeply affects the accuracy of its application. Considering this, the present paper introduced a novel method for detecting the state of the Duffing oscillator. Binary outputs, simple calculation, high precision and fast response time were the main advantages of the phase space trajectory autocorrelation. Also, this study explained the largest Lyapunov exponent as well as a number of other methods commonly employed in detecting the state of the Duffing oscillator. The precision and effectiveness of the method introduced was compared with other well-known state detection methods such as the 0-1 test and the largest Lyapunov exponent.

scholarly journals Multi-Baker Map as a Model of Digital PD Control

2016 ◽  
Vol 26 (02) ◽  
pp. 1650023 ◽  
Author(s):  
Gábor Csernák ◽  
Gergely Gyebrószki ◽  
Gábor Stépán
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Linear Systems ◽  
Small Amplitude ◽  
Chaotic Dynamics ◽  
Largest Lyapunov Exponent ◽  
Two Dimensional ◽  
Smale Horseshoe ◽  
Analytical Proof ◽  
Baker Map

Digital stabilization of unstable equilibria of linear systems may lead to small amplitude stochastic-like oscillations. We show that these vibrations can be related to a deterministic chaotic dynamics induced by sampling and quantization. A detailed analytical proof of chaos is presented for the case of a PD controlled oscillator: it is shown that there exists a finite attracting domain in the phase-space, the largest Lyapunov exponent is positive and the existence of a Smale horseshoe is also pointed out. The corresponding two-dimensional micro-chaos map is a multi-baker map, i.e. it consists of a finite series of baker’s maps.

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