A New Route to Strange Nonchaotic Attractors in an Interval Map

2020 ◽  
Vol 30 (04) ◽  
pp. 2050063 ◽  
Author(s):  
Yongxiang Zhang ◽  
Yunzhu Shen
Keyword(s):  
Lyapunov Exponents ◽  
Control Parameter ◽  
Power Spectra ◽  
Phase Sensitivity ◽  
Recurrence Analysis ◽  
Grazing Bifurcation ◽  
Remarkable Feature ◽  
Finite Time Lyapunov Exponents ◽  
Interval Map ◽  
Strange Nonchaotic Attractors

We identify an unusual route to the creation of a strange nonchaotic attractor (SNA) in a quasiperiodically forced interval map. We find that the smooth quasiperiodic torus becomes nonsmooth due to the grazing bifurcation of the torus. The nonsmooth points on the torus increase more and more with the change of control parameter. Finally, the torus gets extremely fractal and becomes a SNA which is termed the grazing bifurcation route to the SNA. We characterize the SNA by maximal Lyapunov exponents and their variance, phase sensitivity exponents and power spectra. We also describe the transition between a torus and a SNA by the recurrence analysis. A remarkable feature of the route to SNAs is that the positive tails decay linearly and the negative tails exhibit recurrent fluctuations in the distribution of the finite-time Lyapunov exponents.

STRANGE NONCHAOTIC ATTRACTORS IN A QUASIPERIODICALLY-FORCED PIECEWISE SMOOTH SYSTEM WITH FAREY TREE

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950118 ◽  
Author(s):  
YUNZHU SHEN ◽  
YONGXIANG ZHANG
Keyword(s):  
Lyapunov Exponents ◽  
Scaling Laws ◽  
Power Spectra ◽  
Piecewise Smooth ◽  
Largest Lyapunov Exponent ◽  
Piecewise Smooth System ◽  
Finite Time Lyapunov Exponents ◽  
Farey Tree ◽  
Strange Nonchaotic Attractors ◽  
Smooth System

The existence of strange nonchaotic attractors (SNAs) is verified in a simple quasiperiodically-forced piecewise smooth system with Farey tree. It can be seen that more and more jumping discontinuities appear on the smooth torus and the torus becomes extremely fragmented with the change of control parameter. Finally, the torus becomes an SNA with fractal property. In order to confirm the existence of SNAs in this system, we preliminarily use the estimation of the phase sensitivity exponent, estimation of the largest Lyapunov exponent and rational approximation. SNAs are further characterized by power spectra, recurrence plots, the largest Lyapunov exponents and their variance, the distribution of the finite-time Lyapunov exponents, the spectral distribution function and scaling laws.

Observation of Strange Nonchaotic Dynamics in the Frame of State-Controlled Cellular Neural Network-Based Oscillator

2019 ◽  
Vol 14 (11) ◽  
Author(s):  
P. Megavarna Ezhilarasu ◽  
K. Suresh ◽  
K. Thamilmaran
Keyword(s):  
Neural Network ◽  
Lyapunov Exponents ◽  
Laboratory Experiments ◽  
Cellular Neural Network ◽  
Phase Sensitivity ◽  
Chaotic Circuit ◽  
Finite Time Lyapunov Exponents ◽  
Periodically Driven ◽  
Largest Lyapunov Exponents ◽  
Strange Nonchaotic Attractors

Abstract In this paper, the strange nonchaotic dynamics of a quasi-periodically driven state-controlled cellular neural network (SC-CNN) based on a simple chaotic circuit is investigated using hardware experiments and numerical simulations. We report here two different routes to strange nonchaotic attractors (SNAs) taken by this SC-CNN based circuit system. These routes were confirmed using rational approximation (RA) theory, finite time Lyapunov exponents, spectrum of the largest Lyapunov exponents and their variance, and phase sensitivity exponent. It is observed that the results from both computer simulations as well as laboratory experiments have spectacular resemblance.

scholarly journals Hidden Strange Nonchaotic Attractors

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 652
Author(s):  
Marius-F. Danca ◽  
Nikolay Kuznetsov
Keyword(s):  
Finite Time ◽  
Initial Conditions ◽  
Power Spectra ◽  
Recurrence Plot ◽  
Chaotic Attractors ◽  
Largest Lyapunov Exponent ◽  
Hidden Attractor ◽  
Test Power ◽  
Finite Time Lyapunov Exponents ◽  
Strange Nonchaotic Attractors

In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed.

Coexistence of Strange Nonchaotic Attractors in a Quasiperiodically Forced Dynamical Map

2020 ◽  
Vol 30 (13) ◽  
pp. 2050183
Author(s):  
Yunzhu Shen ◽  
Yongxiang Zhang ◽  
Sajad Jafari
Keyword(s):  
Power Spectrum ◽  
Lyapunov Exponents ◽  
Basins Of Attraction ◽  
Initial Condition ◽  
Phase Sensitivity ◽  
Coexisting Attractors ◽  
Sensitive Dependence ◽  
Largest Lyapunov Exponents ◽  
The Creation ◽  
Strange Nonchaotic Attractors

In this paper, we investigate coexisting strange nonchaotic attractors (SNAs) in a quasiperiodically forced system. We also describe the basins of attraction for coexisting attractors and identify the mechanism for the creation of coexisting attractors. We find three types of routes to coexisting SNAs, including intermittent route, Heagy–Hammel route and fractalization route. The mechanisms for the creation of coexisting SNAs are investigated by the interruption of coexisting torus-doubling bifurcations. We characterize SNAs by the largest Lyapunov exponents, phase sensitivity exponents and power spectrum. Besides, the SNAs with extremely fractal basins exhibit sensitive dependence on the initial condition for some particular parameters.

EXPERIMENTAL REALIZATION OF STRANGE NONCHAOTIC ATTRACTORS IN A NONLINEAR SERIES LCR CIRCUIT WITH NONSINUSOIDAL FORCE

2009 ◽  
Vol 19 (12) ◽  
pp. 4131-4163 ◽  
Author(s):  
K. SRINIVASAN ◽  
D. V. SENTHILKUMAR ◽  
R. SURESH ◽  
K. THAMILMARAN ◽  
M. LAKSHMANAN
Keyword(s):  
Lyapunov Exponents ◽  
Electronic Circuit ◽  
Numerical Data ◽  
Electronic System ◽  
Wave Force ◽  
Distribution Functions ◽  
Experimental Realization ◽  
Square Wave ◽  
Finite Time Lyapunov Exponents ◽  
Strange Nonchaotic Attractors

We have identified several prominent routes, namely, fractalization, fractalization followed by intermittency, intermittency and Heagy–Hammel routes, for the birth of strange nonchaotic attractors (SNAs) in a quasiperiodically forced electronic system with nonsinusoidal (square wave) force as one of the quasiperiodic forces [Senthilkumar et al., 2008]. In addition, a new bubbling route has also been identified in this circuit. Although some of these prominent routes have been reported experimentally [Thamilmaran et al., 2006] in a quasiperiodically forced electronic circuit with both the forcings as sinusoidal forces, experimental identification of all these routes is reported here in a quasiperiodically forced electronic circuit with one of the forcings as a nonsinusoidal (square wave) force. The birth of SNAs by these routes are characterized from both the experimental and numerical data by the maximal Lyapunov exponents and their variance, Poincaré maps, Fourier amplitude spectra, spectral distribution functions and the distribution of finite-time Lyapunov exponents.

scholarly journals A preliminary survey of the broadband seismic wavefield at Puu Oo, the active vent of Kilauea volcano, Hawaii

1996 ◽  
Vol 39 (2) ◽  
Author(s):  
D. Seidl ◽  
M. Hellweg ◽  
P. Okubo ◽  
H. Rademacher
Keyword(s):  
Frequency Band ◽  
Power Spectra ◽  
Volcanic Tremor ◽  
Digital Data ◽  
Periodic Pattern ◽  
Data Logger ◽  
Remarkable Feature ◽  
Wide Range ◽  
Active Vent ◽  
Seismic Wavefield

The seismic wavefield near an active volcanic vent consists of superimposed signals in a wide range of frequency bands from sources inside and outside the volcano. To characterize the broadband wavefield near Puu Oo, we deployed a profile of three three-component broadband sensors in a 200 m long line about 1.5 km WSW of the active vent. During this period, Puu Oo maintained a constant, but very low level of activity. The digital data logger recorded the wavefield continuously in the frequency band between 0.01 and 40 Hz between June 25 and July 9, 1994. At the same time, local wind conditions along with air temperature and pressure were monitored by a portable digital weather station. On the basis of characteristic elements, such as waveform, spatial coherence between stations, particle motion and power spectra, the wavefield can be divided into three bands. The dominant signals in the frequency band between 0.01 and 0.1 Hz are not coherent among the stations. Their ground velocities correlate with the wind speed. The signals in the 0.1 to 0.5 Hz band are coherent across the profile and most probably represent a superposition of volcanic tremor and microseisms from the Pacific Ocean. Much of the energy above 0.5 Hz can be attributed to activity at the vent. Power spectra from recordings of the transverse components show complex peaks between 0.5 and 3 Hz which vary in amplitude due to site effects and distance. On the other hand, power spectra calculated from the radial components show a clearly periodic pattern of peaks at 1 Hz intervals for some time segments. A further remarkable feature of the power spectra is that they are highly stationary.

scholarly journals Surface Ocean Mixing Inferred from Different Multisatellite Altimetry Measurements

2010 ◽  
Vol 40 (11) ◽  
pp. 2466-2480 ◽  
Author(s):  
Francisco J. Beron-Vera ◽  
María J. Olascoaga ◽  
Gustavo J. Goni
Keyword(s):  
Lyapunov Exponents ◽  
Coherent Structures ◽  
Diagnostic Tool ◽  
Turbulent Mixing ◽  
Finite Time ◽  
The Other ◽  
Sea Surface ◽  
Anomaly Field ◽  
Finite Time Lyapunov Exponents ◽  
Surface Ocean

Abstract Two sea surface height (SSH) anomaly fields distributed by Archiving, Validation, and Interpretation of Satellite Oceanographic (AVISO) Altimetry are evaluated in terms of the effects that they produce on mixing. One SSH anomaly field, tagged REF, is constructed using measurements made by two satellite altimeters; the other SSH anomaly field, tagged UPD, is constructed using measurements made by up to four satellite altimeters. Advection is supplied by surface geostrophic currents derived from the total SSH fields resulting from the addition of these SSH anomaly fields to a mean SSH field. Emphasis is placed on the extraction from the currents of Lagrangian coherent structures (LCSs), which, acting as skeletons for patterns formed by passively advected tracers, entirely control mixing. The diagnostic tool employed to detect LCSs is provided by the computation of finite-time Lyapunov exponents. It is found that currents inferred using UPD SSH anomalies support mixing with characteristics similar to those of mixing produced by currents inferred using REF SSH anomalies. This result mainly follows from the fact that, being more easily characterized as chaotic than turbulent, mixing as sustained by currents derived using UPD SSH anomalies is quite insensitive to spatiotemporal truncations of the advection field.

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