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.

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.

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.

scholarly journals Global Dynamics of a Piecewise Smooth System for Brain Lactate Metabolism

2018 ◽  
Vol 18 (1) ◽  
pp. 315-332
Author(s):  
J.-P. Françoise ◽  
Hongjun Ji ◽  
Dongmei Xiao ◽  
Jiang Yu
Keyword(s):  
Piecewise Smooth ◽  
Global Dynamics ◽  
Lactate Metabolism ◽  
Piecewise Smooth System ◽  
Smooth System ◽  
Brain Lactate

Dynamics of Human Spine in Jumps Using Lyapunov Exponents

2009 ◽  
Author(s):  
Eliza A. Banu ◽  
Dan Marghitu ◽  
P. K. Raju
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Largest Lyapunov Exponent ◽  
Kinematic Data ◽  
Finite Time Lyapunov Exponents ◽  
Human Spine ◽  
Chaotic Nature ◽  
Male Subjects ◽  
Lyapunov Exponent Spectrum

Characterizing and quantifying the local dynamics of the human spine during various athletic exercises is important for therapeutic and physiotherapy reasons or athletic related effects on the human spine. In this study we computed the largest finite-time Lyapunov exponents for the spine of a human subject during an athletic exercise for several volunteers. The kinematic data was collected using the 3D motion capture system (Motion Realty Inc.). Four healthy male subjects were asked to perform two sets of 30 jumps. In order to draw a conclusion about the chaotic nature of the dynamics of the lumbar spine Lyapunov exponent spectrum was calculated for each volunteer which included four Lyapunov exponents, one negative, one very close to zero and two positive. Subjects 1, 2 and 4 registered a significant increase in the largest Lyapunov exponent from the first set jumps to the next. Subject 3 registered no change in the values of Lyapunov exponents. Nonlinear analysis of the human spine demonstrates the chaotic nature of the system. The computation of the largest Lyapunov exponents enables a more precise characterization of the dynamics of the human spine.

Melnikov Method for a Three-Zonal Planar Hybrid Piecewise-Smooth System and Application

2016 ◽  
Vol 26 (01) ◽  
pp. 1650014
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Periodic Orbits ◽  
Melnikov Method ◽  
Perturbation Parameter ◽  
Melnikov Function ◽  
Displacement Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
First Order ◽  
Piecewise Smooth System ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three domains separated by two switching manifolds [Formula: see text] and [Formula: see text]. The dynamics in each domain is governed by a smooth system. When an orbit reaches the separation lines, then a reset map describing an impacting rule applies instantaneously before the orbit enters into another domain. We assume that the unperturbed system has a continuum of periodic orbits transversally crossing the separation lines. Then, we wish to study the persistence of the periodic orbits under an autonomous perturbation and the reset map. To achieve this objective, we first choose four appropriate switching sections and build a Poincaré map, after that, we present a displacement function and carry on the Taylor expansion of the displacement function to the first-order in the perturbation parameter [Formula: see text] near [Formula: see text]. We denote the first coefficient in the expansion as the first-order Melnikov function whose zeros provide us the persistence of periodic orbits under perturbation. Finally, we study periodic orbits of a concrete planar hybrid piecewise-smooth system by the obtained Melnikov function.

A Lorenz-type attractor in a piecewise-smooth system: Rigorous results

2019 ◽  
Vol 29 (10) ◽  
pp. 103108 ◽  
Author(s):  
Vladimir N. Belykh ◽  
Nikita V. Barabash ◽  
Igor V. Belykh
Keyword(s):  
Piecewise Smooth ◽  
Rigorous Results ◽  
Piecewise Smooth System ◽  
Smooth System
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