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.

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.

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

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.

scholarly journals Predictability of extreme values in geophysical models

2012 ◽  
Vol 19 (5) ◽  
pp. 529-539 ◽  
Author(s):  
A. E. Sterk ◽  
M. P. Holland ◽  
P. Rabassa ◽  
H. W. Broer ◽  
R. Vitolo
Keyword(s):  
Extreme Value Theory ◽  
Finite Time ◽  
Lead Time ◽  
Extreme Values ◽  
Initial Conditions ◽  
Value Theory ◽  
Wind Speeds ◽  
Finite Time Lyapunov Exponents ◽  
Deterministic Systems ◽  
Time Predictability

Abstract. Extreme value theory in deterministic systems is concerned with unlikely large (or small) values of an observable evaluated along evolutions of the system. In this paper we study the finite-time predictability of extreme values, such as convection, energy, and wind speeds, in three geophysical models. We study whether finite-time Lyapunov exponents are larger or smaller for initial conditions leading to extremes. General statements on whether extreme values are better or less predictable are not possible: the predictability of extreme values depends on the observable, the attractor of the system, and the prediction lead time.

scholarly journals Fractional-Order Hidden Attractor Based on the Extended Liu System

2020 ◽  
Vol 2020 ◽  
pp. 1-22 ◽  
Author(s):  
Yaoyu Wang ◽  
Ling Liu ◽  
Xinshan Cai ◽  
Chongxin Liu ◽  
Yan Wang ◽  
...  
Keyword(s):  
Equilibrium Point ◽  
Fractional Order ◽  
Finite Time ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Chaotic Oscillator ◽  
Hidden Attractor ◽  
Fractional System ◽  
Line Equilibrium ◽  
Liu System

In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.

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.

Dynamical Effects of Offset Terms on a Modified Chua’s Oscillator and Its Circuit Implementation

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Léandre Kamdjeu Kengne ◽  
Karthikeyan Rajagopal ◽  
Nestor Tsafack ◽  
Paul Didier Kamdem Kuate ◽  
Balamurali Ramakrishnan ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Phenomena ◽  
Hidden Attractor ◽  
Chua’S Oscillator ◽  
Kirchhoff’S Laws ◽  
Attraction Basins ◽  
The Mathematical Model

This paper addresses the effects of offset terms on the dynamics of a modified Chua’s oscillator. The mathematical model is derived using Kirchhoff’s laws. The model is analyzed with the help of the maximal Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The investigations show that the offset terms break the symmetry of the system, generating more complex nonlinear phenomena like coexisting asymmetric bifurcations, coexisting asymmetric attractors, asymmetric double-scroll chaotic attractors and asymmetric attraction basins. Also, a hidden attractor (period-1 limit cycle) is found when varying the initial conditions. More interestingly, this latter attractor coexists with all other self-excited ones. A microcontroller-based implementation of the circuit is carried out to verify the numerical investigations.

scholarly journals STRANGE NONCHAOTIC ATTRACTORS

2001 ◽  
Vol 11 (02) ◽  
pp. 291-309 ◽  
Author(s):  
AWADHESH PRASAD ◽  
SURENDRA SINGH NEGI ◽  
RAMAKRISHNA RAMASWAMY
Keyword(s):  
Lyapunov Exponent ◽  
Initial Conditions ◽  
Localized States ◽  
Largest Lyapunov Exponent ◽  
Electronic Systems ◽  
Plasma Discharges ◽  
Driven System ◽  
Important Means ◽  
Strange Nonchaotic Attractors ◽  
Novel Applications

Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic Attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schrödinger equation for a particle in a related quasiperiodic potential, showing a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, have emerged as an important means of studying fractal attractors, and this analysis finds useful application here. The ubiquity of such attractors, in conjunction with their several unusual properties, suggests novel applications.

scholarly journals The anisotropy of the power spectrum in periodic cosmological simulations

2021 ◽  
Vol 503 (4) ◽  
pp. 5638-5645
Author(s):  
Gábor Rácz ◽  
István Szapudi ◽  
István Csabai ◽  
László Dobos
Keyword(s):  
Dark Matter ◽  
Power Spectrum ◽  
Large Scale ◽  
Cold Dark Matter ◽  
Initial Conditions ◽  
Power Spectra ◽  
Cosmological Simulations ◽  
Spherical Collapse ◽  
Lower Amplitude ◽  
Hydrodynamical Simulations

ABSTRACT The classical gravitational force on a torus is anisotropic and always lower than Newton’s 1/r2 law. We demonstrate the effects of periodicity in dark matter only N-body simulations of spherical collapse and standard Lambda cold dark matter (ΛCDM) initial conditions. Periodic boundary conditions cause an overall negative and anisotropic bias in cosmological simulations of cosmic structure formation. The lower amplitude of power spectra of small periodic simulations is a consequence of the missing large-scale modes and the equally important smaller periodic forces. The effect is most significant when the largest mildly non-linear scales are comparable to the linear size of the simulation box, as often is the case for high-resolution hydrodynamical simulations. Spherical collapse morphs into a shape similar to an octahedron. The anisotropic growth distorts the large-scale ΛCDM dark matter structures. We introduce the direction-dependent power spectrum invariant under the octahedral group of the simulation volume and show that the results break spherical symmetry.

scholarly journals MORSE SPECTRUM FOR NONAUTONOMOUS DIFFERENTIAL EQUATIONS

2008 ◽  
Vol 08 (03) ◽  
pp. 351-363 ◽  
Author(s):  
FRITZ COLONIUS ◽  
PETER E. KLOEDEN ◽  
MARTIN RASMUSSEN
Keyword(s):  
Differential Equations ◽  
Finite Time ◽  
Accumulation Point ◽  
Morse Decomposition ◽  
Finite Time Lyapunov Exponents ◽  
The Past ◽  
Accumulation Points ◽  
Morse Spectrum ◽  
Nonautonomous Differential Equations ◽  
Linear Cocycle

The concept of a Morse decomposition consisting of nonautonomous sets is reviewed for linear cocycle mappings w.r.t. the past, future and all-time convergences. In each case, the set of accumulation points of the finite-time Lyapunov exponents corresponding to points in a nonautonomous set is shown to be an interval. For a finest Morse decomposition, the Morse spectrum is defined as the union of all of the above accumulation point intervals over the different nonautonomous sets in such a finest Morse decomposition. In addition, Morse spectrum is shown to be independent of which finest Morse decomposition is used, when more than one exists.

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