scholarly journals Experimentally Viable Techniques for Accessing Coexisting Attractors Correlated with Lyapunov Exponents

2021 ◽  
Vol 11 (21) ◽  
pp. 9905
Author(s):  
Joshua Ray Hall ◽  
Erikk Kenneth Tilus Burton ◽  
Dylan Michael Chapman ◽  
Donna Kay Bandy
Keyword(s):  
Lyapunov Exponents ◽  
Critical Points ◽  
Control Parameter ◽  
Experimental System ◽  
Coexisting Attractors ◽  
Shift Method ◽  
Large Domain ◽  
Lag Effects ◽  
Power Shift ◽  
Small Domain

Universal, predictive attractor patterns configured by Lyapunov exponents (LEs) as a function of the control parameter are shown to characterize periodic windows in chaos just as in attractors, using a coherent model of the laser with injected signal. One such predictive pattern, the symmetric-like bubble, foretells of an imminent bifurcation. With a slight decrease in the gain parameter, we find the symmetric-like bubble changes to a curved trajectory of two equal LEs in one attractor, while an increase in the gain reverses this process in another attractor. We generalize the power-shift method for accessing coexisting attractors or periodic windows by augmenting the technique with an interim parameter shift that optimizes attractor retrieval. We choose the gain as our parameter to interim shift. When interim gain-shift results are compared with LE patterns for a specific gain, we find critical points on the LE spectra where the attractor is unlikely to survive the gain shift. Noise and lag effects obscure the power shift minimally for large domain attractors. Small domain attractors are less accessible. The power-shift method in conjunction with the interim parameter shift is attractive because it can be experimentally applied without significant or long-lasting modifications to the experimental system.

scholarly journals Dynamical Complexity and Multistability in a Novel Lunar Wake Plasma System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Bo. Yan ◽  
Punam K. Prasad ◽  
Sayan Mukherjee ◽  
Asit Saha ◽  
Santo Banerjee
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponents ◽  
Alpha Particles ◽  
Coexisting Attractors ◽  
Plasma System ◽  
Electrostatic Waves ◽  
Lunar Wake ◽  
Dynamical Complexity ◽  
Suprathermal Electrons ◽  
Different Types

Dynamical complexity and multistability of electrostatic waves are investigated in a four-component homogeneous and magnetized lunar wake plasma constituting of beam electrons, heavier ions (alpha particles, He++), protons, and suprathermal electrons. The unperturbed dynamical system of the considered lunar wake plasma supports nonlinear and supernonlinear trajectories which correspond to nonlinear and supernonlinear electrostatic waves. On the contrary, the perturbed dynamical system of lunar wake plasma shows different types of coexisting attractors including periodic, quasiperiodic, and chaotic, investigated by phase plots and Lyapunov exponents. To confirm chaotic and nonchaotic dynamics in the perturbed lunar wake plasma, 0−1 chaos test is performed. Furthermore, a weighted recurrence-based entropy is implemented to investigate the dynamical complexity of the system. Numerical results show existence of chaos with variation of complexity in the perturbed dynamics.

scholarly journals Complex Dynamics in a Memcapacitor-Based Circuit

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 188 ◽  
Author(s):  
Fang Yuan ◽  
Yuxia Li ◽  
Guangyi Wang ◽  
Gang Dou ◽  
Guanrong Chen
Keyword(s):  
Complex System ◽  
Lyapunov Exponents ◽  
Dynamic Characteristics ◽  
Complex Dynamics ◽  
Basins Of Attraction ◽  
Chaotic Oscillator ◽  
Initial Condition ◽  
Circuit Implementation ◽  
Coexisting Attractors ◽  
Extreme Multistability

In this paper, a new memcapacitor model and its corresponding circuit emulator are proposed, based on which, a chaotic oscillator is designed and the system dynamic characteristics are investigated, both analytically and experimentally. Extreme multistability and coexisting attractors are observed in this complex system. The basins of attraction, multistability, bifurcations, Lyapunov exponents, and initial-condition-triggered similar bifurcation are analyzed. Finally, the memcapacitor-based chaotic oscillator is realized via circuit implementation with experimental results presented.

ANALYSIS OF NEW FOUR-DIMENSIONAL CHAOTIC CIRCUITS WITH EXPERIMENTAL AND NUMERICAL METHODS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250042 ◽  
Author(s):  
GUO-QING HUANG ◽  
XIN WU
Keyword(s):  
Numerical Methods ◽  
Lyapunov Exponents ◽  
Control Parameter ◽  
Threshold Value ◽  
Qualitative Properties ◽  
Nonlinear Circuit ◽  
Chaotic Circuits ◽  
Fast Lyapunov Indicators

Dynamically qualitative properties of individual orbits in a new four-dimensional nonlinear circuit are observed on an oscilloscope. Meanwhile, they are also traced numerically with the help of some methods for finding chaos. Comparisons show that the observed results are consistent with the computed ones to a great extent. In addition, the bifurcation, Lyapunvon spectra, fast Lyapunov indicators and small alignment indexes represent almost the same rules of transitivity to chaos on a control parameter. It is found when the parameter has a threshold value from order to chaos, and the chaos gets stronger and stronger, the parameter is smoothly varied from small to large. In particular, the entire set of Lyapunov exponents can lead to another threshold value of the parameter from chaotic to hyperchaotic behaviors.

scholarly journals New Order-Revealing Encryption with Shorter Ciphertexts

Information ◽  
2020 ◽  
Vol 11 (10) ◽  
pp. 457
Author(s):  
Kee Sung Kim
Keyword(s):  
Computable Function ◽  
Equal Size ◽  
New Order ◽  
Data Outsourcing ◽  
Large Domain ◽  
Encrypted Data ◽  
Domain Extension ◽  
Small Domain ◽  
Large Domains

As data outsourcing services have been becoming common recently, developing skills to search over encrypted data has received a lot of attention. Order-revealing encryption (OREnc) enables performing a range of queries on encrypted data through a publicly computable function that outputs the ordering information of the underlying plaintexts. In 2016, Lewi et al. proposed an OREnc scheme that is more secure than the existing practical (stateless and non-interactive) schemes by constructing an ideally-secure OREnc scheme for small domains and a “domain-extension” scheme for obtaining the final OREnc scheme for large domains. They encoded a large message into small message blocks of equal size to apply them to their small-domain scheme, thus their resulting OREnc scheme reveals the index of the first differing message block. In this work, we introduce a new ideally-secure OREnc scheme for small domains with shorter ciphertexts. We also present an alternative message-block encoding technique. Combining the proposed constructions with the domain-extension scheme of Lewi et al., we can obtain a new large-domain OREnc scheme with shorter ciphertexts or with different leakage information, but longer ciphertexts.

scholarly journals Invariant measures for interval maps without Lyapunov exponents

2021 ◽  
pp. 1-29
Author(s):  
JORGE OLIVARES-VINALES
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Critical Points ◽  
Invariant Measures ◽  
Full Measure ◽  
Interval Maps ◽  
Interval Map

Abstract We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.

BISTABILITY AND DYNAMICAL TRANSITIONS IN A PHASE-MODULATED DELAYED SYSTEM

2009 ◽  
Vol 23 (23) ◽  
pp. 2733-2743 ◽  
Author(s):  
YONGXIANG ZHANG ◽  
GUIQIN KONG ◽  
JIANNING YU
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Basins Of Attraction ◽  
Parameter Study ◽  
Global Bifurcations ◽  
Coexisting Attractors ◽  
System Parameters ◽  
Different Types ◽  
Delayed System

We study a delayed system with feedback modulation of the nonlinear parameter. Study of the system as a function of nonlinearity and modulation parameters reveals complex dynamical phenomena: different types of coexisting attractors, local or global bifurcations and transitions. Bistability and dynamical attractors can be distinguished in some parameter-space regions, which may be useful to drive chaotic dynamics, unstable attractors or bistability towards regular dynamics. At the bifurcation to bistability, two striking features are that they lead to fundamentally unpredictable behavior of orbits and crisis of attractors as system parameters are varied slowly through the critical curve. Unstable attractors are also investigated in bistable regions, which are easily mistaken for true multi-periodic orbits judging merely from zero measure local basins. Lyapunov exponents and basins of attraction are also used to characterize the phenomenon observed.

Dynamical Bifurcation of the Generalized Swift–Hohenberg Equation

2015 ◽  
Vol 25 (08) ◽  
pp. 1550095 ◽  
Author(s):  
Yuncherl Choi ◽  
Jongmin Han ◽  
Jungho Park
Keyword(s):  
Critical Points ◽  
Control Parameter ◽  
Center Manifolds ◽  
Static Solutions ◽  
Invariant Circles ◽  
Instability Parameter

In this paper, we prove that the generalized Swift–Hohenberg equation bifurcates from the trivial states to an attractor as the control parameter α passes through critical points. The bifurcation is divided into two groups according to the dimension of the center manifolds. We show that the bifurcated attractor is homeomorphic to S1 or S3 and it contains invariant circles of static solutions. We provide a criterion on the quadratic instability parameter μ which determines the bifurcation to be supercritical or subcritical.

ROBUST CHAOS IN A SMOOTH SYSTEM

2001 ◽  
Vol 15 (02) ◽  
pp. 177-189 ◽  
Author(s):  
M. ANDRECUT ◽  
M. K. ALI
Keyword(s):  
Parameter Space ◽  
Random Number ◽  
Random Number Generation ◽  
Coexisting Attractors ◽  
Piecewise Smooth Systems ◽  
Large Domain ◽  
One Dimensional ◽  
Counter Example ◽  
Smooth Map ◽  
Smooth System

Robust chaos is defined by the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space. The occurrence of robust chaos has been discussed [Phys. Rev. Lett.78, 4561 (1997); ibid.80, 3049 (1998)]. It has been shown that robust chaos can occur in piecewise smooth systems. Also, it has been conjectured that robust chaos cannot occur in smooth systems. However, here we give a counter example to this conjecture. We present a one-dimensional smooth map xt + 1 = f(xt, α) that generates robust chaos in a large domain of the parameter space (α). An application to random number generation and cryptography is also presented.

scholarly journals On Two-Dimensional Fractional Chaotic Maps with Symmetries

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 756 ◽  
Author(s):  
Fatima Hadjabi ◽  
Adel Ouannas ◽  
Nabil Shawagfeh ◽  
Amina-Aicha Khennaoui ◽  
Giuseppe Grassi
Keyword(s):  
Fixed Points ◽  
Lyapunov Exponents ◽  
Chaotic Maps ◽  
Chaotic Behavior ◽  
Closed Curve ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Coexisting Attractors

In this paper, we propose two new two-dimensional chaotic maps with closed curve fixed points. The chaotic behavior of the two maps is analyzed by the 0–1 test, and explored numerically using Lyapunov exponents and bifurcation diagrams. It has been found that chaos exists in both fractional maps. In addition, result shows that the proposed fractional maps shows the property of coexisting attractors.

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