scholarly journals Circuit Implementation of a Modified Chaotic System with Hyperbolic Sine Nonlinearities Using Bi-Color LED

Technologies ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 15
Author(s):  
Christos K. Volos ◽  
Lazaros Moysis ◽  
George D. Roumelas ◽  
Aggelos Giakoumis ◽  
Hector E. Nistazakis ◽  
...  
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Original System ◽  
Dynamical Analysis ◽  
Phase Portraits ◽  
Circuit Realization ◽  
Hyperbolic Sine ◽  
Route To Chaos ◽  
Sine Functions

In this paper, a chaotic three dimansional dynamical system is proposed, that is a modification of the system in Volos et al. (2017). The new system has two hyperbolic sine nonlinear terms, as opposed to the original system that only included one, in order to optimize system’s chaotic behavior, which is confirmed by the calculation of the maximal Lyapunov exponents and Kaplan-Yorke dimension. The system is experimentally realized, using Bi-color LEDs to emulate the hyperbolic sine functions. An extended dynamical analysis is then performed, by computing numerically the system’s bifurcation and continuation diagrams, Lyapunov exponents and phase portraits, and comparing the numerical simulations with the circuit simulations. A series of interesting phenomena are unmasked, like period doubling route to chaos, coexisting attractors and antimonotonicity, which are all verified from the circuit realization of the system. Hence, the circuit setup accurately emulates the chaotic dynamics of the proposed system.

Constructing a Novel No-Equilibrium Chaotic System

2014 ◽  
Vol 24 (05) ◽  
pp. 1450073 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Zhouchao Wei ◽  
Xiong Wang
Keyword(s):  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Poincaré Map ◽  
Period Doubling ◽  
Circuit Realization ◽  
Chaotic Flow ◽  
Route To Chaos ◽  
Line Equilibrium

This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincaré map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.

CHAOTIC DYNAMICS OF A SIMPLE PARAMETRICALLY DRIVEN DISSIPATIVE CIRCUIT

2011 ◽  
Vol 21 (07) ◽  
pp. 1927-1933 ◽  
Author(s):  
P. PHILOMINATHAN ◽  
M. SANTHIAH ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
S. RAJASEKAR
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classic Period ◽  
Control Signal ◽  
Period Doubling Bifurcation ◽  
Chaotic Circuit ◽  
Route To Chaos ◽  
Parameter Values

We introduce a simple parametrically driven dissipative second-order chaotic circuit. In this circuit, one of the circuit parameters is varied by an external periodic control signal. Thus by tuning the parameter values of this circuit, classic period-doubling bifurcation route to chaos is found to occur. The experimentally observed phenomena is further validated through corresponding numerical simulation of the circuit equations. The periodic and chaotic dynamics of this model is further characterized by computing Lyapunov exponents.

scholarly journals Autonomous Jerk Oscillator with Cosine Hyperbolic Nonlinearity: Analysis, FPGA Implementation, and Synchronization

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Sifeu Takougang Kingni ◽  
Gaetan Fautso Kuiate ◽  
Victor Kamdoum Tamba ◽  
Viet-Thanh Pham
Keyword(s):  
Control Method ◽  
Sliding Mode ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Fpga Implementation ◽  
Phase Portraits ◽  
Adaptive Sliding Mode Control ◽  
Coexistence Of Attractors ◽  
Route To Chaos ◽  
Two Parameters

A two-parameter autonomous jerk oscillator with a cosine hyperbolic nonlinearity is proposed in this paper. Firstly, the stability of equilibrium points of proposed autonomous jerk oscillator is investigated by analyzing the characteristic equation and the existence of Hopf bifurcation is verified using one of the two parameters as a bifurcation parameter. By tuning its two parameters, various dynamical behaviors are found in the proposed autonomous jerk oscillator including periodic attractor, one-scroll chaotic attractor, and coexistence between chaotic and periodic attractors. The proposed autonomous jerk oscillator has period-doubling route to chaos with the variation of one of its parameters and reverse period-doubling route to chaos with the variation of its other parameter. The proposed autonomous jerk oscillator is modelled on Field Programmable Gate Array (FPGA) and the FPGA chip statistics and phase portraits are derived. The chaotic and coexistence of attractors generated in the proposed autonomous jerk oscillator are confirmed by FPGA implementation of the proposed autonomous jerk oscillator. A good qualitative agreement is illustrated between the numerical and FPGA results. Finally synchronization of unidirectional coupled identical proposed autonomous jerk oscillators is achieved using adaptive sliding mode control method.

scholarly journals Analysis, adaptive control and circuit simulation of a novel finance system with dissaving

2016 ◽  
Vol 26 (1) ◽  
pp. 95-115 ◽  
Author(s):  
Ourania I. Tacha ◽  
Christos K. Volos ◽  
Ioannis N. Stouboulos ◽  
Ioannis M. Kyprianidis
Keyword(s):  
Adaptive Control ◽  
Nonlinear Theory ◽  
Circuit Simulation ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Dynamical Analysis ◽  
Phase Portraits ◽  
The Novel ◽  
Finance System ◽  
Simulation Tools

In this paper a novel 3-D nonlinear finance chaotic system consisting of two nonlinearities with negative saving term, which is called ‘dissaving’ is presented. The dynamical analysis of the proposed system confirms its complex dynamic behavior, which is studied by using wellknown simulation tools of nonlinear theory, such as the bifurcation diagram, Lyapunov exponents and phase portraits. Also, some interesting phenomena related with nonlinear theory are observed, such as route to chaos through a period doubling sequence and crisis phenomena. In addition, an interesting scheme of adaptive control of finance system’s behavior is presented. Furthermore, the novel nonlinear finance system is emulated by an electronic circuit and its dynamical behavior is studied by using the electronic simulation package Cadence OrCAD in order to confirm the feasibility of the theoretical model.

CHAOS, BIFURCATION AND ROBUSTNESS OF A CLASS OF HOPFIELD NEURAL NETWORKS

2011 ◽  
Vol 21 (03) ◽  
pp. 885-895 ◽  
Author(s):  
WEN-ZHI HUANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Chaotic Behavior ◽  
Robustness Analysis ◽  
Original System ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Computer Assisted ◽  
New Class

Chaos, bifurcation and robustness of a new class of Hopfield neural networks are investigated. Numerical simulations show that the simple Hopfield neural networks can display chaotic attractors and limit cycles for different parameters. The Lyapunov exponents are calculated, the bifurcation plot and several important phase portraits are presented as well. By virtue of horseshoes theory in dynamical systems, rigorous computer-assisted verifications for chaotic behavior of the system with certain parameters are given, and here also presents a discussion on the robustness of the original system. Besides this, quantitative descriptions of the complexity of these systems are also given, and a robustness analysis of the system is presented too.

EFFECT OF DISSIPATION ON THE HAMILTONIAN CHAOS IN COUPLED OSCILLATOR SYSTEMS

2002 ◽  
Vol 12 (04) ◽  
pp. 859-867 ◽  
Author(s):  
V. SHEEJA ◽  
M. SABIR
Keyword(s):  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Dissipation Function ◽  
Hamiltonian Chaos ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Route To Chaos ◽  
Period Doubling Bifurcations ◽  
External Driving

We study the effect of linear dissipative forces on the chaotic behavior of coupled quartic oscillators with two degrees of freedom. The effect of quadratic Rayleigh dissipation functions, one with diagonal coefficients only and the other with nondiagonal coefficients as well are studied. It is found that the effect of Rayleigh Dissipation function with diagonal coefficients is to suppress chaos in the system and to lead the system to its equilibrium state. However, with a dissipation function with nondiagonal elements, other types of behaviors — including fixed point attractor, periodic attractors and even chaotic attractors — are possible even when there is no external driving. In such a system the route to chaos is through period-doubling bifurcations. This result contradicts the view that linear dissipation always causes decay of oscillations in oscillator models.

An Autonomous Simple Chaotic Jerk System with Stable and Unstable Equilibria Using Reverse Sine Hyperbolic Functions

2020 ◽  
Vol 30 (05) ◽  
pp. 2050070 ◽  
Author(s):  
Manoj Joshi ◽  
Ashish Ranjan
Keyword(s):  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Phase Portraits ◽  
Hidden Attractors ◽  
Hyperbolic Functions ◽  
Simple Implementation ◽  
Route To Chaos ◽  
Jerk System

This article introduces a new simple jerk system with sine hyperbolic nonlinearity which gives the hidden attractor. An autonomous simple implementation of jerk system experiences an important and striking feature of hidden attractors with both stable equilibrium and unstable equilibrium using a reverse nonlinearity function with parametrically controlled approach. Some basic properties of the system are well studied and analyzed in terms of route to chaos, basins of attraction, Lyapunov exponent (LE), bifurcation sequences, coexistence of attractor and phase portraits. The chaotic behavior of the new system is investigated through numerical simulation and their equivalent electrical circuit implementation using single amplifier with few passive elements. The justification of theoretical observation of the proposed chaotic system is perfectly observed in PSPICE simulation and laboratory experiment.

Bifurcations and Chaos of Duffing–van der Pol Equation with Nonsymmetric Nonlinear Restoring and Two External Forcing Terms

2014 ◽  
Vol 24 (03) ◽  
pp. 1430011 ◽  
Author(s):  
Zhiyan Yang ◽  
Tao Jiang ◽  
Zhujun Jing
Keyword(s):  
Numerical Simulation ◽  
Periodic Orbits ◽  
Period Doubling ◽  
Original System ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
External Forcing ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Period Doubling Bifurcations

Bifurcations and chaos of Duffing–van der Pol equation with nonsymmetric nonlinear restoring and two external forcing terms are investigated. The threshold values of the existence of chaotic motion are obtained under periodic perturbation. By the second-order averaging method, we prove the criteria of the existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 1, 2, 3, 5, and cannot prove the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 4, 6, 7, …, where σ is not rational to ω1, but can show the occurrence of chaos in the original system by numerical simulation. Numerical simulation including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincaré maps, not only show the consistence with the theoretical analysis but also exhibit more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations lead to interleaving occurrence of chaotic behaviors and quasi-periodic orbits, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos occurring more than once, chaos suddenly disappearing to periodic orbits, strange nonchaotic attractor, nonattracting chaotic set and nice chaotic attractors.

Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing

2017 ◽  
Vol 27 (08) ◽  
pp. 1750125 ◽  
Author(s):  
Tao Jiang ◽  
Zhiyan Yang ◽  
Zhujun Jing
Keyword(s):  
Parametric Excitation ◽  
Chaotic Motion ◽  
Period Doubling ◽  
Melnikov Method ◽  
Original System ◽  
Duffing Equation ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
External Forcing ◽  
Dynamical Behaviors

We study the Duffing equation with parametric excitation and single external forcing and obtain abundant dynamical behaviors of bifurcations and chaos. The criteria of chaos of the Duffing equation under periodic perturbation are obtained through the Melnikov method. And the existence of chaos of the averaged system of the Duffing equation under the quasi-periodic perturbation [Formula: see text] (where [Formula: see text] is not rational relative to [Formula: see text]) and [Formula: see text] is shown, but the existence of chaos of averaged system of the Duffing equation cannot be proved when [Formula: see text],[Formula: see text]7–15, whereas the occurrence of chaos in the original system can be shown by numerical simulation. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit some new complex dynamical behaviors, including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent diagrams, phase portraits and Poincaré maps. We find a large chaotic region with some solitary period parameter points, a large period and quasi-period region with some solitary chaotic parameter points, period-doubling to chaos and chaos to inverse period-doubling, nondense curvilinear chaotic attractor, nonattracting chaotic motion, nonchaotic attracting set, fragmental chaotic attractors. Almost chaotic motion and almost nonchaotic motion appear through adjusting the parameters of the Duffing system, which can be taken as a strategy of chaotic control or a strategy of chaotic motion to nonchaotic motion.

scholarly journals A New 4-D Chaotic System with Hidden Attractor and its Circuit Implementation

2018 ◽  
Vol 7 (3) ◽  
pp. 1245 ◽  
Author(s):  
Aceng Sambas ◽  
Mustafa Mamat ◽  
Sundarapandian Vaidyanathan ◽  
Muhammad Mohamed ◽  
Mada Sanjaya
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Hidden Attractor ◽  
Circuit Realization ◽  
New Chaotic System ◽  
Quadratic Nonlinearities ◽  
Work First

In the chaos literature, there is currently significant interest in the discovery of new chaotic systems with hidden chaotic attractors. A new 4-D chaotic system with only two quadratic nonlinearities is investigated in this work. First, we derive a no-equilibrium chaotic system and show that the new chaotic system exhibits hidden attractor. Properties of the new chaotic system are analyzed by means of phase portraits, Lyapunov chaos exponents, and Kaplan-Yorke dimension. Then an electronic circuit realization is shown to validate the chaotic behavior of the new 4-D chaotic system. Finally, the physical circuit experimental results of the 4-D chaotic system show agreement with numerical simulations.

Sign in / Sign up

Export Citation Format

Share Document