Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos

2005 ◽  
Vol 128 (3) ◽  
pp. 282-293 ◽  
Author(s):  
J. C. Chedjou ◽  
K. Kyamakya ◽  
I. Moussa ◽  
H.-P. Kuchenbecker ◽  
W. Mathis
Keyword(s):  
Electronic Circuit ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Electromechanical System ◽  
Phase Portraits ◽  
Coupling Coefficients ◽  
Oscillatory Solutions ◽  
Electromechanical Transducer ◽  
Design Engineers

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

PERTURBED BEHAVIOR FOR THE MODEL OF WAVE PROPAGATION IN A NONLINEAR DISPERSIVE NONCONSERVATIVE MEDIA

2012 ◽  
Vol 22 (08) ◽  
pp. 1250193 ◽  
Author(s):  
GUOXIANG YU ◽  
JUN YU
Keyword(s):  
Wave Propagation ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Type Equation ◽  
Phase Portraits ◽  
Lyapunov Dimension ◽  
Dissipative Media ◽  
Bifurcation Structures

As a governing equation of wave propagation in a nonlinear, dispersive and dissipative media, KdV–Burgers type equation has received great attention. In this paper, the dynamical behavior of the generalized KdV–Burgers equation under a periodic perturbation is investigated numerically in detail. It is shown that dynamical chaos can occur when we choose appropriately systematic parameters and initial conditions. Abundant bifurcation structures and different routes to chaos such as period-doubling and inverse period-doubling cascades, intermittent bifurcation and crisis, are found by applying bifurcation diagrams, the Poincaré maps and phase portraits. To characterize chaotic behavior of this system, the spectrum of Lyapunov exponent and Lyapunov dimension of the attractor are also employed.

Dynamical Analysis and Circuit Simulation of a New Fractional-Order Hyperchaotic System and Its Discretization

2016 ◽  
Vol 26 (13) ◽  
pp. 1650222 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
A. Elsonbaty ◽  
A. A. Elsadany ◽  
A. E. Matouk
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Control Technique ◽  
Order System ◽  
Qualitative Behavior ◽  
Hyperchaotic System

This paper presents an analytical framework to investigate the dynamical behavior of a new fractional-order hyperchaotic circuit system. A sufficient condition for existence, uniqueness and continuous dependence on initial conditions of the solution of the proposed system is derived. The local stability of all the system’s equilibrium points are discussed using fractional Routh–Hurwitz test. Then the analytical conditions for the existence of a pitchfork bifurcation in this system with fractional-order parameter less than 1/3 are provided. Conditions for the existence of Hopf bifurcation in this system are also investigated. The dynamics of discretized form of our fractional-order hyperchaotic system are explored. Chaos control is also achieved in discretized system using delay feedback control technique. The numerical simulation are presented to confirm our theoretical analysis via phase portraits, bifurcation diagrams and Lyapunov exponents. A text encryption algorithm is presented based on the proposed fractional-order system. The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.

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.

Dynamical behavior of nonlinear wave solutions of the generalized Newell–Whitehead–Segel equation

2020 ◽  
Vol 31 (04) ◽  
pp. 2050059
Author(s):  
Asit Saha ◽  
Amiya Das
Keyword(s):  
Nonlinear Wave ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Phase Portraits ◽  
Phase Plane Analysis ◽  
Linear Coefficient ◽  
Wave Solutions ◽  
Nonlinear Wave Solutions

Dynamical behavior of nonlinear wave solutions of the perturbed and unperturbed generalized Newell–Whitehead–Segel (GNWS) equation is studied via analytical and computational approaches for the first time in the literature. Bifurcation of phase portraits of the unperturbed GNWS equation is dispensed using phase plane analysis through symbolic computation and it shows stable oscillation of the traveling waves. Chaotic behavior of the perturbed GNWS equation is obtained by applying different computational tools, like phase plot, time series plot, Poincare section, bifurcation diagram and Lyapunov exponent. A period-doubling bifurcation behavior to chaotic behavior is shown for the perturbed GNWS equation and again it shows chaotic to periodic motion through inverse period-doubling bifurcation. The perturbed GNWS equation also shows chaotic motion through a sequence of periodic motions (period-1, period-3 and period-5) depending on the variation of the parameter of linear coefficient. Thus, the parameter of linear coefficient plays the role of a controlling parameter in the chaotic dynamics of the perturbed GNWS equation.

Multistability and Bubbling Route to Chaos in a Deterministic Model for Geomagnetic Field Reversals

2019 ◽  
Vol 29 (12) ◽  
pp. 1930034
Author(s):  
Paulo C. Rech ◽  
Sudarshan Dhua ◽  
N. C. Pati
Keyword(s):  
Fixed Point ◽  
Geomagnetic Field ◽  
Deterministic Model ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Multiple Attractors ◽  
System A ◽  
Two Parameters

We report coexisting multiple attractors and birth of chaos via period-bubbling cascades in a model of geomagnetic field reversals. The model system comprises a set of three coupled first-order quadratic nonlinear equations with three control parameters. Up to seven kinds of multistable attractors, viz. fixed point-periodic, fixed point-chaotic, periodic–periodic, periodic-chaotic, chaotic–chaotic, fixed point-periodic–periodic, fixed point-periodic-chaotic are obtained depending on the initial conditions for critical parameter sets. Antimonotonicity is a striking characteristic feature of nonlinear systems through which a full Feigenbaum tree corresponding to creation and annihilation of period-doubling cascades is developed. By analyzing the two-parameters dependent dynamics of the system, a critical biparameter zone is identified, where antimonotonicity comes into existence. The complex dynamical behaviors of the system are explored using phase portraits, bifurcation diagrams, Lyapunov exponents, isoperiodic diagram, and basins of attraction.

On the modeling of some triodes-based nonlinear oscillators with complex dynamics: case of the Van der Pol oscillator

2021 ◽  
Author(s):  
Joakim Vianney Ngamsa Tegnitsap ◽  
Merlin Brice Saatsa Tsefack ◽  
Elie Bertrand Megam Ngouonkadi ◽  
Hilaire Bertrand Fotsin
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponent ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Oscillator ◽  
Phase Portraits ◽  
Van Der Pol ◽  
Physical Consideration

Abstract In this work, the dynamic of the triode-based Van der Pol oscillator coupled to a linear circuit is investigated (Triode-based VDPCL oscillator). Towards this end, we present a mathematical model of the triode, chosen from among the many different ones present in literature. The dynamical behavior of the system is investigated using classical tools such as two-parameter Lyapunov exponent, one-parameter bifurcation diagram associated with the graph of largest Lyapunov exponent, phase portraits, and time series. Numerical simulations reveal rather rich and complex phenomena including chaos, transient chaos, the coexistence of solutions, crisis, period-doubling followed by reverse period-doubling sequences (bubbles), and bursting oscillation. The coexistence of attractors is illustrated by the phase portraits and the cross-section of the basin of attraction. Such triode-based nonlinear oscillators can find their applications in many areas where ultra-high frequencies and high powers are demanded (radio, radar detection, satellites communication, etc.) since triode can work with these performances and are often used in the aforementioned areas. In contrast to some recent work on triode-based oscillators, LTSPICE simulations, based on real physical consideration of the triode, are carried out in order to validate the theoretical results obtained in this paper as well as the mathematical model adopted for the triode.

Nonlinear Dynamics of Composite Laminated Circular Cylindrical Shell With Membranes in Thermal Environment

2018 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yufei Zhang ◽  
Qian Wang
Keyword(s):  
Cylindrical Shell ◽  
Multiple Scales ◽  
Thermal Environment ◽  
Initial Conditions ◽  
Circular Cylindrical Shell ◽  
Period Doubling ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Temperature Parameter ◽  
Intermittent Chaos

This paper is focused on the chaotic dynamics of a composite laminated circular cylindrical shell with radially pre-stretched membranes at both ends and clamped along a generatrix. Based on the two-degree-of-freedom non-autonomous nonlinear equations of this system, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equation. The resonant case considered here is the primary parametric resonance-1/2 subharmonic resonance and 1:1 internal resonance. Corresponding to several selected parameters, the periodic and chaotic motions of the composite laminated circular cylindrical shell clamped along a generatrix are demonstrated by the bifurcation diagrams, the maximum Lyapunov exponents, the phase portraits, the waveforms, the power spectrums and the Poincaré map. The temperature parameter excitation shows that the Pomeau-Manneville type intermittent chaos occur under the certain initial conditions. It is also found that there exist the twin phenomena between the Pomeau-Manneville type intermittent chaos and the period-doubling bifurcation.

scholarly journals Chaotic Path Planning for 3D Area Coverage Using a Pseudo-Random Bit Generator from a 1D Chaotic Map

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1821
Author(s):  
Lazaros Moysis ◽  
Karthikeyan Rajagopal ◽  
Aleksandra V. Tutueva ◽  
Christos Volos ◽  
Beteley Teka ◽  
...  
Keyword(s):  
Path Planning ◽  
Chaotic Map ◽  
Dynamical Behavior ◽  
Area Coverage ◽  
Phase Portraits ◽  
Planning Strategy ◽  
Spline Curves ◽  
Key Space ◽  
Aerial Vehicle ◽  
Discrete Motion

This work proposes a one-dimensional chaotic map with a simple structure and three parameters. The phase portraits, bifurcation diagrams, and Lyapunov exponent diagrams are first plotted to study the dynamical behavior of the map. It is seen that the map exhibits areas of constant chaos with respect to all parameters. This map is then applied to the problem of pseudo-random bit generation using a simple technique to generate four bits per iteration. It is shown that the algorithm passes all statistical NIST and ENT tests, as well as shows low correlation and an acceptable key space. The generated bitstream is applied to the problem of chaotic path planning, for an autonomous robot or generally an unmanned aerial vehicle (UAV) exploring a given 3D area. The aim is to ensure efficient area coverage, while also maintaining an unpredictable motion. Numerical simulations were performed to evaluate the performance of the path planning strategy, and it is shown that the coverage percentage converges exponentially to 100% as the number of iterations increases. The discrete motion is also adapted to a smooth one through the use of B-Spline curves.

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

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