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.

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.

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.

DYNAMICS AND MODULATED CHAOS FOR TWO COUPLED OSCILLATORS

2004 ◽  
Vol 14 (01) ◽  
pp. 337-346 ◽  
Author(s):  
QINSHENG BI
Keyword(s):  
Coupled Oscillators ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
The Other ◽  
Van Der Pol ◽  
Parametrically Excited ◽  
Van Der Pol Oscillators ◽  
Period Doubling Bifurcations

The dynamical behavior of two coupled parametrically excited van der Pol oscillators is investigated in this paper. A special road to chaos is explored in detail. Period-doubling bifurcation associated with one of the frequencies of the system may be observed, the other frequency of the coupled oscillators plays a role in the evolution. It is found that one of the frequencies of the system contributes to the cascade of period-doubling bifurcations associated with the other frequency, which leads to a generalized modulated chaos.

DYNAMICAL BEHAVIOR OF A VAN DER POL OSCILLATOR CIRCUIT WITH INTERNAL DELAY

2020 ◽  
Vol 8 (2) ◽  
pp. 16
Author(s):  
CHAKRABORTY SAUMEN ◽  
SANKAR DE SARKAR SAUMENDRA ◽  
◽  
Keyword(s):  
Dynamical Behavior ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Oscillator Circuit

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.

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.

Two timescale harmonic balance. I. Application to autonomous one-dimensional nonlinear oscillators

1992 ◽  
Vol 340 (1659) ◽  
pp. 473-501 ◽  
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Phase Locking ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Wide Range

Two timescale harmonic balance is a semi-analytical/numerical method for deriving periodic solutions and their stability to a class of nonlinear autonomous and forced oscillator equations of the form ẍ + x = f(x,ẋ,λ) and ẍ + x = f(x,ẋ,λ,t) , where λ is a control parameter. The method incorporates salient features from both the method of harmonic balance and multiple scales, and yet does not require an explicit small parameter. Essentially periodic solutions are formally derived on the basis of a single assumption: ‘that an N harmonic, truncated, Fourier series and its first two derivatives can represent x(t) , ẋ(t) and ẍ(t) respectively’. By seeking x(t) as a series of superharmonics, subharmonics, and ultrasubharmonics it is found that the method works over a wide range of parameter space provided the above assumption holds which, in practice, imposes some ‘problem dependent’ restriction on the magnitude of the nonlinearities. Two timescales, associated with the amplitude and phase variations respectively, are introduced by means of an implicit parameter Є . These timescales permit the construction of a set of amplitude evolution equations together with a corresponding stability criterion. In Part I the method is formulated and applied to three autonomous equations, the van der Pol equation, the modified van der Pol equation, and the van der Pol equation with escape. In this case an expansion in superharmonics is sufficient to reveal Hopf, saddle node and homoclinic bifurcations which are compared with results obtained by numerical integration of the equations. In Part II the method is applied to forced nonlinear oscillators in which the solution for x(t) includes superharmonics, subharmonics, and ultrasubharmonics. The features of period doubling, symmetry breaking, phase locking and the Feigenbaum transition to chaos are examined.

On the Van Der Pol Oscillator: an Overview

2013 ◽  
Vol 430 ◽  
pp. 3-13 ◽  
Author(s):  
Livija Cveticanin
Keyword(s):  
Complex Systems ◽  
Nonlinear Term ◽  
Nonlinear Oscillators ◽  
Elastic Force ◽  
The Self ◽  
Van Der Pol Oscillator ◽  
Damping Force ◽  
Van Der Pol ◽  
Linear Elastic ◽  
Linear Damping

In this paper an overview of the self-sustained oscillators is given. The standard van der Pol and the Rayleigh oscillators are considered as basic ones. The cubic nonlinear term of Duffing type is included. The special attention is given to the various complex systems based on the Rayleighs and van der Pols oscillator which are extended with the nonlinear oscillators of Duffing type and also excited with a periodical force. The connection is with the linear elastic force or with linear damping force. The objectives for future investigation are given in this matter.

scholarly journals Diffusion Dynamics and Impact of Noise on a Discrete-Time Ratio-Dependent Model: An Analytical and Numerical Approach

2019 ◽  
Vol 24 (4) ◽  
pp. 103 ◽  
Author(s):  
Kalyan Das ◽  
M. N. Srinivas ◽  
Nurul Huda Gazi
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponent ◽  
Discrete Time ◽  
Discrete System ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Time Ratio ◽  
Diffusion Dynamics ◽  
Ratio Dependent

The paper deals with the dynamical behavior of a discrete-time ratio-dependent predator–prey system. The predator dependence is one of the main concerns of the system. The stability analysis of this 2-dimensional map was carried out analytically. Numerical simulation was carried out to verify the analytical results. We analyzed some specific features that could arise in discrete system. Basin of attraction was found for the endemic equilibrium state. We extended the numerical simulation for the maximal Lyapunov exponent. The presence of positive Lyapunov exponent indicated chaotic behavior of the map. The sensitive dependence on initial condition is one of the criteria for a discrete system. We showed that the system is sensitive on the initial conditions. We also carried out the analysis of diffusion and impact of noise.

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