Analogue Electrical Circuit for Simulation of the Duffing-Holmes Equation (original) (raw)
The Study of a Nonlinear Duffing–Type Oscillator Driven by Two Voltage Sources
In the present work, a detailed study of a nonlinear electrical oscillator with damping and external excitation is presented. The system under study consists of a Duffing-type circuit driven by two sinusoidal voltage sources having different frequencies. The dynamical behavior of the proposed system is investigated numerically, by solving the system of state equations and simulating its behavior as a circuit using MultiSim. The tools of the theoretical approach are the bifurcation diagrams, the Poincaré sections, the phase portraits, and the maximum Lyapunov exponent. The numerical investigation showed that the system has rich complex dynamics including phenomena such as quasiperiodicity, 3-tori, and chaos.
Strong chaotification and robust chaos in the Duffing oscillator induced by two-frequency excitation
Nonlinear Dynamics, 2021
In this work, we demonstrate numerically that two-frequency excitation is an effective method to produce chaotification over very large regions of the parameter space for the Duffing oscillator with singleand double-well potentials. It is also shown that chaos is robust in the last case. Robust chaos is characterized by the existence of a single chaotic attractor which is not altered by changes in the system parameters. It is generally required for practical applications of chaos to prevent the effects of fabrication tolerances, external influences, and aging that can destroy chaos. After showing that very large and continuous regions in the parameter space develop a chaotic dynamics under twofrequency excitation for the double-well Duffing oscillator, we demonstrate that chaos is robust over these regions. The proof is based upon the observation of the monotonic changes in the statistical properties of the chaotic attractor when the system parameters are varied and by its uniqueness, demonstrated by changing the initial conditions. The effects of a second frequency in the single-well Duffing oscillator is also investigated. While a quite significant chaotification is observed, chaos is generally not robust in this case.
Lithuanian Journal of Physics
We consider a second order linear resonator inserted in the negative feedback loop of the chaotic Duffing-Holmes oscillator for stabilizing unstable periodic orbit. Mathematical model is discussed and numerical simulations are presented. An analogue electronic controller is described. Experiments have been performed with an electronic version of the Duffing-Holmes oscillator. Stabilization of periodic oscillations can be achieved with a small control force.
Hyperchaos and bifurcations in a driven Van der Pol–Duffing oscillator circuit
International Journal of Dynamics and Control, 2014
We investigate the dynamics of a driven Van der Pol-Duffing oscillator circuit and show the existence of higher-dimensional chaotic orbits (or hyperchaos), transient chaos, strange-nonchaotic attractors, as well as quasiperiodic orbits born from Hopf bifurcating orbits. By computing all the Lyapunov exponent spectra, scanning a wide range of the driving frequency and driving amplitude parameter space, we explore in two-parameter space the regimes of different dynamical behaviours.
Comparative Analysis of Numerically Computed Chaos Diagrams in Duffing Oscillator
This study utilised optimum fractal disk dimension algorithms to characterize the evolved strange attractor (Poincare section) when adaptive time steps Runge-Kutta fourth and fifth order algorithms are employed to compute simultaneously multiple trajectories of a harmonically excited Duffing oscillator from very close initial conditions. The challenges of insufficient literature that explore chaos diagrams as visual aids in dynamics characterization strongly motivate this study. The object of this study is to enable visual comparison of the chaos diagrams in the excitation amplitude versus frequency plane. The chaos diagrams obtained at two different damp coefficient levels conforms generally in trend to literature results[1] and qualitatively the same for all algorithms. The chances of chaotic behaviour are higher for combined higher excitation frequencies and amplitudes in addition to smaller damp coefficient. Fourth and fifth order Runge-Kutta algorithms indicates respectively 62.3% and 53.3% probability of chaotic behaviour at 0.168 damp coefficient and respectively 77.9% and 78.9% at 0.0168 damp coefficient. The chaos diagrams obtained by fourth order algorithms is accepted to be more reliable than its fifth order counterpart, its utility as tool for searching possible regions of parameter space where chaotic behaviour/motion exist may require additional dynamic behaviour tests.
Chaotic electronic oscillator from single amplifier biquad
AEU-International Journal of …, 2011
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This study exploited the computation accuracy of governing equations of linearly or periodically behaves dynamic system with fourth and fifth order Runge-Kutta algorithms to develop chaos diagrams of harmonically excited Duffing oscillator. The study adopt the fall to tolerance of absolute deviation between two independently sought solutions of governing equation to characterise excitation frequencies and amplitude parameter point of Duffing oscillator as either chaotic or not. Displacement and Velocity time history, Phase plot and Poincare were used to validate FORTRAN coded programmes used for this study and chaos diagrams developed at two different damping coefficients. The validation results agreed perfectly with those obtained in the literature. The chaos diagrams predicted by computation at two different damp coefficient levels conforms generally in trend to literature results by and qualitatively the same for three different combination of constant time based Runge-Kutta algorithms. The chances of chaotic behaviour of Duffing oscillator under the combined driven force of parameters becomes more than double at 0.0168 damping coefficient when compared with corresponding results at 0.168 damping coefficient. The probability that selected excitation frequencies and amplitudes will drive Duffing oscillator chaotically at 0.168 damp coefficients is 29.4%, 27.8% and 29.4% respectively. This study demonstrated the significant utility of numerical techniques in dealing with real-world problems that are dominantly nonlinear and shows that in addition to being sensitive to initial conditions, chaos is equally sensitivity to appropriate simulation time steps. In addition, the present chaos diagram generating numerical tool is uniquely characterised by being faster and predicting reliably than that earlier reported by the authors.
A Universal Circuit for Studying and Generating Chaos-Part 11: Strange Attractors
In this introductory tutorial paper, we demonstrate the generality of Chua's oscillator in generating chaos and bifurcation phenomena by electronic laboratory experiments which illustrate the standard routes to chaos, and by giving a result which shows that Chua's oscillator can generate the same qualitative behavior as any member of a 21-parameter family C of continuous, odd-symmetric, piecewise-linear vector field in R3. This result is of fundamental importance because it unifies many previously published papers on chaotic circuits and systems (e.g. examples from Brockett, Sparrow, Arnkodo, Nishio, Ogorzalek, etc.) under one umbrella, thereby obviating the need to analyze these circuits and systems as separate and unrelated systems. Indeed, every bifurcation and chaotic phenomena exhibited by any member of the family C is also exhibited by this universal circuit. In a companion paper [l], we show how the generality of Chua's oscillator can be used to approximate other chaotic systems in the literature which are not necessarily piecewiselinear.
Electronic Realization Of Chaotic Systems
Using the Undergraduate Research Award provided by the national office of the Society of Physics Students, the CWU chapter of SPS investigated electronic realizations of chaotic systems. The CWU chapter of SPS selected this topic to advance its members’ understanding of chaotic systems. J.C. Sprott [2] reported on a class of chaotic differential equations that can, in principle, be simply realized using discrete electronic components. These circuits can be used to experimentally investigate chaotic behavior in a simple system. This report presents a comparison of the computational and experimental data collected from one simple chaotic circuit realizing the differential equation: &x& = −α&x& − x& − x + x2 .