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# On the periodic and chaotic responses of Duffing"s oscillator by Ameer Hassan

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Written in English

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## Subjects:

• Chaotic behavior in systems -- Mathematical models.,
• Nonlinear oscillators.,
• Duffing equations.,
• Perturbation.

Edition Notes

## Book details

The Physical Object ID Numbers Statement by Ameer Hassan. Pagination xxvii, 235 leaves, bound : Number of Pages 235 Open Library OL16841636M

Download On the periodic and chaotic responses of Duffing"s oscillator

The van der Pol-Duffing oscillator does show regions of potentially chaotic response to periodic and quasiperiodic forcing depending on the amplitude of the forcing.

Figure 1. Cited by: 4. In this paper, chaotic dynamics of a cubic-quintic-septic Duffing oscillator subjected to periodic excitation is investigated. The multiple scales method is used to determine the various resonance states of the model.

It is found that the considered model posses thirteen resonance states whose seven are thoroughly studied.

The steady-state solutions and theirs stabilities are by: 1. Duffing Oscillator behaves chaotically. In a study of mechanical systems,[9] reported certain values of amplitude of excitation that could produce various responses – chaotic and periodic. To ensure that the Duffing oscillator is chaotic at the.

We express no doubt that the Duffing Oscillator behaves chaotically. In a study of mechanical systems,[9] reported certain values of amplitude of excitation that could produce various responses – chaotic and periodic.

To ensure that the Duffing oscillator is chaotic at the first instance, we chose Dowell’s amplitude of excitation. We study the bifurcations and the chaotic behaviour of a periodically forced double-well Duffing oscillator coupled to a single-well Duffing oscillator.

The chaotic motions of the Duffing-Van der Pol oscillator with external and parametric excitations are investigated both analytically and numerically in this paper.

The critical curves separating the chaotic and nonchaotic regions are obtained. The chaotic feature on the system parameters is discussed in detail. Some new dynamical phenomena including the controllable frequency are presented.

This paper is devoted to the development of a new approach to distinguish between a chaotic and a periodic motion of a dynamical On the periodic and chaotic responses of Duffings oscillator book. A criterion namely periodicity ratio is introduced to distinguish a periodic motion from a nonperiodic motion and differentiate chaos from a regular motion by a numerical procedure without plotting any figures.

A Duffing’s equation and a nonlinear driven. Response of a lightly damped hard Duffing oscillator to harmonic excitations has been investigated experimentally as well as numerically.

A single degree-of-freedom torsional vibratory system has been fabricated as a mechanical analogue of Duffing equation with strong nonlinearity. A stable Duffing system is examined by numerical simulations in order to obtain a better understanding of the behavior of periodic and chaotic responses to sinusoidal excitations.

It is found that beside the multiplicity of responses, there is a duality for both periodic and chaotic responses.

Based on the homotopy analysis method (HAM), the high accuracy frequency response curve and the stable/unstable periodic solutions of the Van der Pol-Duffing forced oscillator with the variation of the forced frequency are obtained and studied.

The stability of the periodic solutions obtained is analyzed by use of Floquet theory. When the periodic force that drives the system is large, the motion can become chaotic and the phase space diagram can develop a strange attractor.A Poincaré section can be plotted by taking one phase space point in each period of the driving force.

In the simplest cases, when the system enters a limit cycle, the Poincaré section reduces to a single point. However, for bifurcations of periodic to chaotic motions encountered in the low-frequency range, the corresponding variations in TAIP of the double-well potential systems are small.

For a chaotic response, the associated TAIP is insensitive to the initial conditions but tends to an asymptotic value as the averaging time increases, and thus can. A Partial Differential Equation with Infinitely Many Periodic Orbits: Chaotic Oscillations of a Forced Beam (P Holmes & J Marsden) Classical Nonlinear Oscillators: Duffing, Van der Pol and Pendulum: Universal Scaling Property in Bifurcation Structure of Duffing's and Generalized Duffing's.

36 Hervé Lucas Koudahoun et al.: Chaotic Dynamics of an Extended Duffing Oscillator Under Periodic Excitation where Ais the complex conjugate of tuting the solution (8) into (7)yields to 0 0 2 3 2 4 3 0 1 1 1 3 35 2 10 2 4 2.

2 iT i T Du u iDA i A AA AA AA e. The OGY method has been applied to a Duffing oscillator subjected to a partially stochastic excitat while the application of Pyragas' method to the Duffing oscillator has been considered.

Abstract. An oscillator with asymmetric (Helmholtz) and symmetric (Duffing) nonlinearities, representative of an interesting problem in nonlinear structural dynamics, is discussed as a simple but prototypical model to describe bifurcation from regular to chaotic behaviour.

The chaotic oscillator has become an important tool in the analysis of harmonic signals with low signal to noise ratio. On the other hand, traditionally, frequency modulated (FM) signals have always been studied through conventional techniques of time-frequency analysis.

Recently, the Duffing oscillator has shown to be a powerful tool to detect frequencies for periodic signals because of its. The Duffing oscillator is a non-linear system that can exhibit both a periodic and a chaotic response.

It is a damped, driven oscillator. It is modelled by a non-linear, second order differential equation known as the “Duffing Equation”. The Duffing equation for the system in Fig.

1 is Figure 2. Block diagram of electronic analogue of the. Chaotic motions of a Rayleigh -Duffing oscillator with periodically external and parametric excitations are investigated rigorously.

Chaos arising from intersections of homoclinic orbits is analyzed with the Melnikov method. The critical curves separating the chaotic and nonchaotic - regions are obtained.

In this paper, asymmetric periodic motions in a periodically forced, softening Duffing oscillator are presented analytically through the generalized harmonic balance method. For the softening Duffing oscillator, the symmetric periodic motions with jumping phenomena were understood very well.

5 Forced Harmonic Vibration of a Duffing Oscillator with Linear Viscous Damping (Tamas Kalmar-Nagy and Balakumar Balachandran). Introduction. Free and forced responses of the linear oscillator. Amplitude and phase responses of the Duffing oscillator. Periodic solutions, Poincare sections, and bifurcations.

Global dynamics. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): 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. The Duffing equation (or Duffing oscillator), named after Georg Duffing (–), is a non-linear second-order differential equation used to model certain damped and driven equation is given by ¨ + ˙ + + = ⁡ where the (unknown) function = is the displacement at time, ˙ is the first derivative of with respect to time, i.e.

velocity, and ¨ is the second time. Analysis of Dynamics of a Duffing Oscillator (with specific conditions) - andyj1/chaotic-duffing-oscillator. The forced Duffing oscillator exhibits behavior ranging from limit cycles to chaos due to its nonlinear dynamics.

When the periodic force that drives the system is large, chaotic behavior emerges and the phase space diagram is a strange attractor. In that case the behavior of the system is sensitive to the initial condition. excited Duffing oscillator. Renjung Chang, Junfu Liu and Chengtang, Fan Department of Mechanical Engineering, National Cheng Kung University, Taiwan, ROC.

Summary. A novel stochastic linearization approach is proposed for predicting periodic and/or chaotic response of Duffing oscillator subjected to sinusoidal and weaknoise excitations. Duffing oscillator is an example of a periodically forced oscillator with a nonlinear elasticity, written as $\tag{1} \ddot x + \delta \dot x + \beta x + \alpha x^3 = \gamma \cos \omega t \,$.

where the damping constant obeys $$\delta\geq 0\,$$ and it is also known as a simple model which yields chaos, as well as van der Pol oscillator.

A new type of responses called as periodic-chaotic motion is found by numerical simulations in a Duffing oscillator with a slowly periodically parametric excitation. The periodic-chaotic motion is an attractor, and simultaneously possesses the feature of periodic and chaotic oscillations, which is a.

In this paper, the existence of chaotic behavior in the single-well Duffing Oscillator was examined under parametric excitations using Melnikov method and Lyapunov exponents. The minimum and maximum values were obtained and the dynamical behaviors showed the intersections of manifold which was illustrated using the MATCAD software.

This extends some results in the literature. ~ Book Bifurcation And Chaos In Coupled Oscillators ~ Uploaded By James Michener, system upgrade on fri jun 26th at 5pm et during this period our website will oscillators are numerically investigated as a function of the strength of nonlinear coupling and the amplitude f of the periodic driving forcethe influence of the chaotic.

Duffing equation [1]. E Babourina-Brooks, et al. determined the noise response of a quantum nanomechanical resonator using a Duffing oscillator-based model due to the dynamical equivalency [2]. Heung-Ryoul Noh indicated that the Duffing oscillator can be realized in an intensity-modulated magneto-optical trap [3].

der Pol oscillator. The simulation results show that the high-order subharmonic and chaotic responses and their bifurcations can be effectively observed. Key-Words: Poincaré section method, Chaotic motion, Response integration, Bifurcation, Duffing-Van der Pol oscillator 1 Introduction In recent twenty years, the problem of identifying.

The Duffing Equation Introduction We have already seen that chaotic behavior can emerge in a system as simple as the logistic map.

In that case the "route to chaos" is called period-doubling. In practice one would like to understand the route to chaos in systems described by partial differential equations, such as flow in a randomly stirred fluid.

The paper starts by discussing the Duffing oscillator which features a second order non-linear differential equation describing complex motion whereas the second model is the Van der Pol oscillator with non-linear damping. A first order actuator is added to both models to expand on the chaotic behavior of the oscillators.

Professor Luo is currently a Distinguished Research Professor at Southern Illinois University Edwardsville. He is an international renowned figure in the area of nonlinear dynamics and mechanics.

For about 30 years, Dr. Luo’s contributions on nonlinear dynamical systems and mechanics lie in (i) the local singularity theory for discontinuous dynamical systems, (ii) Dynamical systems. Stochastic chaos discussed here means a kind of chaotic responses in a Duffing oscillator with bounded random parameters under harmonic excitations.

A system with random parameters is usually called a stochastic system. The modifier 'stochastic' here implies dependent on some random parameter.

As the system itself is stochastic, so is the response, even under harmonic excitations alone. Chua's circuit (also known as a Chua circuit) is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never "repeats".It was invented in by Leon O.

Chua, who was a visitor at Waseda University in Japan at that time. potential for chaotic behavior in the Du ng system) two additional terms must be included in the system. A term, x_, must be included to allow damping in the system and a term, cos(!t), must be included to allow external forcing of the system.

The result of including these terms is the general Du ng Equation: x + x_ + x+ x3 = cos(!t). is introduced the basic principle of weak periodic signal detection based on chaotic vibration shortcomings of the chaotic Duffing oscillator are presented by analyzingthe traditional detection results and detection principle based on Duffing oscillator, To make up the shortage of Duffing chaotic oscillator,it used multiple auto correlation.

The Duffing oscillator from big cycle motion to chaotic motion is more sensitively to same or near periodic signal than the traditional Duffing oscillator.

Simulation shows that the Duffing oscillator from big cycle motion to chaotic motion is more suitable to be used to detect the 1 Hz characteristic current precisely and sensitively.

A formulation of statistical linearization for multi-degree-of-freedom (M-D-O-F) systems subject to combined mono-frequency periodic and stochastic excitations is presented. The proposed technique is based on coupling the statistical linearization and the harmonic balance concepts.The capability of conventional detection method such as power spectral density is limited while detecting periodic signals buried in the noise and deciding Weak Signal Detection Based on Duffing Oscillator - IEEE Conference Publication.Fig.

1. Phase space of the unexcited Duffing oscillator. For the excited case (A = 10 and = 1 rad/s), the nom-inal frequency of the system will be dominated by the excitation function, as shown in Fig. (2). This fact will prove crucial when using TDAS for controlling the chaot-ic response of the Duffing oscillator.

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