# Chaotic Behaviour Of Forced Oscillator Containing A Square Nonlinear Term On Principal Resonance Curves

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### Backbone Curve Analysis of Nonlinear Mechanical Systems

by X Liu 2018 is my own work and contains nothing which is the outcome of work done in collaboration tions are used to demonstrate this nonlinear behaviour, which shows that the frequency 3.3 The response stability of a forced, 1-DoF Duffing oscillator. An {N ×1} vector of resonant nonlinear terms and linear damping terms,.

### Past, present and future of nonlinear system identification in

by G Kerschen 2006 Cited by 1161 Inertia nonlinearity derives from nonlinear terms containing velocities System identification in the presence of chaotic vibrations (Moon, the principle of linear superposition to nonlinear systems, it has been transitions are typical in forced resonances of nonlinear oscillators. Black-box modeling.

### Subharmonic resonances of the parametrically driven pendulum

by EI Butikov Cited by 64 Depending on the frequency and amplitude of forced oscillations of the suspension point Besides the principal parametric resonance, excited when two driving cycles Chaotic behaviour of this simple nonlinear system has been a subject of Indeed, the angular amplitude of the rapid (second) term in equation (4) is the.

### Duffing Equation - NPTEL

6 Sep 2019 31-42 SDOF Free and Forced Vibration: Duffing Equation, non-conservative systems: Negative damping, van der Pol oscillator, Let us consider the Duffing equation with cubic nonlinear term for the free vibration study of a we may consider the behaviour of the system near the resonance condition,.Missing: square Must include: square

### Nonlinear Oscillation - UCSB Physics

Figure 1: A plot of the unstable potential energy function (blue curve), for k = γ = 1. a solution is that in the presence of a nonlinear term, the principle of super- position no longer Again, we find an equation for y1 which contains an undamped resonance certain forcing functions will result in chaotic behaviour. While a

### Self-oscillation

by A Jenkins 2011 Cited by 222 In a self-oscillator, the driving force is controlled by the oscillation itself more commonly the term perpetual motion is reserved for machines that can tuner works by having the listener adjust the resonant frequency of The dashed curve were the first to observe the onset of chaos in a simple nonlinear

### Modelling of autoresonant control of a parametrically - DORA

Keyword: parametric resonance, screen oscillation, autoresonant control. 1. close to the natural frequency of a system, the system will experience forced resonance. the excitation term appears as the time dependence parameter in the and nonlinear-velocity feedback control to suppress the principal PR in a flexible.

### THREE PROBLEMS OF RESONANCE IN COUPLED OR

by L Lazarus 2016 driving terms are discussed, with particular regard to their resonance patterns. In our first research in the nonlinear dynamics of oscillator systems under the direction of. Professor 2 Dynamics of a System of Two Coupled Oscillators Driven by a Third Behaviors at the drift/lock boundary curve; not to scale. 22. 3.1.

### Chaos and Non-Linear Dynamics - MIT OpenCourseWare

define chaos as aperiodic long-term behavior that exhibits sensitive dependence on initial For our damped nonlinear driven oscillator example we have: ∂ω.

### Author template for journal articles - Research Square

by X Hu 2021 Abstract Nonlinear Quasi-zero-stiffness (QZS) vibration isolation systems with linear sometimes chaotic behavior, and the jump-through phenomenon, which can resonance in the amplitude-frequency curve. Substituting Eq. (23) into Eq. (2) and by comparing the coefficients of the terms containing cos( ) and.

### Chaos in mechanical systems — - Indian Academy of

by P SEKAR Cited by 10 predict certain bifurcation behaviour leading to chaos. eventually be attracted to infinity if the nonlinear term is absent. But as z increases tion of a forced self oscillatory system as early as 1961 he could publish his results only after he chaos occurs at frequencies in the region of the principal resonance and the route to.

### Robust identification of backbone curves using control-based

by L Renson 2015 Cited by 77 such that the resonance frequency generally varies with the oscillation amplitude. Following the principle of linear phase separation techniques, the method The nonlinear force term can be decomposed in a similar cosine series with the excitation is usually limited to a few number of DOFs, i.e. Fk contains only a few.

### Dynamical systems and Introduction to Chaos - ESPCI

11 Trajectories inside a resonance. 12 quired for a physical system to exhibit chaotic behavior. We shall see that it is necessary The motion is an oscillation having a frequency a nonlinear term of X. This is for instance the case if the force acting on the combination of two closed curves : it has the topology of a torus.

### Chaotic breakdown of a periodically forced, weakly damped

by PJ Bryant 1992 Cited by 1 described in terms of resonance curves and Poincare cross-sections oscillations of a forced pendulum is that there is then only one but as is indicated above, a large variety of types of nonlinear tions have a very small window of stability, with the resonance curve having This behaviour is similar to.

### Instabilities and chaos in nonlinear dynamic systems - Wiley

of instabilities and chaotic behavior in nonlinear dynamic systems, with the unstable equilibrium point C (positive x), followed by a few oscillations about small population in the following year, then Eq. 2 contains an extraordinarily rich simultaneous values of x and dxldt for each cycle of the forcing term, i.e., for t = 27rn.

### Shanmuganathan Rajasekar Miguel AF Sanjuan - National

by S Rajasekar Cited by 94 modified Chua's circuit model equation with periodic characteristic curve of The resonance behaviour found in the Duffing oscillator can be realized in the coefficient of the nonlinear term becomes negative the response amplitude lean forced self-excited systems, it is reasonable to expect the response to contain both.

### Classical & Nonlinear Dynamics - Oregon State University

Hamilton's principle. Note that Chapter 8 contains a number of problems dealing with the several discrete maps that lead to chaotic behavior in biological systems. Left: The potentials of an harmonic oscillator (solid curve) and of an turbation that introduces a nonlinear term to the force for large x values: If αx ≪ 1,.

### The Emergence of Chaos in Quantum Mechanics - MDPI

by E Fiordilino 2020 Cited by 1 As examples we quote the equation for the Duffing oscillator (1918) Almost by definition, Quantum Mechanics lays its foundation in the study In principle these equations might display chaotic behaviour when by a nonlinear Schrödinger equation and driven by a resonant laser of on a regular curve.

### Nonlinear effect of forced harmonic oscillator subject to sliding

by Q Xu 2018 Cited by 4 We study the nonlinear behaviors of mass-spring systems damped undergraduates.15 19 Usually, nonlinearity comes from the restoring force containing a cubic term (Duffing's type)15,16 or piecewise linear terms.17 These papers observe the nonlinear oscillation curves and the frequency spectrum

### Feedback Control and Stability of the Van der Pol Equation

by M Sayed 2018 Cited by 3 analyze the nonlinear behavior of this model. The stability parameters, the curves are bent to right or left leading to the parametric excitation amplitude on the routes to chaos is The general form of the Van der Pol Oscillator model with forcing frequency of the system. principal parametric resonance case where. 1.

### Microelectromechanical Oscillators - What is Materials

will contain background and preliminary material on quartz crystal oscillators. buffer amplifier, placed directly after the oscillator, improves this behaviour by Examples of nonlinear amplitude-frequency curves are depicted in Fig. 3.14. An extensive treatment of resonant sensors used as force or strain gauge is found in

### ANALYSIS OF WIND TURBINE BLADE VIBRATION AND

by V Ramakrishnan 2017 Cited by 4 Embedded in the ODE's are terms of a forced Mathieu equation with cubic nonlinearity. When the turbine is on idle, there can be self-induced oscillations in the standby terms of partial differential equations using the principle of virtual work. lead to a softening type behavior (the resonance curve bends to the left).

### 10 Years of Advances in Nonlinear System Identification in

by JP Noël Cited by 4 Because this nonlinear behaviour was an important concern as for the In the latter example, structural resonances showing significant peak skewness were Figure 2: Isolated response curve situated inside the nonlinear frequency response [31] characterise competing attractors, i.e. periodic, quasiperiodic and chaotic.

### A study of four problems in nonlinear vibrations via the method

by K Nandakumar 2009 Cited by 1 oscillator, but with an added fractional damping term. We use the behavior in the transition regions. view near junction of branches A, B and E (right box in Fig. 9.5). Now the right hand side of Eq. 2.7 contains resonant forcing. curve (a geometrical object), here we represent the limit cycle using a simpler geometrical.

### DOCTORAL THESIS 2019 NONLINEAR AND CHAOTIC

by J Barceló Aguiló That the thesis titles Nonlinear and chaotic behavior in CMOS- MEMS non-axially forced bistable cc-beam resonator operating, in addition, in the MHz reducing its resonance frequency, and adds constraints in terms of fabrication reliability. In 1918, Georg Duffing introduced an equation for a nonlinear oscillator with a

### Some elements for a history of the dynamical systems theory

by C Letellier 2021 term chaos was used by Li and Yorke in a very suggestive title Period- aperiodic behavior he was studying in the state space.3 6 With the inset and outset curves for the forced Duffing equation, two preprints on nonlinear oscillations by Floris Takens and things contains Smale horseshoes (see Fig.

### Using the transient trajectories of an optically - Nature

by J Flajšmanová 2020 Cited by 1 Despite long-term experimental investigation, new diverse effects have been A precise description of transient effects in nonlinear oscillators is therefore essential nonlinearity without any external driving force which could affect the system Ŵ and the resonant peak is not visibly influenced by the nonlinear behaviour

### NONLINEAR DYNAMICS OF HYSTERETIC OSCILLATORS A

by A SHEKHAWAT 2008 Cited by 1 resonance is derived. An oscillator with hysteretic restoring force and sinusoidal excitation. Response curves obtained from Poincaré maps. ǫ = 0.6, A = 0.6. In a nutshell we aim to study the transient and long-term behavior of an the long-term response of weakly nonlinear systems with damping contains only the.

### Forced Oscillators

terms, the lhs of the Duffing equation can be thought of as a damped nonlinear spring. rotating the observational box in file MF09 for the above run so that x versus In particular, how does the nonlinear resonance response curve compare? pected behavior of the oscillator as w is varied from small values to large.

### M 17e Pohl's wheel (linear and nonlinear oscillations)

Plot the logarithmic decrement as a function of the square of the current IW Record the resonance curve of the rotating pendulum at one of the given principles of nonlinear dynamics, transition into chaotic states, bifurcations In this experiment free, damped and forced oscillations of a rotating or chaotic behaviour).

### SYNCHRONIZATION AND RESONANCE EFFECTS IN

by MAG Vellisca 2017 4.3 Chaos in unidirectionally coupled neural oscillators 72 dynamics; nonlinear systems; Chua's circuit; Hindmarsh-Rose model; syn- include not only linear terms but also, at least, one term that contains the as an example of elementary forced resonance in physics textbooks [77]; it (curve) for coupling σ = 0.

### Nonlinear Waves in Granular Lattices - UCLA Math

by MP Byrne Cited by 4 Studies of nonlinear oscillator However, the KdV equation is not alone in having solitary wave solutions system can also show chaotic behaviour depending on the initial conditions. We term such a resonance as Fano-like because Fano resonance Solitary waves, however, are nonlinear waves - the return force.

### Performance and Robustness of Nonlinear Systems - ORBi

by T Detroux 2016 Cited by 4 1.3 Computation of Nonlinear Forced Response Curves 16 3 Relating Nonlinear Resonances and Nonlinear Normal Modes. 67 curves, quasiperiodic oscillations, and even chaos, are directly related to the presence of The term harmonic balance was first introduced by Krylov and Bogoliubov [102].

### The Dynamics of Coupled Oscillators - CORE

by NL Holland Cited by 4 It is seen that driven oscillators can be used as a model of 2 coupled oscillators in 2 and 3 incommensurate, at higher coupling chaotic behaviour is possible with a εω0 is the first order linear term of the damping of a nonlinear oscillator A resonance peak in the curve of S vs Ω shows the phase synchronisation.

### Forced Oscillations of a Self-Oscillating Bimolecular - JSTOR

by MA McKarnin 1988 Cited by 24 in the sides of some resonance horns, period doubling, Hopf bifurcations including frequency) or even chaotic, when the time series of the response appears to be The study of forced oscillations in nonlinear systems crosses disciplinary devices to AC voltages, of the heart to a pace-maker, the behaviour of RF plasma.

### Nonlinear vibration control with nanocapacitive sensor for

by Q Gong 2018 Cited by 12 Nonlinear vibration control, nanobeam, primary resonance, superharmonic chaos, bifurcation phenomena, and other relatively complex behaviors of vibration. Creative Commons CC-BY: This article is distributed under the terms of the The square of the excitation voltage to determine the electrostatically driving force

### Direct Normal Form Analysis of Oscillators with Different

by A Nasir 2021 (conservative) backbone curves (e.g. frequency-amplitude curves) of In the field of nonlinear dynamics, the DNF technique has been used to conduct a polynomial stiffness nonlinear terms, a system containing four types of forcing transformation involves the removal of any non-resonant forcing terms.

### The use of normal forms for analysing nonlinear mechanical

by SA Neild 2015 Cited by 43 normal forms, where forcing and damping terms can be treated as specific forms of unfoldings. resonant responses, leading to so-called backbone curves in structural analysis, )2 contains resonant terms where αj(iωj)2 + βj(−iωj)2 = 0 with αj = βj = 2. in general will include quasi-periodic and possibly chaotic motion.

### On the nonlinear Helmholtz response of almost-enclosed tidal

Netherlands Institute for Sea Research, PO Box 59, 1790 AB Texel, The modulation of its tidal amplitude, or 'bent resonance horns', implying that resonant as the forced and damped nonlinear Helmholtz oscillator will be addressed, both term on the right) may contain an additional multiplication factor, reflecting the.

### Reconstructing force from harmonic motion - DiVA portal

by D Platz 2013 Cited by 5 band are often not detectable and the total force acting on the oscillator remains The simplest model which exhibits oscillating behaviour is a point mass particle the accuracy of a measurement often the principle of resonant detection is used which is defined in equation (3.1) since the response y(t) contains terms like.

### General principles of chaotic dynamics

by PB Persson 1996 Cited by 55 should clarify the potential effects of resonances in the solar system: do chaotic ones, are the non-linear terms in the equation of motion. odic to periodic motion and finally to chaotic behaviour. With regard to the harmonically forced oscillator, defining since the attractor is a closed curve. sions is by box-counting.

### Backbone curves, Neimark-Sacker boundaries and

31 May 2021 The behaviour of these modes, when the detuning be- tween the chaos related to the stability of the period-three solu- tions [3, 4, 21, in nonlinear oscillators featuring 1:2 internal resonance. frequency of the forcing term with intensity F. Since analysis contains isolated branches of solutions, as al-.

### Non-linear modal analysis - Imperial College London

by HRE Siller 2004 Cited by 47 term explicit , yielding the nonlinear FRF at a selected DOF as a taken from a test rig, and it was found that the nonlinear behaviour Index associated with a resonant modal quantity, e.g. λs. 5.9 Force-displacement curves for different values of β Several approaches, such as the Square and Rectangular.

### Lecture Notes on Nonlinear Vibrations - Vibrationdata

by RH Rand 2003 Cited by 125 13 Melnikov's Method for Predicting Chaos The differential equation describing many nonlinear oscillators can be written in the to (3) may be pictured as a curve in the x-y phase plane passing through the This key step is called removal of resonance or secular terms. is absent from the steady state forced behavior.

### Technological applications of deterministic chaos - University

by AR Murch 1989 The use of a hierarchy of nonlinear recursive equations to generate coloured noise and the tailoring Chaotic behaviour in generalisations of an oscillator circuit the general mathematical terms and concepts required to provide the proper setting organisational principles having application to a wide range of systems.

### Renson, L., Gonzalez-Buelga, A., Barton, DAW, & Neild, SA

by L Renson Cited by 77 Keywords: nonlinear normal modes, backbone curve, phase quadrature, quasi-periodic oscillations, chaos, and bifurcations. A problem well the evolution of the resonance frequencies of the damped forced system. term (k = 0) to account for the constant term arising in the Fourier Check if the root-mean-square error.

### UvA-DARE (Digital Academic Repository)

by GM Terra Cited by 5 ondary tides and quasi-periodically forced nonlinear oscillators binnen de project- changes may occur in the resonance characteristics (amplification and phase lag) of on response curves and modulation of tidal amplitudes is discussed in is simply the footprint of chaotic behaviour in the meteorological forcing.

### Minimal Realizations of Autonomous Chaotic Oscillators

by J Petrzela 2019 Cited by 13 that represent the current standard in the field of nonlinear dynamics, i.e., INDEX TERMS Analog oscillator, filtering two-port, chaos, immittance first sight, chaos seems to be noise-like dynamical behavior ing about driven systems high-Q active frequency filters tion of the resonant tunneling diodes.

### Taming Spatiotemporal Chaos by Impurities in the

by NV ALEXEEVA 2001 Cited by 5 Solitons of the parametrically driven, damped nonlinear Schrödinger The ability to synchronize populations of coupled nonlinear oscillators to tame the chaotic behaviour and produce simple spatiotemporal patterns in very long is this variable amplitude that contains the information on how disorganized the array is.

### NONLINEAR RESONANCE - the FAU Digital Library!

nonlinear restoring forces, i.e., Duffing-type oscillators, resonant frequency changes its amplitude, plotted here as the nonlinear backbone (F = 0) curve Oscillations are a universal mode of behavior in simple and complex systems. term) to the restoring force resulted in up to a several hundred-fold amplitude increase.