scholarly journals Control of Bistability in a Delayed Duffing Oscillator

2012 ◽  
Vol 2012 ◽  
pp. 1-5 ◽  
Author(s):  
Mustapha Hamdi ◽  
Mohamed Belhaq
Keyword(s):  
Hopf Bifurcation ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Stable Limit Cycle ◽  
Slow Flow ◽  
Stable Limit ◽  
Nontrivial Solutions ◽  
Slow Dynamic ◽  
High Frequency Excitation ◽  
Frequency Excitation

The effect of a high-frequency excitation on nontrivial solutions and bistability in a delayed Duffing oscillator with a delayed displacement feedback is investigated in this paper. We use the technique of direct partition of motion and the multiple scales method to obtain the slow dynamic of the system and its slow flow. The analysis of the slow flow provides approximations of the Hopf and secondary Hopf bifurcation curves. As a result, this study shows that increasing the delay gain, the system undergoes a secondary Hopf bifurcation. Further, it is indicated that as the frequency of the excitation is increased, the Hopf and secondary Hopf bifurcation curves overlap giving birth in the parameter space to small regions of bistability where a stable trivial steady state and a stable limit cycle coexist. Numerical simulations are carried out to validate the analytical finding.

scholarly journals Levitation Stability and Hopf Bifurcation of EMS Maglev Trains

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Junxiong Hu ◽  
Weihua Ma ◽  
Xiaohao Chen ◽  
Shihui Luo
Keyword(s):  
Hopf Bifurcation ◽  
Stable Limit Cycle ◽  
Coupled System ◽  
Stable Limit ◽  
Maglev Train ◽  
Real Time System ◽  
Levitation System ◽  
The Stability ◽  
Control Feedback ◽  
Track Beam

This paper analyzed the mechanical characteristics of single electromagnet system and elastic track beam of EMS maglev train and established a five-dimensional dynamics model of single electromagnet-track beam coupled system with classical PD control strategy adopted for its levitation system. Then, based on the Hurwitz criterion and the high-dimensional Hopf bifurcation theory, the stability of the coupled system is analyzed; the existence of the Hopf bifurcation is discussed and the bifurcation direction and the stability of the periodic solution are determined with levitation control feedback coefficient kp as the bifurcation parameter; and numerical simulation is conducted to verify the validity of the theoretical analysis results. The results show that the Hurwitz algebra criterion can directly determine the eigenvalues and symbols of the dynamics system to facilitate the analysis on the stability of the system and the Hopf bifurcation without the necessity of calculating the specific eigenvalues; supercritical Hopf bifurcation will occur under the given parameters, that is, when kp<kp0, the real-time system is asymptotically stable, yet Hopf bifurcation occurs as kp increases gradually beyond kp0, with the stability of the system lost and a stable limit cycle branched.

Dynamics of a Rocking Horizontal Pendulum Under High Frequency Excitation

2012 ◽  
Author(s):  
Si Mohamed Sah ◽  
Clark McGehee ◽  
Brian P. Mann
Keyword(s):  
Dynamic Equation ◽  
High Frequency ◽  
Slow Dynamic ◽  
Numerical Investigations ◽  
Stable Equilibria ◽  
High Frequency Excitation ◽  
Frequency Excitation ◽  
Parameter Values ◽  
Horizontal Pendulum ◽  
Good Agreement

In this work, we investigate the effect of high frequency excitation on a pendulum positioned at a given distance from a tilted platform. The tilted platform is subjected to high frequency excitation. The method of direct partition of motion is applied to the governing equation to obtain the slow dynamic equation. It is shown that two stable equilibria may coexist for certain parameter values. The analytical results are in a good agreement with the numerical investigations.

scholarly journals Quasi-periodic intermittency in oscillating cylinder flow

2017 ◽  
Vol 828 ◽  
pp. 680-707 ◽  
Author(s):  
Bryan Glaz ◽  
Igor Mezić ◽  
Maria Fonoberova ◽  
Sophie Loire
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Stable Limit Cycle ◽  
Forced Flow ◽  
Stable Limit ◽  
Decomposition Approach ◽  
New Normal ◽  
Koopman Operator ◽  
Mode Decomposition ◽  
Approach Zero

Fluid dynamics induced by periodically forced flow around a cylinder is analysed computationally for the case when the forcing frequency is much lower than the von Kármán vortex shedding frequency corresponding to the constant flow velocity condition. By using the Koopman mode decomposition approach, we find a new normal form equation that extends the classical Hopf bifurcation normal form by a time-dependent term for Reynolds numbers close to the Hopf bifurcation value. The normal form describes the dynamics of an observable and features a forcing (control) term that multiplies the state, and is thus a parametric – i.e. not an additive – forcing effect. We find that the dynamics of the flow in this regime is characterized by alternating instances of quiescent and strong oscillatory behaviour and that this pattern persists indefinitely. Furthermore, the spectrum of the associated Koopman operator is shown to possess quasi-periodic features. We establish the theoretical underpinnings of this phenomenon – that we name quasi-periodic intermittency – using the new normal form model and show that the dynamics is caused by the tendency of the flow to oscillate between the unstable fixed point and the stable limit cycle of the unforced flow. The quasi-periodic intermittency phenomenon is also characterized by positive finite-time Lyapunov exponents that, over a long period of time, asymptotically approach zero.

Analysis of Hopf Bifurcation of Steering Wheel Shimmy of Automobile

2013 ◽  
Vol 344 ◽  
pp. 61-65
Author(s):  
Li Juan He ◽  
Yu Cun Zhou
Keyword(s):  
Nonlinear Dynamics ◽  
Hopf Bifurcation ◽  
Numerical Analysis ◽  
Limit Cycle ◽  
Critical Speed ◽  
Stable Limit Cycle ◽  
Steering Wheel ◽  
Stable Limit ◽  
Supercritical Hopf Bifurcation ◽  
Wheel Shimmy

It proves that steering wheel shimmy is a vibration of stable limit cycle occurring after Hopf bifurcation, which is elaborated by nonlinear dynamics theory, and the control objectives of shimmy are proposed according to its bifurcation properties. Numerical analysis of bifurcation characteristics has been conducted with a nonlinear shimmy model whose parameters come from a domestic automobile with independent suspension. The results indicate that when the speed reaches 49.98Km/h, supercritical Hopf bifurcation occurs to the system and stable limit cycle appears, i.e. wheels oscillate around the kingpin at the same amplitude; when the speed comes to 76.30 Km/h, Hopf bifurcation occurs again and limit cycle disappears. The bifurcation speed and amplitude of limit cycle match the shimmy speed and amplitude measured from road experiments very well, which confirms the conclusions that shimmy is a vibration of stable limit cycle occurring after Hopf bifurcation at critical speed.

scholarly journals Dynamic Monopolistic Competition

2021 ◽  
Author(s):  
Simon Hoof
Keyword(s):  
Steady State ◽  
Nash Equilibrium ◽  
Hopf Bifurcation ◽  
Product Differentiation ◽  
Monopolistic Competition ◽  
Necessary Conditions ◽  
Stable Limit Cycle ◽  
Open Loop ◽  
Stable Limit ◽  
Bifurcation Theorem

AbstractI study a dynamic variant of the Dixit–Stiglitz (Am Econ Rev 67(3), 1977) model of monopolistic competition by introducing price stickiness à la Fershtman and Kamien (Econometrica 55(5), 1987). The analysis is restricted to bounded quantity and price paths that fulfill the necessary conditions for an open-loop Nash equilibrium. I show that there exists a symmetric steady state and that its stability depends on the degree of product differentiation. When moving from complements to perfect substitutes, the steady state is either a locally asymptotically unstable (spiral) source, a stable (spiral) sink or a saddle point. I further apply the Hopf bifurcation theorem and prove the existence of limit cycles, when passing from a stable to an unstable steady state. Lastly, I provide a numerical example and show that there exists a stable limit cycle.

scholarly journals Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Marluci Cristina Galindo ◽  
Cristiane Nespoli ◽  
Marcelo Messias
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Chaotic Attractor ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Tumour Cells ◽  
Dimensional System ◽  
Stable Limit ◽  
Cancer Model

We study a cancer model given by a three-dimensional system of ordinary differential equations, depending on eight parameters, which describe the interaction among healthy cells, tumour cells, and effector cells of immune system. The model was previously studied in the literature and was shown to have a chaotic attractor. In this paper we study how such a chaotic attractor is formed. More precisely, by varying one of the parameters, we prove that a supercritical Hopf bifurcation occurs, leading to the creation of a stable limit cycle. Then studying the continuation of this limit cycle we numerically found a cascade of period-doubling bifurcations which leads to the formation of the mentioned chaotic attractor. Moreover, analyzing the model dynamics from a biological point of view, we notice the possibility of both the tumour cells and the immune system cells to vanish and only the healthy cells survive, suggesting the possibility of cure, since the interactions with the immune system can eliminate tumour cells.

Applicability of Harmonic Balance for the Determination of Blade Stability Within the Prescribed Motion Approach

2020 ◽  
Author(s):  
Virginie Anne Chenaux ◽  
Matthias Schuff ◽  
David Quero
Keyword(s):  
Time Domain ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Stable Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Stable Limit ◽  
Nonlinear Frequency ◽  
Compressor Rotor ◽  
New Development ◽  
Cycle Oscillation

Abstract To predict blade aerodynamic damping during the design phase, unsteady linearized CFD methods are commonly used as they offer a reasonable accuracy at acceptable computational costs. However, for moderate blade oscillation amplitudes, nonlinear aerodynamic effects may appear, yielding eventually an evolution into a stable, limit cycle oscillation (LCO). In the perspective of raising performance and safety, identifying such scenarios might open new development possibilities. Therefore, a valuable alternative to expensive CFD time domain methods consists in applying the nonlinear frequency domain harmonic balance (HB) approach to determine the aerodynamic response. An appropriate number of higher harmonics have to be retained depending on the severity of the aerodynamic nonlinearity under consideration. This number can be identified using either a convergence study with an increasing number of harmonics, or a direct comparison with time-domain simulations. For weak to moderate aerodynamic nonlinearities, this work proposes a guideline to determine the number of harmonics without additional, comparative simulations. First, the HB convergence properties are derived using the well-known Duffing oscillator. Next, the method is applied to a compressor rotor blade subjected to a prescribed harmonic motion for conditions with and without aerodynamic nonlinearities.

scholarly journals The Effects of Harvesting on the Dynamics of a Leslie–Gower Model

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Jingli Xie ◽  
Hanyan Liu ◽  
Danfeng Luo
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Stable Limit Cycle ◽  
Local Bifurcation ◽  
Stable Limit ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Bifurcation Analysis

In this paper, we study a Leslie–Gower predator-prey model with harvesting effects. We carry out local bifurcation analysis and stability analysis. Under certain conditions, the model is shown to undergo a supercritical Hopf bifurcation resulting in a stable limit cycle. Numerical simulations are presented to illustrate our theoretic results.

scholarly journals SLOW HIGH-FREQUENCY EFFECTS IN MECHANICS: PROBLEMS, SOLUTIONS, POTENTIALS

2005 ◽  
Vol 15 (09) ◽  
pp. 2799-2818 ◽  
Author(s):  
JON JUEL THOMSEN
Keyword(s):  
High Frequency ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mechanical Systems ◽  
Frequency Effects ◽  
Direct Separation ◽  
High Frequency Excitation ◽  
Frequency Excitation ◽  
High Frequency Effects

Strong high-frequency excitation (HFE) may change the "slow" (i.e. effective or average) properties of mechanical systems, e.g. their stiffness, natural frequencies, equilibriums, equilibrium stability, and bifurcation paths. This tutorial describes three general HFE effects: Stiffening — an apparent change in the stiffness associated with an equilibrium; Biasing — a tendency for a system to move towards a particular state which does not exist or is unstable without HFE; and Smoothening — a tendency for discontinuities to be apparently smeared out by HFE. The effects and a method for analyzing them are introduced, first in terms of simple physical examples, and then in generalized form for mathematical models covering broad classes of discrete and continuous mechanical systems. Several application examples are summarized. Three mathematical tools for analyzing HFE effects are described and compared: The Method of Direct Separation of Motions, the Method of Averaging, and the Method of Multiple Scales. The tutorial concludes with a suggestion that more vibration experts, researchers and students should be aware of HFE effects, for the benefit of general vibration troubleshooting, and also for furthering the creation of innovative technical devices and processes utilizing HFE effects.

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