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.

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.

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.

scholarly journals Dynamic mode decomposition analysis of rotating detonation waves

Shock Waves ◽  
2020 ◽  
Author(s):  
M. D. Bohon ◽  
A. Orchini ◽  
R. Bluemner ◽  
C. O. Paschereit ◽  
E. J. Gutmark
Keyword(s):  
Stable Limit Cycle ◽  
Operating Conditions ◽  
Dynamic Mode Decomposition ◽  
Mode Shapes ◽  
Dynamic Mode ◽  
Stable Limit ◽  
Rotating Waves ◽  
Reduced Order ◽  
Mode Decomposition ◽  
Rotating Detonation

Abstract A rotating detonation combustor (RDC) is a novel approach to achieving pressure gain combustion. Due to the steady propagation of the detonation wave around the perimeter of the annular combustion chamber, the RDC dynamic behavior is well suited to analysis with reduced-order techniques. For flow fields with such coherent aspects, the dynamic mode decomposition (DMD) has been shown to capture well the dominant oscillatory features corresponding to stable limit-cycle or quasi-periodic behavior within its dynamic modes. Details regarding the application of the technique to RDC—such as the number of frames, the effect of subtracting the temporal mean from the processed dataset, the resulting dynamic mode shapes, and the reconstruction of the dynamics from a reduced set of dynamic modes—are analyzed and interpreted in this study. The DMD analysis is applied to two commonly observed operating conditions of rotating detonation combustion, viz., (1) a single spinning wave with weak counter-rotating waves and (2) a clapping operating mode with two counter-propagating waves at equal speed and strength. We show that care must be taken when applying DMD to RDC datasets due to the presence of standing waves (expressed as either counter-propagating azimuthal waves or longitudinal pulsations). Without accounting for these effects, the reduced-order reconstruction fails using the standard DMD approach. However, successful application of the DMD allows for the reconstruction and separation of specific wave modes, from which models of the stabilization and propagation of the primary and counter-rotating waves can be derived.

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.

An infinite period bifurcation arising in roll waves down an open inclined channel

1986 ◽  
Vol 405 (1828) ◽  
pp. 103-116 ◽  
Keyword(s):  
Reynolds Number ◽  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Coming Out ◽  
Small Amplitude ◽  
Stable Limit Cycle ◽  
Finite Amplitude ◽  
Stable Limit ◽  
Continuous Solutions ◽  
Roll Waves

The discussion in a previous paper on roll waves is completed by showing how the limit cycles created at small amplitude by a Hopf bifurcation are destroyed. It is shown that there is an infinite period bifurcation creating stable limit cycles at finite amplitude. The conditions under which such a bifurcation coming out of a separatrix loop from a saddle point in the plane can occur are first derived (under the assumption that the Reynolds number is small). The complete evolution of the limit cycles is then deduced. In the subcritical case it is found that there is just one stable limit cycle, created at small amplitude by a Hopf bifurcation and destroyed at finite amplitude by an infinite period bifurcation. In the supercritical case it is shown that there are two limit cycles, one unstable (created by a Hopf bifurcation) and the other stable (created by the infinite period bifurcation) which finally merge and are then both destroyed. The discontinuous roll wave solutions derived by R. F. Dressler ( Communs pure appl. Math. 2, 49-194 (1949)) are compared with the continuous solutions for large values of the Reynolds number. It is shown that there is a difference in the jump condition between Dressler’s solutions and the present ones. It is then shown that this difference could be resolved by a slight modification to the dissipation term, which leaves the basic form of the continuous solutions unaltered. Finally it is then shown that both sets of waves are similar in that they both terminate with a solitary wave.

Limit cycles in a tritrophic food chain model with general functional responses

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Gamaliel Blé ◽  
Iván Loreto-Hernández
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Stable Limit Cycle ◽  
Growth Function ◽  
Logistic Growth ◽  
Chain Model ◽  
Stable Limit ◽  
Functional Responses ◽  
Tritrophic Model ◽  
Food Chain Model

Abstract The conditions to have a stable limit cycle by an Andronov–Hopf bifurcation in a tritrophic model are given. A generalized logistic growth function for the prey is considered, and a general family of functional responses, including the Holling type, are taken for the predators. Some results obtained in previous works for tritrophic models, which consider logistic growth in the prey and Holling functional responses, are generalized.

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