ANALYTICAL CRITERION FOR HOMOCLINIC BIFURCATIONS FOLLOWING A SUPERCRITICAL HOPF BIFURCATION IN TWO SYSTEMS

2016 ◽  
Vol 28 (1) ◽  
pp. 33-61
Author(s):  
Tanushree Roy ◽  
S. Roy Choudhury ◽  
Ugur Tanriver
Keyword(s):  
Hopf Bifurcation ◽  
Supercritical Hopf Bifurcation ◽  
Homoclinic Bifurcations ◽  
Two Systems ◽  
Analytical Criterion

Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake

2016 ◽  
Vol 48 (6) ◽  
pp. 061401 ◽  
Author(s):  
Francois Gallaire ◽  
Edouard Boujo ◽  
Vladislav Mantic-Lugo ◽  
Cristobal Arratia ◽  
Benjamin Thiria ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Cylinder Wake ◽  
Amplitude Equations ◽  
Supercritical Hopf Bifurcation

THE COHERENCE RESONANCE IN VAN DER POL SYSTEM INDUCED BY NOISE RECYCLING

2012 ◽  
Vol 11 (02) ◽  
pp. 1250002 ◽  
Author(s):  
X. Y. LI ◽  
J. H. YANG ◽  
X. B. LIU
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Natural Frequency ◽  
Coherence Resonance ◽  
Van Der Pol ◽  
Input Noise ◽  
Supercritical Hopf Bifurcation ◽  
Induced Motion

The phenomenon of coherence resonance (CR) in a delayed noisy Van der Pol system with supercritical Hopf bifurcation, which is influenced by a recycled noise, is numerically studied. Different from the traditional CR theory, in this paper, the characteristics of CR is affected by the time delay in the input noise. Namely, the CR is weakened or enhanced by the time delay feedback. Moreover, we find that several characteristics of this particular system vary periodically and its period has some certain relation with the natural frequency. By using the results given by the paper, we can control the noise-induced motion by modulating the time delay in noise.

New results in rotating Hagen–Poiseuille flow

2000 ◽  
Vol 417 ◽  
pp. 103-126 ◽  
Author(s):  
D. R. BARNES ◽  
R. R. KERSWELL
Keyword(s):  
Hopf Bifurcation ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Axial Rotation ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Rotation Rates ◽  
Supercritical Hopf Bifurcation

New three-dimensional finite-amplitude travelling-wave solutions are found in rotating Hagen–Poiseuille flow (RHPF[Ωa, Ωp]) where fluid is driven by a constant pressure gradient along a pipe rotating axially at rate Ωa and at Ωp about a perpendicular diameter. For purely axial rotation (RHPF[Ωa, 0]), the two-dimensional helical waves found by Toplosky & Akylas (1988) are found to become unstable to three-dimensional travelling waves in a supercritical Hopf bifurcation. The addition of a perpendicular rotation at low axial rotation rates is found only to stabilize the system. In the absence of axial rotation, the two-dimensional steady flow solution in RHPF[0, Ωp] which connects smoothly to Hagen–Poiseuille flow as Ωp → 0 is found to be stable at all Reynolds numbers below 104. At high axial rotation rates, the superposition of a perpendicular rotation produces a ‘precessional’ instability which here is found to be a supercritical Hopf bifurcation leading directly to three-dimensional travelling waves. Owing to the supercritical nature of this primary bifurcation and the secondary bifurcation found in RHPF[Ωa, 0], no opportunity therefore exists to continue these three-dimensional finite-amplitude solutions in RHPF back to Hagen–Poiseuille flow. This then contrasts with the situation in narrow-gap Taylor–Couette flow where just such a connection exists to plane Couette flow.

Complex Dynamics in the Unified Lorenz-Type System

2014 ◽  
Vol 24 (04) ◽  
pp. 1450055 ◽  
Author(s):  
Qigui Yang ◽  
Yuming Chen
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Dynamics ◽  
Type System ◽  
Global Dynamics ◽  
Two Systems ◽  
The Lorenz System ◽  
The Stability

This paper is devoted to the analysis of complex dynamics of the unified Lorenz-type system (ULTS) with six parameters, which contain common chaotic systems as its particular cases. First, some important local dynamics such as pitchfork bifurcation, Hopf bifurcation, and the stability of nondegenerate and double-zero equilibria are systematically investigated using the parameter-dependent center manifold theory combined with some bifurcation theories. Some adequate conditions for guaranteeing the occurrence of degenerate Hopf bifurcation (DHB) and the stability of the equilibria are given. Second, it is found that if DHB does not generate at the trivial equilibrium but generates at two symmetric nontrivial equilibria, then a small perturbation can lead that ULTS to exhibit a chaotic attractor. Interestingly, such a case can take place in the Chen and Lü systems (two common chaotic systems) but cannot take place in the Lorenz and Yang systems (the other two common chaotic systems), essentially distinguishing the Lorenz system from the Chen system. In addition, it is numerically verified that both of the latter two systems can exhibit the coexistence of both a chaotic attractor and multiple limit cycles but the former two systems seem not to have this property. If DHB takes place simultaneously at three equilibria of ULTS, then this system has an invariant algebraic surface, and rigorously prove the existence of some global dynamics such as periodic orbit, center, homoclinic/heteroclinic orbits. Third, it is shown that a singularly degenerate heteroclinic cycle can exist in the case of b = 0 (where b is a parameter of ULTS, like that in the Lorenz system), and a chaotic attractor can be generated by perturbing this cycle for small b > 0. These results altogether indicate that the ULTS can exhibit complex dynamics, and provide a more reasonable classification for chaos in the 3D autonomous chaotic ODE systems that were developed based on the Lorenz system, in contrast to the previous studies.

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.

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