scholarly journals Splitting of the separatrices after a Hamiltonian–Hopf bifurcation under periodic forcing

Nonlinearity ◽  
2019 ◽  
Vol 32 (4) ◽  
pp. 1440-1493 ◽  
Author(s):  
E Fontich ◽  
C Simó ◽  
A Vieiro
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Forcing ◽  
Hamiltonian Hopf Bifurcation

scholarly journals Hamiltonian Hopf Bifurcation with Symmetry

2002 ◽  
Vol 163 (1) ◽  
pp. 1-33 ◽  
Author(s):  
Pascal Chossat ◽  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian Hopf Bifurcation

The paradox of enrichment, spatial heterogeneity, community effects and the phenomenon of apparent disappearance in the marine bacteriophage dynamics

2021 ◽  
Vol 62 ◽  
pp. 453-468
Author(s):  
Andrei Korobeinikov ◽  
Elena Shchepakina ◽  
Vladimir Sobolev Sobolev
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
External Factors ◽  
Loss Of Stability ◽  
Simple Mathematical Model ◽  
Community Effects ◽  
Periodic Forcing ◽  
Internal Factors ◽  
High Magnitude ◽  
Microbial Systems

In aquatic microbial systems, high magnitude variations in abundance, such as sudden blooms alternating with comparatively long periods of very low abundance ("apparent disappearance'') are relatively common. The authors suggest that, in order to occur, such variations in abundance in microbial systems and, in particular, the apparent disappearance of species do not require seasonal or periodic forcing of any kind, or external factors of any other nature, and can be caused by internal factors, and in particular by bacteria-phage interaction. Specifically, the authors suggest that the variations in abundance and the apparent disappearance phenomenon can be a result of phage infection and the lysis of infected bacteria. To illustrate this idea, the authors consider a reasonably simple mathematical model of bacteria-phage interaction based on the model suggested by Edoardo Beretta and Yang Kuang, which assumes neither periodic forcing, nor action of other external factors. The model admits a loss of stability via Andronov-Hopf bifurcation and exhibits dynamics which is able to explains the phenomenon. These properties of the model are especially distinctive for spatially non-homogeneous biosystems as well as biosystem with some sorts of cooperation or community effects. doi:10.1017/S1446181120000085

Correction to ``Hamiltonian Hopf Bifurcation with Symmetry''

2003 ◽  
Vol 167 (1) ◽  
pp. 83-84
Author(s):  
Pascal Chossat ◽  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
Hopf Bifurcation ◽  
Hamiltonian Hopf Bifurcation

scholarly journals THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION

2011 ◽  
Vol 21 (08) ◽  
pp. 2321-2330 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
G. CONTOPOULOS
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Transition Point ◽  
Initial Conditions ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Surface Of Section ◽  
Autonomous Hamiltonian System ◽  
Hamiltonian Hopf Bifurcation ◽  
Stable Periodic

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

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