File Name: theory and applications of hopf bifurcation .zip
Metrics details. We consider a time delay predator-prey model with Holling type-IV functional response and stage-structured for the prey. Our aim is to observe the dynamics of this model under the influence of gestation delay of the predator.
In the mathematical theory of bifurcations , a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point.
Curator: John Guckenheimer. Eugene M. Yuri A. The Hopf-Hopf bifurcation is a bifurcation of an equilibrium point in a two-parameter family of autonomous ODEs at which the critical equilibrium has two pairs of purely imaginary eigenvalues.
This phenomenon is also called the double-Hopf bifurcation. The bifurcation point in the parameter plane lies at a transversal intersection of two curves of Andronov-Hopf bifurcations. Generically, two branches of torus bifurcations emanate from the Hopf-Hopf HH point. Depending on the system, other bifurcations occur for nearby parameter values, including bifurcations of Shilnikov's homoclinic orbits to the focus-focus equilibrium, and bifurcations of a heteroclinic structure connecting saddle limit cycles and equilibria.
This bifurcation, therefore, can imply a local birth of "chaos". Also quasi-periodicity is involved Braaksma and Broer, In such a family 1 :. Each torus bifurcation of these limit cycles generates an invariant two-dimensional torus with periodic or quasiperiodic orbits. The 2D invariant torus can be accompanied by an invariant set resembling a 3D torus, which can disappear via either a "heteroclinic destruction" or a "blow-up".
Here two cases should be distinguished:. Limit cycles of the amplitude system correspond to invariant 3D tori. In the "difficult case", heteroclinic bifurcation in the amplitude system 3 suggests the breakdown of an invariant 3D torus and the appearance of chaotic invariant sets in the full 4D-system 2.
Nearby, various homo- and heteroclinic orbits connecting the equilibrium and saddle limit cycles exist Guckenheimer and Holmes, ; Broer, ; Broer and Vegter, Also compare with Broer, To analyze bifurcations of 2D tori, one has to normalize the fourth- and fifth-order terms.
The resulting normal form is not unique. If the following nondegeneracy conditions hold:. Local bifurcation diagrams of this system satisfying some extra genericity conditions can be found in Kuznetsov , Sec. In 5 , the positive equilibrium exhibits the Andronov-Hopf bifurcation generating a limit cycle.
This limit cycle corresponds to a 3D invariant torus in the truncated normal form 4. John Guckenheimer and Yuri A. Kuznetsov , Scholarpedia, 3 8 Jump to: navigation , search. Post-publication activity Curator: John Guckenheimer Contributors:. Elias August. Sponsored by: Eugene M. Namespaces Page Discussion. Views Read View source View history. Contents 1 Definition 2 Four-dimensional case 3 Multidimensional case 4 Cubic normal form coefficients 5 Gavrilov normal form 6 Other cases 7 References 8 External links 9 See also.
Izhikevich , Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia. Reviewed by : Anonymous. Accepted on: GMT.
Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems Abstract: One of the most powerful methods for studying periodic solutions In autonomous nonlinear systems is the theory which has developed from a proof by Hopf. He showed that oscillations near an equilibrium point can be understood by looking at the eigenvalues of the linearized equations for perturbations from equilibrium, and at certain crucial derivatives of the equations. A good deal of work has been done recently on this theory and the present paper summarizes recent results, presents some new ones, and shows how they can be used to study almost sinusoidal oscillations in nonlinear circuits and systems. The new results are a proof of the basic part of the Hopf theorem using only elementary methods, and a graphical interpretation of the theorem for nonlinear multiple-loop feedback systems.
x. PREFACE. The other main new result here is our proof of the validity of the Hopf bifurcation theory for nonlinear partial differential equations of parabolic type.
The Hopf Bifurcation and Its Applications
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Curator: John Guckenheimer. Eugene M. Yuri A. The Hopf-Hopf bifurcation is a bifurcation of an equilibrium point in a two-parameter family of autonomous ODEs at which the critical equilibrium has two pairs of purely imaginary eigenvalues. This phenomenon is also called the double-Hopf bifurcation.
Забудьте об. Поехали. Свет от фары пробежал по цементным стенам. - В главный банк данных попал вирус, - сказал Бринкерхофф. - Я знаю, - услышала Сьюзан собственный едва слышный голос.
Однако тот не подавал никаких признаков жизни. Сьюзан перевела взгляд на помост перед кабинетом Стратмора и ведущую к нему лестницу. - Коммандер. Молчание. Тогда она осторожно двинулась в направлении Третьего узла.
Смерть ее веры в. Любовь и честь были забыты. Мечта, которой он жил все эти годы, умерла.