scholarly journals An analysis of two degenerate double-Hopf bifurcations

2022 ◽  
Vol 30 (1) ◽  
pp. 382-403
Author(s):  
Gheorghe Moza ◽  
◽  
Mihaela Sterpu ◽  
Carmen Rocşoreanu ◽  
◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Hopf Bifurcations ◽  
Bifurcation Diagrams ◽  
Double Hopf Bifurcation

<abstract><p>The generic double-Hopf bifurcation is presented in detail in literature in textbooks like references. In this paper we complete the study of the double-Hopf bifurcation with two degenerate (or nongeneric) cases. In each case one of the generic conditions is not satisfied. The normal form and the corresponding bifurcation diagrams in each case are obtained. New possibilities of behavior which do not appear in the generic case were found.</p></abstract>

Delay-Induced Double Hopf Bifurcations in a System of Two Delay-Coupled van der Pol–Duffing Oscillators

2015 ◽  
Vol 25 (04) ◽  
pp. 1550058 ◽  
Author(s):  
Heping Jiang ◽  
Tonghua Zhang ◽  
Yongli Song
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Hopf Bifurcations ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Double Hopf Bifurcation ◽  
Codimension Two ◽  
Form Theory ◽  
Weak Resonance ◽  
Delay Differential

In this paper, we investigate the codimension-two double Hopf bifurcation in delay-coupled van der Pol–Duffing oscillators. By using normal form theory of delay differential equations, the normal form associated with the codimension-two double Hopf bifurcation is calculated. Choosing appropriate values of the coupling strength and the delay can result in nonresonance and weak resonance double Hopf bifurcations. The dynamical classification near these bifurcation points can be explicitly determined by the corresponding normal form. Periodic, quasi-periodic solutions and torus are found near the bifurcation point. The numerical simulations are employed to support the theoretical results.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

DYNAMICS AND DOUBLE HOPF BIFURCATIONS OF THE ROSE–HINDMARSH MODEL WITH TIME DELAY

2009 ◽  
Vol 19 (11) ◽  
pp. 3733-3751 ◽  
Author(s):  
SUQI MA ◽  
ZHAOSHENG FENG ◽  
QISHAI LU
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Value Of Time ◽  
Asymptotically Stable ◽  
Universal Unfolding ◽  
Double Hopf Bifurcation ◽  
Bifurcation Points ◽  
The Rose

In this paper, we are concerned with the Rose–Hindmarsh model with time delay. By applying the generalized Sturm criterion, a number of imaginary roots of the characteristic equation are classified. The absolutely stable regions for any value of time delay are detected. By the continuous software DDE-Biftool, both the Hopf bifurcation curves and double Hopf bifurcation points are illustrated in parametric spaces. The normal form and universal unfolding at double Hopf bifurcation points are considered by the center manifold method. Some examples also indicate that the corresponding unique attractor near each double Hopf point is asymptotically stable.

scholarly journals Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Finite Delay ◽  
The Stability ◽  
Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

Analysis of Zero-Hopf Bifurcation in Two Rössler Systems Using Normal Form Theory

2020 ◽  
Vol 30 (16) ◽  
pp. 2030050
Author(s):  
Bing Zeng ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Orbits ◽  
Critical Points ◽  
Normal Forms ◽  
Averaging Method ◽  
Hopf Bifurcations ◽  
Time Averaging ◽  
Normal Form Theory ◽  
Form Theory

In recent publications [Llibre, 2014; Llibre & Makhlouf, 2020], time-averaging method was applied to studying periodic orbits bifurcating from zero-Hopf critical points of two Rössler systems. It was shown that the averaging method is successful for a certain type of zero-Hopf critical points, but fails for some type of such critical points. In this paper, we apply normal form theory to reinvestigate the bifurcation and show that the method of normal forms is applicable for all types of zero-Hopf bifurcations, revealing why the time-averaging method fails for some type of zero-Hopf bifurcation.

Local Stability and Hopf Bifurcation Analysis of the Arneodo’s System

2011 ◽  
Vol 130-134 ◽  
pp. 2550-2557
Author(s):  
Yi Jing Liu ◽  
Zhi Shu Li ◽  
Xiao Mei Cai ◽  
Ya Lan Ye
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Theoretical Analysis ◽  
Local Stability ◽  
Hopf Bifurcations ◽  
Explicit Formulas ◽  
Normal Form Theory ◽  
Numerical Examples ◽  
Form Theory ◽  
The Stability

The chaotic behaviors of the Arneodo’s system are investigated in this paper. Based on the Arneodo's system characteristic equation, the equilibria of the system and the conditions of Hopf bifurcations are obtained, which shows that Hopf bifurcations occur in this system. Then using the normal form theory, we give the explicit formulas which determine the stability of bifurcating periodic solutions and the direction of the Hopf bifurcation. Finally, some numerical examples are employed to demonstrate the effectiveness of the theoretical analysis.

scholarly journals FROM FIRST LYAPUNOV COEFFICIENTS TO MAXIMAL CANARDS

2010 ◽  
Vol 20 (05) ◽  
pp. 1467-1475 ◽  
Author(s):  
CHRISTIAN KUEHN
Keyword(s):  
Asymptotic Expansion ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Differential Equations ◽  
Rapid Growth ◽  
Slow Variable ◽  
Hopf Bifurcations ◽  
First Order ◽  
Numerical Software ◽  
Lyapunov Coefficient

Hopf bifurcations in fast–slow systems of ordinary differential equations can be associated with a surprisingly rapid growth of periodic orbits. This process is referred to as canard explosion. The key step in locating a canard explosion is to calculate the location of a special trajectory, called a maximal canard, in parameter space. A first-order asymptotic expansion of this location was found by Krupa and Szmolyan [2001a, 2001b, 2001c] in the framework of a "canard point"-normal-form for systems with one fast and one slow variable. We show how to compute the coefficients in this expansion using the first Lyapunov coefficient at the Hopf bifurcation thereby avoiding the use of this normal form. Our results connect the theory of canard explosions with existing numerical software, enabling easier calculations of where canard explosions occur.

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