Codimension two bifurcations and hopf bifurcations of an impacting vibrating system

1996 ◽  
Vol 17 (1) ◽  
pp. 65-75 ◽  
Author(s):  
Xie Jianhua
Keyword(s):  
Hopf Bifurcations ◽  
Codimension Two ◽  
Vibrating System

Forced oscillations of a self-oscillating bimolecular surface reaction model

1988 ◽  
Vol 417 (1853) ◽  
pp. 363-388 ◽  
Keyword(s):  
Surface Reaction ◽  
Period Doubling ◽  
Reaction Model ◽  
Forced Oscillations ◽  
Hopf Bifurcations ◽  
Codimension Two ◽  
Homoclinic Tangles ◽  
Two Parameter ◽  
Parameter Bifurcation ◽  
Surface Reaction Model

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-periodic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations.

ON THE BEHAVIOR OF A SELF-EXCITING FARADAY DISK HOMOPOLAR DYNAMO WITH A VARIABLE NONLINEAR SERIES MOTOR

2002 ◽  
Vol 12 (10) ◽  
pp. 2123-2135 ◽  
Author(s):  
IRENE M. MOROZ
Keyword(s):  
General Problem ◽  
Eddy Currents ◽  
Steady States ◽  
Equilibrium States ◽  
Hopf Bifurcations ◽  
Multiple Steady States ◽  
Double Zero ◽  
Oscillatory Solutions ◽  
Codimension Two ◽  
Special Case

In this paper we seek to bridge the gap between the study of a self-exciting Faraday disk homopolar dynamo with a linear series motor [Hide et al., 1996] and the case when the torque acting on the armature of the motor is proportional to the square of the current flowing through the dynamo [Hide, 1998]. We also focus on the issue of when the nonlinear quenching of oscillatory solutions can occur. The present study is a special case of the more general problem when azimuthal eddy currents are permitted to flow [Moroz & Hide, 2000] and shares with that problem the existence of multiple steady states and Hopf bifurcations. This results in distinct double-zero bifurcations for the trivial and the nontrivial equilibrium states as well as other codimension-two bifurcations, which leads to the suppression of oscillatory solutions.

HOPF BIFURCATIONS AND THEIR DEGENERACIES IN CHUA'S EQUATION

2011 ◽  
Vol 21 (09) ◽  
pp. 2749-2763 ◽  
Author(s):  
ANTONIO ALGABA ◽  
MANUEL MERINO ◽  
FERNANDO FERNÁNDEZ-SÁNCHEZ ◽  
ALEJANDRO J. RODRÍGUEZ-LUIS
Keyword(s):  
Analytical Study ◽  
Numerical Results ◽  
Hopf Bifurcations ◽  
Codimension Two ◽  
Strong Agreement

We perform an analytical study of the Hopf bifurcations and their degeneracies in Chua's equation. In the case of the equilibrium at the origin only codimension-two Hopf bifurcations appear. However, for the nontrivial equilibria we prove the existence of codimension-three Hopf bifurcations. Numerical results are in strong agreement with the analytical ones.

STRONG AND WEAK RESONANCES IN DELAYED DIFFERENTIAL SYSTEMS

2013 ◽  
Vol 23 (07) ◽  
pp. 1350119 ◽  
Author(s):  
WANYONG WANG ◽  
JIAN XU ◽  
XIUTING SUN
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Amplitude Equation ◽  
Complex Amplitude ◽  
Hopf Bifurcations ◽  
Differential Systems ◽  
Codimension Two ◽  
Dynamical Behaviors ◽  
Systematic Scheme ◽  
Analytical Expressions

In this paper, a general and systematic scheme is provided to research strong and weak resonances derived from delay-induced various double Hopf bifurcations in delayed differential systems. The method of multiple scales is extended to obtain a common complex amplitude equation when the double Hopf bifurcation with frequency ratio k1:k2 occurs in the systems under consideration. By analyzing the complex amplitude equation, we give the conditions of the strong and weak resonances respectively in the analytical expressions. The weak resonances correspond to the codimension-two double Hopf bifurcations since the amplitudes and the phases may be decoupled, but the strong resonances to the codimension-three double Hopf bifurcations in the system. It is seen that the weak resonances happen in the system even for a lower-order ratio, i.e. k1 + k2 ≤ 4. As applications, two examples are displayed. Three cases of the delay-induced resonance with 1:2, 1:3, 1:5 and [Formula: see text] are discussed in detail and the corresponding normals are represented. Thus, the relative dynamical behaviors can be easily classified in the physical parameter space in terms of nonlinear dynamics. The results show the provided conditions may be used to determine that a resonance is strong or weak.

scholarly journals Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Daifeng Duan ◽  
Ben Niu ◽  
Junjie Wei
Keyword(s):  
Generation Time ◽  
Three Dimensional ◽  
Spatiotemporal Patterns ◽  
Hopf Bifurcations ◽  
Dimensional Torus ◽  
Periodic Oscillations ◽  
One Dimensional ◽  
Codimension Two ◽  
Spatially Inhomogeneous ◽  
Delayed System

<p style='text-indent:20px;'>We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.</p>

Autonomous bifurcations of a simple bimolecular surface-reaction model

1988 ◽  
Vol 415 (1849) ◽  
pp. 363-387 ◽  
Keyword(s):  
Simple Model ◽  
Surface Reaction ◽  
Analytical Techniques ◽  
Reaction Model ◽  
Hopf Bifurcations ◽  
Codimension Two ◽  
Sustained Oscillations ◽  
Langmuir Hinshelwood Mechanism ◽  
Double Eigenvalues ◽  
Surface Reaction Model

A pair of ordinary differential equations describing the Langmuir-Hinshelwood mechanism of a bimolecular surface reaction is presented as a simple model for a heterogeneously catalysed chemical reaction. Numerical and analytical techniques are combined to determine the possible dynamics of this model, which, even though it consists only of low-order polynomials, exhibits very complicated behaviour. This includes sustained oscillations, which appear or disappear at Hopf bifurcations, turning points on periodic branches, global homoclinic and metacritical Hopf bifurcations and codimension-two bifurcations with double eigenvalues of zero. The emphasis of the presentation is on geometrical representation of the observed dynamics.

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.

PATTERN FORMATION IN A RING NETWORK WITH DELAY

2009 ◽  
Vol 19 (10) ◽  
pp. 1797-1852 ◽  
Author(s):  
SHANGJIANG GUO ◽  
YUAN YUAN
Keyword(s):  
Periodic Solutions ◽  
Hopf Bifurcations ◽  
Coupling Coefficients ◽  
Ring Network ◽  
Center Manifold Reduction ◽  
Codimension Two ◽  
Codimension One ◽  
Fold Bifurcation ◽  
Manifold Reduction ◽  
Bifurcation And Stability

We consider a ring network of three identical neurons with delayed feedback. Regarding the coupling coefficients as bifurcation parameters, we obtain codimension one bifurcation (including a Fold bifurcation and Hopf bifurcation) and codimension two bifurcations (including Fold–Fold bifurcations, Fold–Hopf bifurcations and Hopf–Hopf bifurcations). We also give concrete formulas for the normal form coefficients derived via the center manifold reduction that provide detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, and quasi-periodic solutions.

Sign in / Sign up

Export Citation Format

Share Document