STEADY-STATE, HOPF AND STEADY-STATE-HOPF BIFURCATIONS IN DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO A DAMPED HARMONIC OSCILLATOR WITH DELAY FEEDBACK

2012 ◽  
Vol 22 (12) ◽  
pp. 1250286 ◽  
Author(s):  
YONGLI SONG ◽  
JIAO JIANG
Keyword(s):  
Steady State ◽  
Normal Form ◽  
Differential Equations ◽  
Harmonic Oscillator ◽  
Delay Differential Equations ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Damped Harmonic Oscillator ◽  
Form Theory ◽  
Delay Differential

In this paper, employing the normal form theory of delay differential equations due to Faria and Magalhães, we present explicit formulas of the coefficients of a normal form associated with the flow on a center manifold with the unfolding for general delay differential equations under the cases of steady-state, Hopf and steady-state-Hopf singularities. The explicit conditions determining the transcritical and pitchfork bifurcations for steady-state singularity, determining the direction and stability of Hopf bifurcations, and determining the coefficients of a normal form with universal unfolding for steady-state-Hopf singularity up to third order are obtained. Using the obtained results, we give a complete description of bifurcation scenario of the damped harmonic oscillator with delay feedback near the zero equilibrium. Finally, numerical simulations are given to illustrate our theoretical results and some numerical extensions are obtained as a supplement to our theoretical analysis.

scholarly journals Equivariant Hopf Bifurcation in a Ring of Identical Cells with Delay

2009 ◽  
Vol 2009 ◽  
pp. 1-34 ◽  
Author(s):  
Dejun Fan ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Form Theory ◽  
Functional Differential ◽  
Delay Differential ◽  
Equivariant Hopf Bifurcation

A kind of delay neural network withnelements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.

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.

Bogdanov–Takens and Triple Zero Bifurcations of Coupled van der Pol–Duffing Oscillators with Multiple Delays

2017 ◽  
Vol 27 (09) ◽  
pp. 1750133 ◽  
Author(s):  
Xia Liu ◽  
Tonghua Zhang
Keyword(s):  
Delay Differential Equations ◽  
Time Delays ◽  
Normal Forms ◽  
Multiple Delays ◽  
Third Order ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Form Theory ◽  
Delay Differential ◽  
Theoretical Results

In this paper, the Bogdanov–Takens (B–T) and triple zero bifurcations are investigated for coupled van der Pol–Duffing oscillators with [Formula: see text] symmetry, in the presence of time delays due to the intrinsic response and coupling. Different from previous works, third order unfolding normal forms associated with B–T and triple zero bifurcations are needed, which are obtained by using the normal form theory of delay differential equations. Numerical simulations are also presented to illustrate the theoretical results.

On the Equivalence of Normal Form Theory and Multiple Time Scale Method

2007 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Differential Equations ◽  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Perturbation Scheme ◽  
Normal Form Theory ◽  
Weakly Nonlinear ◽  
Form Theory

Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales are introduced that serve as the independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear systems as simple as possible. The “simplest” differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the equivalence of these two methods for constructing periodic solutions is proven, and it is explained why some studies have found the results obtained by the two techniques to be inconsistent.

SIMPLEST NORMAL FORMS OF HOPF AND GENERALIZED HOPF BIFURCATIONS

1999 ◽  
Vol 09 (10) ◽  
pp. 1917-1939 ◽  
Author(s):  
P. YU
Keyword(s):  
Normal Form ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Normal Forms ◽  
Amplitude Equation ◽  
Polar Coordinates ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Odd Order ◽  
Form Theory

The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form. In this paper, three theorems are presented to show that the conventional normal forms of Hopf and generalized Hopf bifurcations can be further simplified. The forms obtained in this paper for Hopf and generalized Hopf bifurcations are shown indeed to be the "simplest", and at most only two terms remain in the amplitude equation of the "simplest normal form" up to any order. An example is given to illustrate the applicability of the theory. A computer algebra system using Maple is used to derive all the formulas and verify the results presented in this paper.

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.

On the Formal Equivalence of Normal Form Theory and the Method of Multiple Time Scales

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Differential Equations ◽  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Normal Form Theory ◽  
Weakly Nonlinear ◽  
Formal Equivalence ◽  
Form Theory

Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales is introduced that serve as independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear terms as simple as possible. The simplest differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the formal equivalence of these two methods for constructing periodic solutions and amplitude evolution equations is proven for autonomous as well as harmonically excited nonlinear vibratory dynamical systems. The reasons as to why some studies have found the results obtained by the two techniques to be inconsistent are also pointed out.

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