Studying the focal value of ordinary differential equations by normal form theory

1995 ◽  
Vol 16 (9) ◽  
pp. 891-900
Author(s):  
Zhang Qichang ◽  
A. Y. T. Leung
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Normal Form Theory ◽  
Focal Value ◽  
Form Theory

scholarly journals Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

2008 ◽  
Vol 237 (8) ◽  
pp. 1029-1052 ◽  
Author(s):  
R.E. Lee DeVille ◽  
Anthony Harkin ◽  
Matt Holzer ◽  
Krešimir Josić ◽  
Tasso J. Kaper
Keyword(s):  
Renormalization Group ◽  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Group Method ◽  
Renormalization Group Method ◽  
Normal Form Theory ◽  
Form Theory

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.

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.

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.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

On the Analysis of Time-Periodic Nonlinear Hamiltonian Dynamical Systems

1995 ◽  
Author(s):  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Normal Form ◽  
Linear Part ◽  
Normal Form Theory ◽  
Mathieu’S Equation ◽  
Form Theory ◽  
Hamiltonian Dynamical Systems ◽  
Time Invariant ◽  
Time Periodic

Abstract In this paper, some analysis techniques for general time-periodic nonlinear Hamiltonian dynamical systems have been presented. Unlike the traditional perturbation or averaging methods, these techniques are applicable to systems whose Hamiltonians contain ‘strong’ parametric excitation terms. First, the well-known Liapunov-Floquet (L-F) transformation is utilized to convert the time-periodic dynamical system to a form in which the linear pan is time invariant. At this stage two viable alternatives are suggested. In the first approach, the resulting dynamical system is transformed to a Hamiltonian normal form through an application of permutation matrices. It is demonstrated that this approach is simple and straightforward as opposed to the traditional methods where a complicated set of algebraic manipulations are required. Since these operations yield Hamiltonians whose quadratic parts are integrable and time-invariant, further analysis can be carried out by the application of action-angle coordinate transformation and Hamiltonian perturbation theory. In the second approach, the resulting quasilinear time-periodic system (with a time-invariant linear part) is directly analyzed via time-dependent normal form theory. In many instances, the system can be analyzed via time-independent normal form theory or by the method of averaging. Examples of a nonlinear Mathieu’s equation and coupled nonlinear Mathieu’s equations are included and some preliminary results are presented.

scholarly journals Chaos and bifurcation in the controlled chaotic system

2018 ◽  
Vol 16 (1) ◽  
pp. 1255-1265
Author(s):  
Yongjian Liu ◽  
Xiezhen Huang ◽  
Jincun Zheng
Keyword(s):  
Periodic Solution ◽  
Normal Form ◽  
Chaotic System ◽  
Variable Parameter ◽  
System Behavior ◽  
Normal Form Theory ◽  
Controlled System ◽  
Hybrid Strategy ◽  
Form Theory ◽  
Bifurcating Periodic Solution

AbstractIn this paper, chaos and bifurcation are explored for the controlled chaotic system, which is put forward based on the hybrid strategy in an unusual chaotic system. Behavior of the controlled system with variable parameter is researched in detain. Moreover, the normal form theory is used to analyze the direction and stability of bifurcating periodic solution.

Spatiotemporal Dynamics in Reaction–Diffusion Neural Networks Near a Turing–Hopf Bifurcation Point

2019 ◽  
Vol 29 (11) ◽  
pp. 1950154 ◽  
Author(s):  
Jiazhe Lin ◽  
Rui Xu ◽  
Xiaohong Tian
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Point ◽  
Reaction Diffusion ◽  
Spatiotemporal Dynamics ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Codimension Two ◽  
Form Theory

Since the electromagnetic field of neural networks is heterogeneous, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate the existence of Turing–Hopf bifurcation in a reaction–diffusion neural network. By the normal form theory for partial differential equations, we calculate the normal form on the center manifold associated with codimension-two Turing–Hopf bifurcation, which helps us understand and classify the spatiotemporal dynamics close to the Turing–Hopf bifurcation point. Numerical simulations show that the spatiotemporal dynamics in the neighborhood of the bifurcation point can be divided into six cases and spatially inhomogeneous periodic solution appears in one of them.

Studies on Codimension-3 Degenerate Bifurcations of the Flexible Beam

2001 ◽  
Author(s):  
Wei Zhang ◽  
Feng-Xia Wang ◽  
Hong-Bo Wen
Keyword(s):  
Normal Form ◽  
Flexible Beam ◽  
Main Attention ◽  
Double Zero ◽  
Normal Form Theory ◽  
Simply Supported ◽  
Axial Excitation ◽  
Averaged Equations ◽  
Homoclinic Bifurcations ◽  
Form Theory

Abstract We present the analysis of codimension-3 degenerate bifurcations of a simply supported flexible beam subjected to harmonic axial excitation. The equation of motion with quintic nonlinear terms and the parametrical excitation for the simply supported flexible beam is derived. The main attention is focused on the dynamical properties of the global bifurcations including homoclinic bifurcations. With the aid of normal form theory, the explicit expressions of normal form associated with a double zero eigenvalues and Z2-symmetry for the averaged equations are obtained. Based on the normal form, it has been shown that a simply supported flexible beam subjected to the harmonic axial excitation can exhibit homoclinic bifurcations, multiple limit cycles, and jumping phenomena in amplitude modulated oscillations. Numerical simulations are also given to verify the good analytical predictions.

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