Counter-Examples in Dynamical Systems via Normal Form Theory

SIAM Review ◽  
1986 ◽  
Vol 28 (1) ◽  
pp. 41-51 ◽  
Author(s):  
K. R. Meyer
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Normal Form Theory ◽  
Form Theory

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.

An Efficient Computation Method for Hopf Bifurcation of High Dimensional Systems

2005 ◽  
Author(s):  
Songhui Zhu ◽  
Pei Yu ◽  
Stacey Jones
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Critical Conditions ◽  
High Dimensional ◽  
Computation Method ◽  
Efficient Computation ◽  
Computationally Efficient ◽  
Normal Form Theory ◽  
Form Theory ◽  
Stability And Bifurcations

Normal form theory is a powerful tool in the study of nonlinear systems, in particular, for complex dynamical behaviors such as stability and bifurcations. However, it has not been widely used in practice due to the lack of efficient computation methods, especially for high dimensional engineering problems. The main difficulty in applying normal form theory is to determine the critical conditions under which the dynamical system undergoes a bifurcation. In this paper a computationally efficient method is presented for determining the critical condition of Hopf bifurcation by calculating the Jacobian matrix and the Hurwitz condition. This method combines numerical and symbolic computation schemes, and can be applied to high dimensional systems. The Lorenz system and the extended Malkus-Robbins dynamo system are used to show the applicability of the method.

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.

scholarly journals The Existence of Periodic Orbits and Invariant Tori for Some 3-Dimensional Quadratic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yanan Jiang ◽  
Maoan Han ◽  
Dongmei Xiao
Keyword(s):  
Normal Form ◽  
Limit Cycles ◽  
Averaging Method ◽  
Integral Manifold ◽  
Invariant Tori ◽  
Normal Form Theory ◽  
Quadratic Systems ◽  
3 Dimensional ◽  
Volterra Systems ◽  
Form Theory

We use the normal form theory, averaging method, and integral manifold theorem to study the existence of limit cycles in Lotka-Volterra systems and the existence of invariant tori in quadratic systems inℝ3.

Sign in / Sign up

Export Citation Format

Share Document