A New Approach for Computing the Normal Forms of High Dimensional Nonlinear Systems

2010 ◽  
Author(s):  
Shuping Chen ◽  
Wei Zhang ◽  
Minghui Yao
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Cantilever Beam ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Real Life ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. The method developed here uses the lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the theoretical model for the nonplanar nonlinear oscillation of a cantilever beam, the computation method is applied to compute the coefficients of the normal forms for the case of two non-semisimple double zero eigenvalues. The normal forms of the averaged equations and their coefficients for non-planar non-linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated.

Further Reduction of Normal Forms for High Dimensional Nonlinear Systems and Application to a Composite Laminated Piezoelectric Plate

2013 ◽  
Vol 291-294 ◽  
pp. 2662-2665
Author(s):  
Shu Ping Chen ◽  
Wei Zhang
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms ◽  
Nonlinear Oscillations ◽  
Piezoelectric Plate ◽  
Nonlinear Differential Equations ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations ◽  
Form Theory

Normal form theory is robust and useful for direct bifurcation and stability analysis of nonlinear differential equations in real engineering problems. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate, the computation method is applied to compute the coefficients of the normal forms for the case of one double zero and a pair of pure imaginary eigenvalues. The algorithm is implemented in Maple V and the normal forms of the averaged equations and their coefficients for nonlinear oscillations of the composite laminated piezoelectric plate under combined parametric and transverse excitations are calculated.

Application of Normal Forms to a Laminated Composite Piezoelectric Rectangular Plate with One-to-Three Internal Resonance

2013 ◽  
Vol 483 ◽  
pp. 14-17
Author(s):  
Shu Ping Chen
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Real Life ◽  
Laminated Composite ◽  
Computation Method ◽  
Normal Form Theory ◽  
Form Theory ◽  
Pure Imaginary Eigenvalues ◽  
Bifurcation And Stability

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate under the parametrically and externally excitations, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues.

Computation of Normal Forms for High Dimensional Nonlinear Systems and Application to Nonplanar Motions of a Cantilever Beam

2003 ◽  
Author(s):  
Wei Zhang ◽  
Feng-Xia Wang ◽  
Jean W. Zu
Keyword(s):  
Nonlinear Systems ◽  
Cantilever Beam ◽  
Normal Forms ◽  
Adjoint Operator ◽  
Operator Method ◽  
Computational Method ◽  
High Dimensional ◽  
Polynomial Equations ◽  
Efficient Computation ◽  
Averaged Equations

A new and efficient computation of normal forms is developed in this paper for high dimensional nonlinear systems, and the computational method is applied to nonplanar motion of a cantilever beam. The method is based on the adjoint operator method and has the advantage of directly calculating coefficients of normal forms. Moreover, the new method is easy to apply to engineering applications, and the final partial differential equations of various resonant cases appear in a canonical form whose solutions can be conveniently obtained using polynomial equations. With the aid of the Maple software, a symbolic program for computing the normal forms of high dimensional nonlinear systems is developed. Based on the symbolic program, the normal forms and their coefficients of the averaged equations for nonplanar motions of a cantilever beam are calculated for two resonant cases.

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.

scholarly journals An Algorithm for Higher Order Hopf Normal Forms

1995 ◽  
Vol 2 (4) ◽  
pp. 307-319 ◽  
Author(s):  
A.Y.T. Leung ◽  
T. Ge
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Higher Order ◽  
Qualitative Behavior ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
System A ◽  
Form Theory ◽  
Cubic System

Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing–Van der Pol system. We use exact arithmetic and find that the undamped Duffing equation can be represented by an exact polynomial differential amplitude equation in a finite number of terms.

On the Normal Forms Associated with High Dimensional Systems

2000 ◽  
Vol 123 (2) ◽  
pp. 157-169 ◽  
Author(s):  
Koncay Huseyin ◽  
Weiyi Zhang
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Normal Forms ◽  
High Dimensional ◽  
Engineering Applications ◽  
Modified Normal Form Approach ◽  
Form Approach

In this paper, a modified normal form approach is proposed for the analysis of high dimensional nonlinear systems. Using the modified approach, calculations of normal forms and, in particular, the related coefficients are carried out much more conveniently. Certain high dimensional systems, including systems with inner resonances, are investigated. These systems exist widely in engineering applications. To illustrate the approach, five examples are presented.

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.

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.

Effect of Higher Order Terms on the Bifurcation Structure of Coupled Rulkov Neurons

2017 ◽  
Vol 27 (11) ◽  
pp. 1750178
Author(s):  
Lifang Cheng ◽  
Hongjun Cao
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Initial Conditions ◽  
Global Bifurcation ◽  
Original System ◽  
Higher Order ◽  
Bifurcation Structure ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Higher Order Terms

Three kinds of bifurcations of two coupled Rulkov neurons with electrical synapses are investigated in this paper. The critical normal forms are derived based on the center manifold theorem and the normal form theory. For the flip and the Neimark–Sacker bifurcation, the quartic terms and above in the normal forms are defined as higher order terms, which originate from the Taylor expansion of the original system. Then the effects of the quartic and quintic terms on the flip and the Neimark–Sacker bifurcation structure are discussed, which verifies that the normal form is locally topologically equivalent to the original system for the infinitesimal 4-sphere of initial conditions and tiny perturbation on the bifurcation curve. By the flip-Neimark–Sacker bifurcation analysis, a novel firing pattern can be found which is that the orbit oscillates between two invariant cycles. Two disconnected cardioid cycles also appear, which makes one, two, three, four, etc. turns happen before closure. Finally, we present a global bifurcation structure in the parameter space and exhibit the distribution of the periodic, quasi-periodic and chaotic firing patterns of the coupled neuron model.

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.

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