Normal Form Reduction Research Articles

ALGORITHM FOR AMPLITUDE CONTROL OF LIMIT CYCLES EMERGING FROM HOPF BIFURCATIONS IN 3D SYSTEMS

A general algorithm is proposed for controlling the amplitude of limit cycle emerging from the Hopf bifurcation in a 3D nonlinear dynamic system using nonlinear state feedback. An expression of the curvature coefficient of limit cycle and an explicit control formula are derived by utilizing the center manifold theory and normal form reduction. The expression and formula present a convenient approach to obtain an effective analytical control and predict the amplitude of limit cycles. Three examples — the Langford, the Chen and the Chua systems — are used to illustrate the application of this control approach. The simulation results are presented to confirm the analytical predictions.

On the flat remainder in normal forms of families of analytic planar saddles

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter. To cite this article: P. Bonckaert, F. Verstringe, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

FURTHER REDUCTION OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF SMOOTH MAPS

Further reduction for classical normal forms of smooth maps is considered in this paper. Firstly, based on the idea of computation of simplest normal forms for vector fields [Yu & Yuan, 2003], we compute the transformed map of a given smooth map under a near identity formal transformation, and give a recursive formula for the homogeneous terms of the transformed map, which is a powerful tool for further reduction of classical normal forms of smooth maps. Secondly, by using the recursive formula, the idea of [Chen & Della Dora, 1999] for further reduction of normal forms for maps and the method introduced by Kokubu et al. [1996] for further reduction of normal forms of vector fields, we develop the concepts of Nth order normal forms and infinite order normal forms of smooth maps, and give some sufficient conditions for uniqueness of normal forms of smooth maps. As an application, we show the occurrence of the flip–Neimark–Sacker bifurcation in a financial model.

Further reduction of normal forms of formal maps

Further reduction for classical normal forms of formal maps is considered in this note. Based on a recursive formula for computing the transformed map of a formal map under a near identity formal transformation, we develop the concepts of Nth order normal forms and infinite order normal forms for formal maps, and give some sufficient conditions for uniqueness of normal forms of formal maps. To cite this article: D. Wang et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Quasi-periodic solutions in a nonlinear Schrödinger equation

In this paper, one-dimensional (1D) nonlinear Schrödinger equation i u t − u x x + m u + | u | 4 u = 0 with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1 , the equation admits a Whitney smooth family of small amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.

Normal forms of vector fields with perturbation parameters and their application

Abstract This paper is concerned with the normal forms of vector fields with perturbation parameters. Usually, such a normal form is obtained via two steps: first find the normal form of a “reduced” system of the original vector field without perturbation parameters, and then add an unfolding to the normal form. This way, however, it does not yield the relation (transformation) between the original system and the normal form. A study is given in this paper to consider the role of near-identity transformations in the computation of normal forms. It is shown that using only near-identity transformations cannot generate the normal form with unfolding as expected. Such normal forms, which contain many nonlinear terms involving perturbation parameters, are not very useful in bifurcation analysis. Therefore, additional transformations are needed, resulting in a further reduction of normal forms – the simplest normal form. Examples are presented to illustrate the theoretical results.

Elimination of the initial value parameters when identifying a system close to a Hopf bifurcation

One of the biggest problems when performing system identification of biological systems is that it is seldom possible to measure more than a small fraction of the total number of variables. If that is the case, the initial state, from where the simulation should start, has to be estimated along with the kinetic parameters appearing in the rate expressions. This is often done by introducing extra parameters, describing the initial state, and one way to eliminate them is by starting in a steady state. We report a generalisation of this approach to all systems starting on the centre manifold, close to a Hopf bifurcation. There exist biochemical systems where such data have already been collected, for example, of glycolysis in yeast. The initial value parameters are solved for in an optimisation sub-problem, for each step in the estimation of the other parameters. For systems starting in stationary oscillations, the sub-problem is solved in a straight-forward manner, without integration of the differential equations, and without the problem of local minima. This is possible because of a combination of a centre manifold and normal form reduction, which reveals the special structure of the Hopf bifurcation. The advantage of the method is demonstrated on the Brusselator.

Bifurcations of travelling wave solutions in the discrete NLS equations

We study discrete nonlinear Schrödinger (NLS) equations, which include the cubic NLS lattice with on-site interactions and the integrable Ablowitz–Ladik lattice. Standing wave solutions are known to exist in the discrete NLS equations outside of the finite spectral band. We study travelling wave solutions which have nonlinear resonances with unbounded linear spectrum. By using center manifold and normal form reductions, we show that a continuous NLS equation with the third-order derivative term is a canonical normal form for the discrete NLS equation near the zero-dispersion limit. Bifurcations of travelling wave solutions near the zero-dispersion limit are analyzed in the framework of the third-order derivative NLS equation. We show that there exists a continuous two-parameter family of single-humped travelling wave solutions in the third-order derivative NLS equation, when it is derived from the integrable Ablowitz–Ladik lattice. On the contrary, there are no single-humped solutions in the third-order derivative NLS equation, when it is derived from the cubic NLS equation with on-site interactions. Nevertheless, we show that there exists an infinite discrete set of one-parameter families of double-humped travelling wave solutions in the latter case. Our results are valid in the neighborhood of the zero-dispersion point on the two-parameter plane of travelling wave solutions.

Control algorithm for creation of Hopf bifurcations in continuous-time systems of arbitrary dimension

Creation of Hopf bifurcations via appropriate controls is viewed as one way of designing oscillatory behaviors into a dynamical system. A general control algorithm is proposed for creation of Hopf bifurcation at a desired parameter point in nonlinear systems of arbitrary dimension. The linear gains that control the two critical bifurcation conditions are derived according to the suitable criteria without using eigenvalues, and the nonlinear gains that control the stability of the created limit circle are analytically derived by utilizing the center manifold theory and normal form reduction. The control law allows us to develop a desirable oscillatory behavior in high-dimensional systems. Applications are illustrated for an eight-dimensional chaotic system.

Nonlinear System Identification of Systems with Periodic Limit-Cycle Response

A nonlinear system identification methodology based on the principle of harmonic balance and bifurcation theory techniques like center manifold analysis and normal form reduction, is presented for multi-degree-of-freedom systems. The methodology, called Bifurcation Theory System IDentification, (BiTSID), is a general procedure for any nonlinear system that exhibits periodic limit cycle response and can be used to capture the bifurcation behavior of the nonlinear systems. The BiTSID methodology is demonstrated on an experimental system single-degree-of-freedom system that deals with self-excited motions of a fluid-structure system with a sub-critical Hopf bifurcation. It is shown that BiTSID performs excellently in capturing the stable and unstable limit cycles within the experimental regime. Its performance outside the experimental regime is also studied. The application of BiTSID to experimental multi-degree-of-freedom systems has also been very successful. However in this study only the results of the single-degree-of-freedom system are presented.

Assigning vibrational polyads using relative equilibria: application to ozone

We demonstrate how relative equilibria of a vibrating molecule, which are families of principal periodic orbits otherwise known as nonlinear normal modes, can be used to describe the global polyad structure of vibrational energy levels. The classical action integral n ( E) computed along these orbits at different energies E corresponds to the polyad quantum number n so that the energy E ( n) of different relative equilibria describes the splitting of n-polyads. Further information on the internal polyad structure can be driven from the stability analysis of relative equilibria. We use the ozone molecule as a concrete example where n-polyads or “hyperpolyads” should be distinguished from the well-known polyads of the 1:1 stretching mode resonance; the stretching polyads are structural elements of hyperpolyads. We give dynamical interpretation of the relation between relative equilibria and n-polyads based on the normal form reduction in the limit of small vibrations near the equilibrium.

Generalized Lyapunov–Schmidt Reduction Method and Normal Forms for the Study of Bifurcations of Periodic Points in Families of Reversible Diffeomorphisms

We introduce a general reduction method for the study of periodic points near a fixed point in a family of reversible diffeomorphisms. We impose no restrictions on the linearization at the fixed point except invertibility, allowing higher multiplicities. It is shown that the problem reduces to a similar problem for a reduced family of diffeomorphisms, which is itself reversible, but also has an additional ℤ q -symmetry. The reversibility in combination with the ℤ q -symmetry translates to a 𝕋 q -symmetry for the problem, which allows to write down the bifurcation equations. Moreover, the reduced family can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Lyapunov–Schmidt method, and makes repetitive use of the Implicit Function Theorem. As an application we analyze the branching of periodic points near a fixed point in a family of reversible mappings, when for a critical value of the parameters the linearization at the fixed point has either a pair of simple purely imaginary eigenvalues that are roots of unity or a pair of non-semisimple purely imaginary eigenvalues that are roots of unity with algebraic multiplicity 2 and geometric multiplicity 1.

UNIQUE NORMAL FORMS FOR NILPOTENT PLANAR VECTOR FIELDS

Further reduction of normal forms for nilpotent planar vector fields has been considered. Unique normal form for a special case of an unsolved problem for the Takens–Bogdanov singularity is given. Computations in Maple are used to conjecture the main results and some computations in the proof are also done with Maple.

Stabilization of nonlinear systems in compound critical cases

In this paper, we study the stabilization of nonlinear systems in critical cases by using the center manifold reduction technique. Three degenerate cases are considered, wherein the linearized model of the system has two zero eigenvalues, one zero eigenvalue and a pair of nonzero pure imaginary eigenvalues, or two distinct pairs of nonzero pure imaginary eigenvalues; while the remaining eigenvalues are stable. Using a local nonlinear mapping (normal form reduction) and Liapunov stability criteria, one can obtain the stability conditions for the degenerate reduced models in terms of the original system dynamics. The stabilizing control laws, in linear and/or nonlinear feedback forms, are then designed for both linearly controllable and linearly uncontrollable cases. The normal form transformations obtained in this paper have been verified by using code MACSYMA.

Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling

Normal form theory is one of the most power tools for the study of nonlinear differential equations, in particular, for stability and bifurcation analysis. Recently, many researchers have paid attention to further reduction of conventional normal forms (CNF) to so called the simplest normal form (SNF). However, the computation of normal forms has been restricted to systems which do not contain perturbation parameters (unfolding). The computation of the SNF is more involved than that of CNFs, and the computation of the SNF with unfolding is even more complicated than the SNF without unfolding. Although some author mentioned further reduction of the SNF, no results have been reported on the exact computation of the SNF of systems with perturbation parameters. This paper presents an efficient method for computing the SNF of differential equations with perturbation parameters. Unlike CNF theory which uses an independent nonlinear transformation at each order, this approach uses a consistent nonlinear transformation through all order computations. The particular advantage of the method is able to provide an efficient recursive formula which can be used to obtain the nth-order equations containing the nth-order terms only. This greatly saves computational time and computer memory. The recursive formulations have been implemented on computer systems using Maple. As an illustrative example, the SNF for single zero singularity is considered using the new approach.

Normal form methods for symbolic creation of approximate solutions of nonlinear dynamical systems

We present a scheme for the computation (and further reduction) of normal forms of Hamiltonian systems, and we describe the scheme’s computer algebra application to some two-degrees-of-freedom mechanical models to build symbolic approximations for local families of their periodic solutions near the origin. More precisely, we consider one-parameter families of periodic solutions of real Hamiltonian systems, the parameter being the energy. We deal with some interesting cases in the generalized Hénon-Heiles system and another Hamiltonian considered by Churchill et al. The families of periodic solutions are represented as truncated Fourier series in the approximate frequencies and power series in the mechanical energy. It is also shown that some periods of the found families can be given closed expressions in terms of the energy constant in the form of very simple generalized hypergeometric functions.

Unique normal forms for the Takens–Bogdanov singularity in a special case

We consider further reduction of normal forms for nilpotent planar vector fields. We give a unique normal form for a special case of an open problem for the Takens–Bogdanov singularity.

Further Reduction of Normal Forms for Vector Fields

We study Lie's method in the way of Ushiki for further reduction of normal forms for vector fields with singularity at the origin. We give further reduction of normal forms in two typical cases for vector fields in dimension 2: one with a rotation as its linear part and the other with a nilpotent linear part.

Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems

The bifurcation of double-pulse homoclinic orbits under parameter perturbation is analysed for reversible systems having a homoclinic solution that is biasymptotic to a saddle-centre equilibrium. This is a non-hyperbolic equilibrium with two real and two purely imaginary eigenvalues. Reversibility enforces that small perturbations will not change this eigenvalue configuration. Using a Shil'nikov-type analysis, it is found that (generically) an infinite sequence of parameter values exists, on one side of that of the primary homoclinic, for which there are double-pulse homoclinic orbits. Mielke, Holmes and O'Reilly considered the same situation with the additional assumption of Hamiltonian structure. There, double pulses exist on either both or neither side, depending on a sign condition which also determines whether there can be any recurrent dynamics. It is shown how this sign condition occurs in the purely reversible case, via the breaking of a non-degeneracy assumption. Two possible two-parameter bifurcation diagrams are constructed under the addition of a perturbation that keeps reversibility but destroys Hamiltonian structure. The results can be stated rigorously only under a technical hypothesis on the validity of a normal form reduction. Even if this hypothesis fails to be strictly true, then the analysis is shown to be qualitatively and quantitatively correct by careful comparison with two numerical examples. The examples are also of interest in their own right; one of them a generalization of the classical Massive Thirring Model for optical spatial solitons in the presence of linear and nonlinear dispersion, the other is a perturbation to a continuum model of a discrete lattice. The computations agree perfectly with the theory including the prediction of different rates at which double pulses accumulate in the Hamiltonian and nonHamiltonian cases.

An Algorithm for Computing a New Normal Form for Dynamical Systems

We propose in this paper a new normal form for dynamical systems or vector fields which improves the classical normal forms in the sense that it is a further reduction of the classical normal forms. We give an algorithm for an effective computation of these normal forms. Our approach is rational in the sense that if the coefficients of the system are in a field K(which, in practice, is Q, R), so is the normal form and all computations are done inK . As a particular case, if the matrix of the linear part is a companion matrix then we reduce the dynamical system to a single differential equation. Our method is applicable for both the nilpotent and the non-nilpotent cases. We have implemented our algorithm in Maple V and obtained many examples of the further reduced normal forms up to some finite order.