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Nonlinear Dynamics
Exploration Through Normal Forms
By Peter B. Kahn, Yair Zarmi Dover Publications, Inc.
Copyright © 2014 Peter B. Kahn and Yair Zarmi
All rights reserved.
ISBN: 978-0-486-79502-7
CHAPTER 1
THE TEXT: ITS SCOPE, STYLE, AND CONTENT
1.1 INTRODUCTION
In our study of nature we observe phenomena that are strikingly different. Thus, attempting to understand properties of some limited set of observations, we try to group "similar" things together and hopefully develop an adequate systemization. In this endeavor, science has been quite successful over the years, as judged by our understanding of the periodic structure of the elements, development of a classification of biological systems according a hierarchical structure, chemical reactions, models of the solar system, the structure of elementary particles, etc.
At the next level, in our development of models of many physical systems, we have been successful at incorporating the salient features of the system by both designing and analyzing a linear description that serves as the basis for a sound first approximation. Examples of areas of application are classical and quantum mechanics, electric circuits, Newtonian theory of gravitation, Maxwell's equations, etc.
In the study of these problems, the analysis has relied heavily on the linearity of the basic equations and uses the principle of superposition as a fundamental component. Furthermore, classification schemes were developed of the equations themselves since diverse physical phenomena are described by the same mathematical functions and techniques. (For example, in one area of mathematics of interest to scientists, this has led to serious investigations of the properties of the so-called special functions.) This is a successful procedure, in part, because the structure of the solutions to linear equations is present in the equations themselves, independent of the initial conditions. The success in the use of linear equations was so great that our traditional body of scientific knowledge and the associated curriculum that one studies both rely almost exclusively on a linear analysis.
The desire to obtain an approximate, but useful, analytic "picture" of natural processes, has often led scientists to ignore those aspects of the mathematics that are associated with the nonlinearities in the model system. The basic premise has always been that a thorough understanding of the approximate linear system would give insight into the associated nonlinear system.
Alas, this hope is rarely fulfilled. Nonlinear phenomena that are not adequately described by the linear approximation are encountered in all areas of the quantitative sciences. Almost always, the nonlinear system has fundamental aspects that cannot be faithfully approximated by a linear perturbation scheme. Perhaps most important is the loss of the principle of superposition that plays a critical role in almost all aspects of linear systems. By training, one gets used to seeking a solution to a problem by constructing a sequence of functions that, when added together appropriately, yield a satisfactory approximation. The loss of this important principle leads one to realize that the spectrum of behavior of nonlinear systems is appreciably richer than that of linear systems, and, furthermore there generally does not exist the possibility of a systematic classification scheme based on the structure of the equations. This latter point follows in part because the singularity structure of the solution of a nonlinear differential equation is affected by the initial conditions.
Today it is becoming progressively clearer that the linear world, for which mathematics has been developed over a period of 300 years, is but a tiny comer of a much richer world that is being slowly unraveled. Over the years, many attempts have been made to improve our understanding of systems governed by nonlinear equations. Already in the 18th century, Clairot, Lagrange, and Laplace developed perturbation methods for the treatment of nonlinear problems in celestial mechanics. Jacobi, Poincaré, and Lindstedt studied similar problems toward the end of the 19th century. Rayleigh studied "self-sustained" vibrations within the framework of his theory of sound. His equation for the self-sustained oscillations in organ pipes turns out to be equivalent to the equation developed by Van der Pol in the 1920s for the current in a triode. Both systems include nonlinear dissipative terms that drive the system toward a self-sustaining oscillatory mode denoted today as a limit cycle.
Nonlinear models arose in a variety of contexts. At the beginning of this century, Ross proposed a modified exponential growth model, now known as a logistic equation, to describe the spread of malaria. The same model was used by Verhulst and others to account for populations that were "growth-limited." This type of model has played a key role in many studies of biological systems and in the analysis of chemical reactions governed by the law of mass action. Volterra introduced a set of coupled nonlinear equations that were used to describe oscillations in fish populations. It is a beautiful model that has played an important role in the development of mathematical biology. It is interesting that in the construction of these equations, Volterra was greatly influenced by concurrent developments in statistical mechanics. In particular, Volterra's dis-discussion of the so-called encounters is reminiscent of methods used by Boltzmann to study the approach to equilibrium of an ideal gas, the celebrated H theorem. A similar model was introduced by Lotka to study a class of chemical kinetics and rate equations. The equations are now known as the Volterra–Lotka equations, and are used extensively as an idealization of the interaction of two isolated species, a predator and a prey and interactions of "forces in conflict" (i.e., the Lanchester equations). The equations are often generalized to include dissipative terms similar to those in Rayleigh's and Van der Pol's equations, to describe sustained oscillating chemical reactions known as Belousov–Zhabotinskii equations.
In the period from the 1930s to the 1960s Krylov, Bogoliubov, and Mitropolskii developed methods for analyzing oscillatory systems that contained small nonlinear perturbations. Later, interest arose in trying to develop analytic techniques that could be used to study nonlinear problems in which the nonlinearity is not a small perturbation. Important milestones were, among others, the Lorenz equations which arise from a truncated Navier–Stokes equation for fluid motion together with heat and mass transfer, meteorology, chemical kinetics, etc.; May's work on discrete models and his role in drawing attention to the importance of transferring one's focus from the study of linear to the study of nonlinear phenomena; Feigenbaum's analysis of discrete nonlinear maps and their relevance to the behavior of continuous nonlinear systems; and finally the advent of the concepts of chaos and fractals.
In this work we primarily consider the modeling of weakly nonlinear systems by means of ordinary nonlinear differential equations. Our working hypothesis is that the associated linear system has been solved and that our efforts go toward constructing a reliable and systematic perturbation scheme, as a power series in a small parameter. The resulting approximation has a prescribed error, valid for a finite duration (that can be estimated) in time. When the perturbations are included, the basic equation takes the form
dx/dt = Ax + εF(ε, x), (1)
where x is an n-dimensional vector and A is a diagonal n×n constant matrix. (The diagonalization greatly simplifies the algebra.) The interaction or perturbation is written as F (ε, x), where 8 is a small parameter, |ε| << 1, and F is an n-dimensional vector field that may contain linear and nonlinear terms. The nonlinear terms are polynomials or functions with Taylor series expansions in x of degree ≥2. One locates the fixed points, i.e., those values of x(t) such that dx/dt = 0; transforms them so that they are at the origin; and studies the behavior of the solutions as a power series in the small parameter for a finite interval of time. Finding an approximation to the solution x(t) may begin with expanding x in a power series in ε:
x(t) = x0 = x0 + x1 + ε2x2 + ··· (1.2)
The zero-order termx0(t), holds the key to the development of the perturbation expansion. To see this, visualize an n-dimensional sphere with radius O(ε) (defined in some appropriate norm in the vector space), which contains the exact solution as it evolves in time. Then, any choice for the vector function x0 (t) is a valid one as long as it remains within O(ε) of x(t). For example, one can construct x0(t) entirely from the unperturbed part of the equation. This is called naive perturbation theory. It turns out that this procedure often has serious short-comings and it is found that a scheme, such as the method of normal forms, that incorporates in the quantity x0(t) significant aspects of the interaction terms leads to a more effective perturbation expansion.
The "path" connecting x0((t and x(t) (see Fig 1.1) depicts the progressive approach to x(t) due to the successive addition of higher order corrections εnxn to x0 (t). If the series expansion of x(t) as given by Equation (1.2) converges, the "path" will converge to x(t) as t -> ∞. If the series is an asymptotic one, then the "path" will initially approach x(t), but as the successively higher-order terms are included, it will diverge away from it. The choice of x0(t) and the "path" are interrelated, since all xn must satisfy equations resulting from the dynamical equation, and the constraints that are derived from the imposed initial conditions. In the following chapters this program will be realized through the normal form expansion.
Equation (1.1) describes an autonomous system if the right hand side does not explicitly contain the time. Then the rate of evolution of the system is entirely determined by its present state. In this regard, it is important to note that in autonomous systems with one degree of freedom (i.e., two variables, x and y), the motion takes place in the phase plane. The solution is unique and, as a result, trajectories cannot cross, except at fixed points. The analysis is straightforward and will form the basis of most of our discussion. The situation changes when Equation (1.1) includes interaction terms that have explicit time dependence. It then describes a nonautonomous system. The interaction term is changed from εF(ε, x) to εF (ε, x, t). For example, our equation might be
dx/dt = y dy/dt = -x + ε {α + β exp(-t)}x2y (1.3a)
One may study the equation as it is given or introduce a third dependent variable, z, by writing Equation (1.3a) as
dx/dt = y dy/dt = -x + ε{α + βexp(-z)| x2y dz/dt = 1 (1.3b)
We have converted Equation (1.3a), a nonautonomous system with two dependent variables, x and y, into to Equation (1.3b), an autonomous system in three variables (x,y,z). This leads us to conclude that, there is no way to distinguish between the variable we call z and the "time" t. It then follows that as soon as one considers systems with more than two variables, the distinction between autonomous and nonautonomous is no longer clear. [Generally, a nonautonomous system of degree n can be transformed into an autonomous one of degree (n +1) by assigning the variable "time" to be the (n +l)st dimension.] One needs to keep in mind that the solution to Equation (1.3b) is unique. Hence, its trajectory in three-dimensional space cannot cross except at fixed points. However, the projection of the true motion onto a plane may yield crossing trajectories.
This situation arises in the study of conservative systems that are perturbed by time-dependent forces. The latter have the capacity to interact with the natural (or unperturbed) frequency of the system leading to conditions of resonance that may result in excitations and instability. Often one would like to obtain relationships among parameters characterizing such a problem that both describe and separate regions of stable and unstable motions. One learns that the richness of nonautonomous systems requires us to engage in an aspect of nonlinear analysis that is both inadequately explored and full of pitfalls. Thus, it is judicious to begin by having a firm grasp of autonomous systems with two variables before exploring nonautonomous ones.
Finally, it is crucial to realize that, in this work, we concentrate exclusively on problems that are amenable to a perturbation expansion that is a power series in a small parameter. By this restriction, we cannot treat problems associated with chaotic motion.
1.1.1 Near-identity transformations
Investigators have devised a variety of perturbation schemes to attack problems modeled by Equation (1.1). This is not surprising, as the procedure one introduces needs to be tailored to a certain extent to the questions one asks and the form one want these answers to take. For example, the method of averaging (MOA) is applied with success in the studies of perturbed oscillatory systems. The physical idea behind the method is that, for these systems it is possible to separate, by a smoothing or averaging technique the slowly varying component from those components that vary rapidly. Another perturbation scheme is the method of multiple time-scales (MMTS). It is applied with success in the study of many types of equations, including perturbed oscillations, boundary-layer problems, and a class of partial-differential equations. The basic idea is to introduce artificially separated (or independent) timescales that characterize different aspects of solution. In this text, we have chosen to follow the approach known as the method of normal forms (NF) that is based on rather different underlying ideas than either MOA or MMTS. (Although derived from a different set of principles than MOA or MMTS, it yields in appropriate instances the same set of hierarchical equations.)
1.1.2 Basic ideas behind the method of normal forms
There are four aspects of the normal form expansion that we mention briefly at this point, to provide some perspective and motivation for the development of the method. We would like to convey to the reader the flavor of the method through a quick run through some aspects of the method. Critical details and concepts will not be fully developed or explained in this section. The fleshing out of all the essential details and applications will form central core of the book.
A. We begin by referring to Equation (1.1) and introduce a near-identity transformation from the vector x to the variable u:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4a)
The vector u is the zero-order approximation and the T functions are the generators of the transformation. In this work, they will be polynomials or functions that have a Taylor series expansion in u. On occasion, one has to write the transformation in component form:
xi = ui + εTi,1(u1, u2, ..., un) + ε 2Ti,2 (u1, u2, ..., un) + ··· (1.4b)
Each linear component, ui, is derived from the associated vector component, xi. All the components are subsequently related to one another in the perturbation expansion. Furthermore, if one had not diagonalized the matrix A, the i component of x would have the possibility to be related linearly to all the components of u.
A critical aspect of the method is that, in general, the variable udoes not obey the unperturbed equation but rather is "updated" or modified by the perturbation. Thus, its equation of motion must depend on the perturbation parameter e and be of the general form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
The quantity U0 is the unperturbed operator, and the vector functions Un are the "updating" of the equation of motion of the zero order or the fundamental component of the full solution.
B. The T functions are the generators of the expansion and serve as modifications that are added to the zero-order term u, so as to obtain a better approximation to the structure of the full solution. They depend on the u variables. In our analysis, we will see that the T functions may be chosen so as to yield for the vector u equations of a simple form by eliminating as many extraneous terms in the equations.
(Continues...)
Excerpted from Nonlinear Dynamics by Peter B. Kahn, Yair Zarmi. Copyright © 2014 Peter B. Kahn and Yair Zarmi. Excerpted by permission of Dover Publications, Inc..
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