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2010 | Book

Introduction and Prerequisites Read chapter Read first chapter

Practical Bifurcation and Stability Analysis

Author: Rüdiger Seydel

Publisher: Springer New York

Book Series : Interdisciplinary Applied Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Fifteen years have elapsed after the second edition of Practical Bifurcation and Stability Analysis was published. During that time period the ?eld of computational bifurcation has become mature. Today, bifurcation mec- nisms are widely accepted as decisive phenomena for explaining and - derstanding stability and structural change. Along with the high level of sophistication that bifurcation analysis has reached, the research on basic computational bifurcation algorithms is essentially completed, at least in - dinary di?erential equations. The focus has been shifting from mathematical foundations towards applications. The evolution from equilibrium to chaos has become commonplace and is no longer at the cutting edge of innovation. But the corresponding methods of practical bifurcation and stability analysis remain indispensable instruments in all applications of mathematics. This constant need for practical bifur- tion and stability analysis has stimulated an e?ort to maintain this book on a present-day level. The author’s endeavor has resulted in this third edition. It is based on more than three decades of practical experience with the subject, and on many courses given at several universities.

Table of Contents

Frontmatter
1. Introduction and Prerequisites
Abstract
Every day of our lives we experience changes that occur either gradually or suddenly. We often characterize these changes as quantitative or qualitative, respectively. For example, consider the following simple experiment (Figure 1.1). Imagine a board supported at both ends, with a load on top. If the load λ is not too large, the board will take a bent shape with a deformation depending on the magnitude of λ and on the board’s material properties (such as stiffness, K). This state of the board will remain stable in the sense that a small variation in the load λ (or in the stiffness K) leads to a state thatis only slightly perturbed. Such a variation (described by Hooke’slaw) would be referred to as a quantitative change. The board is deformed within its elastic regime and will return to its original shape when the perturbation in λ is removed.
Rüdiger Seydel
2. Basic Nonlinear Phenomena
Abstract
Beginning with this chapter, nonlinearity and parameter dependence willplay a crucial role. We shall assume throughout that λ is a real parameter, and we shall study solutions of a system of ODEs, or solutions of a system of “algebraic” equations, Sometimes boundary conditions must be attached to equation (2.1). As in Chapter 1, the vectors y and f have n components. If a particular example involves more than one parameter, we assume for the time being that all except λ are kept fixed. Clearly, solutions y of equation (2.1) or (2.2) in general vary with λ. We shall assume throughout that f depends smoothly on y and λ—that is, f is to be sufficiently often continuously differentiable. This hypothesis is usually met by practical examples.
Rüdiger Seydel
3. Applications and Extensions
Abstract
Ordinary differential equations are the backbone of this book. Symbolically this class of problems can be represented by the ODE prototype equation which is a short way for with f sufficiently smooth. For this type of equation we discuss parameter dependence, bifurcation, and stability in detail. Many other bifurcation problems are not of this ODE type. For example, a delay may be involved, or the dynamics fails to be smooth. But even then the ODE background is helpful. On the one hand, methods can be applied that are similar as the ODE approaches. On the other hand, the ODE system (3.1) can be used to approximate non-ODE situations, or to characterize certain special cases.
Rüdiger Seydel
4. Principles of Continuation
Abstract
The diagrams in Chapter 2 are idealizations of the results one gets in practice. Numerical reality does not look quite as good as these diagrams. The smooth curves of the bifurcation diagrams are drawn from a skeleton of computed points. It requires—in addition to reliable computational methods— imagination and condence to draw a good diagram from the relatively few points the computer gives us. How to select and construct the points is one concern, and the other is the correct interpretation, which may be summarized by drawing a smooth branch.
Rüdiger Seydel
5. Calculation of the Branching Behavior of Nonlinear Equations
Abstract
Equipped now with some knowledge about continuation, we assume that we are able to trace branches. We take for granted thatthe entirebranch canbe traced, provided one solution on that branch can be found. In this chapter we address problems of locating bifurcation points and switching branches. Essential ideas and methods needed for a practical bifurcation and stability analysis are presented.
Rüdiger Seydel
6. Calculating Branching Behavior of Boundary-Value Problems
Abstract
The main topic of this chapter is the calculation of branching behavior of the one-parameter family of two-point boundary-value problems As usual, the variable y (t) consists of n scalar functions y 1 (t),…,y n (t). The right-hand side f(t, y, λ) is a vector function; the boundary conditions [the second equation (6.1)] consist of n scalar equations, The independent variable t (a ≤ t ≤ b) need not be time; accordingly, the derivative with respect to t is denoted by a prime ′ rather than a dot: y′ = dy/dt. The bifurcation parameter λ can occur in the boundary conditions: However, because the methods discussed in this chapter are not affected by the dependence of r on λ, the notation r(y (a), y (b)) of equation (6.1) will be retained.
Rüdiger Seydel
7. Stability of Periodic Solutions
Abstract
Periodic solutions of a differential equation are time-dependent or space-dependent orbits, cycles, or oscillations. In this chapter a major concern is time-dependent periodicity (“time periodicity” for short) of solutions of autonomous systems of ODEs For time-periodic solutions, there is a minimum time interval T > 0 (the “period”) after which the system returns to its original state: for all t. In Section 7.7 we turn our attention to the non-autonomous case.
Rüdiger Seydel
8. Qualitative Instruments
Abstract
Numerical calculations are primarily quantitative in nature. Appropriate interpretation transforms quantitative information into meaningful results.The diversity of nonlinear phenomena requires more than just the ability to understand numerical output. The preceding chapters have concentrated on one-parameter problems, with only brief excursions (in Sections 2.10 and 5.9) to two-parameter models, mainly to show how one-parameter studies can be applied to investigate certain aspects of two-parameter models. Reducing a parameter space to lower-dimensional subsets will remain a standard approachfor obtainingquantitativeinsight.However,afullqualitativeinterpretation of multiparameter models requires instruments that have not yet been introduced in the previous chapters. These instruments are provided by singularity theory and catastrophetheory. Both these fields are qualitative in nature; in no way do they replace numerical parameter studies. Knowledge of singularity theory and catastrophe theory helps organize a series of partial results into a global picture.
Rüdiger Seydel
9. Chaos
Abstract
The oscillations we have encountered in previous chapters have been periodic. Periodicity refiects a high degree of regularity and order. Frequently, however, one encounters irregular oscillations, like those illustrated in Figure 9.1. Functions or dynamical behavior that is not stationary, periodic or quasiperiodic may be called chaotic, sometimes aperiodic or erratic. We shall use “chaos” in this broad meaning; note that chaos can be defined in a restricted way, characterizing orbits with a positive Liapunov exponent (to be explained later). Such dynamic behavior can be seen as the utmost fiexibility a dynamical system may show. It is expected that systems with chaotic behavior can be easily modulated or stabilized. Note that the “irregularity” of chaos is completely deterministic and not stochastic.
Rüdiger Seydel
Backmatter
Metadata
Title
Practical Bifurcation and Stability Analysis
Author
Rüdiger Seydel
Copyright Year
2010
Publisher
Springer New York
Electronic ISBN
978-1-4419-1740-9
Print ISBN
978-1-4419-1739-3
DOI
https://doi.org/10.1007/978-1-4419-1740-9

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