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Nonlinear Differential Equations and Dynamical Systems

verfasst von: Ferdinand Verhulst

Verlag: Springer Berlin Heidelberg

Buchreihe : Universitext

Enthalten in: Professional Book Archive

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Über dieses Buch

On the subject of differential equations a great many elementary books have been written. This book bridges the gap between elementary courses and the research literature. The basic concepts necessary to study differential equations - critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds - are discussed. Stability theory is developed starting with linearisation methods going back to Lyapunov and Poincaré. The global direct method is then discussed. To obtain more quantitative information the Poincaré-Lindstedt method is introduced to approximate periodic solutions while at the same time proving existence by the implicit function theorem. The method of averaging is introduced as a general approximation-normalisation method. The last four chapters introduce the reader to relaxation oscillations, bifurcation theory, centre manifolds, chaos in mappings and differential equations, Hamiltonian systems (recurrence, invariant tori, periodic solutions). The book presents the subject material from both the qualitative and the quantitative point of view. There are many examples to illustrate the theory and the reader should be able to start doing research after studying this book.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

This section contains material which is basic to the development of the theory in the subsequent chapters. We shall consider differential equations of the form(1.1) $${\dot x}= f(t,x)$$using Newton’s fluxie notation ẋ = dx/dt. The variable t is a scalar, t ∈ ℝ, often identified with time. The vector function f : G → ℝⁿ is continuous in t and x; G is an open subset of ℝⁿ⁺¹, so x ∈ ℝⁿ.

Ferdinand Verhulst

2. Autonomous equations

Abstract

In this chapter we shall consider equations, in which the independent variable t does not occur explicitly:(2.1) $${\dot x} = f(x)$$A vector equation of the form (2.1) is called autonomous. A scalar equation of order n is often written as(2.2) $$x^{(n)}+ F(x^{(n-1)},“ots ,x)=0$$in which x^(k) = d^k x/dt^k, k = 0,1, …, n, x⁽⁰⁾ = xIn characterising the solutions of autonomous equations we shall use three special sets of solutions: equilibrium or stationary solutions, periodic solutions and integral manifolds.

Ferdinand Verhulst

3. Critical points

Abstract

In section 2.2 we saw that linearisation in a neighbourhood of a critical point of an autonomous system ẋ = f(x) leads to the equation(3.1) $${\dot y} = Ay$$ with A a constant n × n-matrix; in this formulation the critical point has been translated to the origin. We exclude in this chapter the case of a singular matrix A, so$${\rm det}\ A \ne 0.$$

Ferdinand Verhulst

4. Periodic solutions

Abstract

The concept of a periodic solution of a differential equation was introduced in section 2.3. We have shown that in the case of an autonomous equation the periodic solutions correspond with closed orbits in phase-space.

Ferdinand Verhulst

5. Introduction to the theory of stability

Abstract

In the chapters three and four we have seen equilibrium solutions and periodic solutions. These are solutions which exist for all time. In applications one is often interested also in the question whether solutions which at t = t₀ are starting in a neighbourhood of such a special solution, will stay in this neighbourhood for t τ; t₀. If this is the case, the special solution is called stable and one expects that this solution can be realised in the practice of the field of application: a small perturbation does not cause the solutions to move away from this special solution. In mathematics these ideas pose difficult questions. In defining stability, this concept turns out to have many aspects. Also there is of course the problem that in investigating the stability of a special solution, one has to characterise the behaviour of a set of solutions. One solution is often difficult enough.

Ferdinand Verhulst

6. Linear equations

Abstract

There is an abundance of theorems for linear equations but still there are many difficult and unsolved problems left. This chapter contains a summary of a number of important results.

Ferdinand Verhulst

7. Stability by linearisation

Abstract

The stability of equilibrium solutions or of periodic solutions can be studied often by analysing the system, linearised in a neighbourhood of these special solutions. In section 5.4 we have discussed linearisation and we have given a summary of the analysis of linear systems. These methods have been in use for a long time but only since around 1900 the justification of linearisation methods has been started by Poincaré and Lyapunov.

Ferdinand Verhulst

8. Stability analysis by the direct method

Abstract

In this chapter we shall discuss a method for studying the stability of a solution, which is very different from the method of linearisation of the preceding chapter. When linearising one starts off with small perturbations of the equilibrium or periodic solution and one studies the effect of these local perturbations. In the so-called direct method one characterises the solution in a way with respect to stability which is not necessarily local.

Ferdinand Verhulst

9. Introduction to perturbation theory

Abstract

This chapter is intended as an introduction for those readers who are not aquainted with the basics of perturbation theory. In that case it serves in preparing for the subsequent chapters.

Ferdinand Verhulst

10. The Poincaré-Lindstedt method

Abstract

In this chapter we shall show how to find convergent series approximations of periodic solutions by using the expansion theorem and the periodicity of the solution. This method is usually called after Poincaré and Lindstedt, it is also called the continuation method.

Ferdinand Verhulst

11. The method of averaging

Abstract

In this chapter we shall consider again equations containing a small parameter ε. The approximation method leads generally to asymptotic series as opposed to the convergent series studied in the preceding chapter; see section 9.2 for the basic concepts. This asymptotic character of the approximations is more natural in many problems; also the method turns out to be very powerful, it is not restricted to periodic solutions.

Ferdinand Verhulst

12. Relaxation oscillations

Abstract

Relaxation oscillations are periodic phenomena with very special features during a period. The characteristics can be illustrated by the following mechanical system.

Ferdinand Verhulst

13. Bifurcation theory

Abstract

In most examples of the preceding chapters, the equations which we have studied are containing parameters. For different values of these parameters, the behaviour of the solutions can be qualitatively very different. Consider for instance equation 7.12 in example 7.3 (population dynamics). When passing certain critical values of the parameters, a saddle changes into a stable node. The van der Polequation which we have used many times, for instance in example 5.1, illustrates another phenomenon. If the parameter μ in this equation equals zero, all solutions are periodic, the origin of the phase-plane is a centre point. If the parameter is positive with 0 < μ < 1, the origin is an unstable focus and there exists an asymptotically stable periodic solution, corresponding with a limit cycle around the origin. Another important illustration of the part played by parameters is the forced Duffing-equation in section 10.3 and example 11.8.

Ferdinand Verhulst

14. Chaos

Abstract

In this chapter we shall sketch a number of complicated phenomena which are tied in with the concept of “chaos” and “strange attraction”. There are scientists who see also a relation with the phenomenon of turbulence in continuum mechanics but this interesting idea involves still many unsolved problems. We shall restrict ourselves to a discussion of two examples from the various domains where these phenomena have been found: autonomous differential equations with dimension n ≥ 3, second-order forced differential equations like the forced van der Pol- or the forced Duffing equation and mappings of ℝ into ℝ, ℝ² into ℝ² etc.

Ferdinand Verhulst

15. Hamiltonian systems

Abstract

In section 2.4 we were introduced to Hamiltonian systems. If H is a C² function of the 2n variables p_i, q_i, i = 1, … n, H : ℝ²ⁿ → ℝ, then the equations of Hamilton(15-x) $${\dot p}_i = -{\partial H\over \partial q_i}, {\dot q}_i = {\partial H\over \partial p_i}, i = 1, “ots , n.$$Now we have for the orbital derivative L_tH = 0, so H(p, g) is a first integral of the equations 15.1. We have seen a number of examples where n = 1; in this case the integral H(p, q) = constant describes the orbits in the phase-plane completely.

Ferdinand Verhulst

Backmatter

Titel: Nonlinear Differential Equations and Dynamical Systems
verfasst von: Ferdinand Verhulst
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-97149-5
Print ISBN: 978-3-540-50628-7
DOI: https://doi.org/10.1007/978-3-642-97149-5

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Autonomous equations

3. Critical points

4. Periodic solutions

5. Introduction to the theory of stability

6. Linear equations

7. Stability by linearisation

8. Stability analysis by the direct method

9. Introduction to perturbation theory

10. The Poincaré-Lindstedt method

11. The method of averaging

12. Relaxation oscillations

13. Bifurcation theory

14. Chaos

15. Hamiltonian systems

Backmatter