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## Inhaltsverzeichnis

### 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
$$\dot{x} = f(t,x)$$
(1.1)
using Newton’s fluxie notation $$\dot{x} = dx/dt$$. 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:
$$\dot{x} = f(x)$$
(2.1)
A vector equation of the form (2.1) is called autonomous. A scalar equation of order n is often written as
$${{x}^{{(n)}}} + F({{x}^{{(n - 1)}}},...,x) = 0$$
(2.2)
in which x(k) = d k x/dt k , k = 0, 1, . . ., n, x(0) = x In 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 $$\dot{x} = f(x)$$ leads to the equation
$$\dot{y} = Ay$$
(3.1)
with 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
$$\det A \ne 0.$$
.
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 > 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. The reader who is not familiar with some of the results may consult Coddington and Levinson (1955), Arnold (1978) or Walter (1976).
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 and more discussion in Sanders and Verhulst (1985), chapter 2. 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 Pol-equation 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 discuss a number of complicated phenomena which are described as “chaotic”, “showing strange attraction” or “sensitive dependence on initial conditions” etc. These phenomena play a part in iterated maps, ordinary differential equations, partial differential equations and, as a number of scientists are believing, in the phenomenon of turbulence in fluid mechanics.
Ferdinand Verhulst

### 15. Hamiltonian systems

Abstract
In dynamical systems theory, conservative systems, in particular Hamiltonian systems, play an important part. Especially in applications from mechanics, the underlying structure is usually Hamiltonian to which dissipative effects have been added. Exploring this underlying structure is usually profitable.
Ferdinand Verhulst

### Backmatter

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