Titlebook: Nonlinear Differential Equations and Dynamical Systems; Ferdinand Verhulst Textbook 1996Latest edition Springer-Verlag Berlin Heidelberg 1

暖昧關(guān)系

Autonomous equations,alled autonomous. A scalar equation of order . is often written as . in which . = . ., . = 0, 1, . . ., ., . = . In characterising the solutions of autonomous equations we shall use three special sets of solutions: . or ., . and ..

Calculus

conquer

Stability analysis by the direct method,eceding chapter. When linearising one starts off with small perturbations of the equilibrium or periodic solution and one studies the effect of these . perturbations. In the so-called direct method one characterises the solution in a way with respect to stability which is not necessarily local.

HEAVY

The method of averaging,osed 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.

Dedication

Confidential

反省

Critical points,In section 2.2 we saw that linearisation in a neighbourhood of a critical point of an autonomous system . leads to the equation. with . constant . × .-matrix; in this formulation the critical point has been translated to the origin. We exclude in this chapter the case of a singular matrix ., so..

咒語

Periodic solutions,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.

比目魚

insurrection

Introduction to perturbation theory,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.