Project Abstract

We considered the evolutional problems in two-dimensional autonomous system. We showed that the bifurcating steady solutions are obtained from the points of intersection of the two conic sections and we used the implicit function theorem to justify their existence, and also we applied the Lyapunov theorem to establish their stability.

Project Overview

INTRODUCTION

Consider a system of differential equations

(1.1)

where  is a parameter. Suppose  for some point  then  is called an equilibrium solution. An equilibrium solution can be found by solving nonlinear algebraic equation (1.1). The equilibrium solutions which form intersecting branches in a suitable space of functions are called bifurcating solutions. For , the bifurcating solution form intersecting branches of the curve  in the  plane. For , the bifurcating solutions form connected interacting surfaces or curves in the three-dimensional  space.

As we shall see later, many stability problems are naturally formulated with respect to equilibrium solutions which form intersecting branches in a suitable space of functions.

Now, we consider evolution equations which are governed by nonlinear differential equations of the form

where  is a given nonlinear function and the unknown is  In one-dimensional problems,  is a scalar which lies in   and in two-dimensional problems,  is a two-dimensional vector with components (, and  is vector-function whose components  are nonlinear functions of the components of . The same notations are adopted for n-dimensional problems with ; in this case the vectors have n components.

Here we emphasize that we are going to confine our attention to problems which are in two dimensions.

We shall see in the next section that a physical system is said to be autonomous if its evolutional equation does not contain the independent variable (time t, say) explicitly. Hence if the evolutional equation is of second order, it is of the form

(1.3)

Here  is the velocity. By the chain rule,

(1.4)

We thus obtain a first-order evolutional equation for  as a function of variable , which now becomes the independent variable. Solutions of this new evolutional equation represent curves in the  plane. The  plane is called the phase plane.

The phase plane can give information about the general behaviour of solutions of equations without actually solving the equations. The more complicated the equations are, the more important this approach becomes.

In chapter three, we shall see that systems of equations can also be studied in the phase plane. This will lead, in a rather natural way, to stability considerations. Stability concepts are suggested by physics, where stability means, roughly speaking, that a small change (small disturbance) of a physical system at some instant changes the behaviour of the system only slightly at all future times.

We first observe that an evolution equation

can be written as a system

and a solution  of this systems represents a vector in the

For our present more general discussion, it is convenient to change our notation, replacing  Then the phase plane is the  plane. And our system is . More generally, we consider systems of the form

or

A solution of represents a curve in (plane. This curve is called a solution curve or path of (1.7)2.

From (1.7)2 we see that the slope of a path passing through a point say

(1.8)

From (1.8), we have  at If but  at P, we can take  instead of (1.8) and conclude from  that the tangent of the curve at P is vertical. However, what can we do if both  are zero at some point? This problem is a part of the main work of this project and will lead to interesting results of practical importance.

Autonomous and Non-autonomous Problems

Linear systems are classified as either time-varying or time-invariant, depending on whether the system matrix varies with time or not. In the case of general context of nonlinear problems, these adjectives are traditionally replaced by “autonomous” and “non-autonomous”. Therefore, the evolution equation (1.2) is said to be autonomous if  does not depend explicitly on time, i.e, if (1.2) can be written as (1.7)1, otherwise, it is called non-autonomous.

Strictly speaking, all physical systems are non-autonomous, because none of their dynamic characteristics is strictly time-invariant. The concept of an autonomous system is an idealized notion, like the concept of a linear system. In practice however, system properties often change very slowly, and we neglect their time variation without causing any practically meaningful error.