1.1 Qualitative analysis

Let us consider the systems of ordinary differential equations (ODEs) having the form

 

x' = f(t,x) (1.1)

where $x=(x_{1}(t), \ldots , x_{n} (t))^{T}$ and $f(t,x) = (f_{1} (t, x_{1}, \ldots , x_{n}), \ldots , f_{n} (t, x_{1}, \ldots , x_{n}))^{T}$. In general, there are no known methods of solving system (1.1), even when f(t,x) is linear, i.e. f(t,x) = A(t) x + b(t), and (1.1) reduces to a linear system of the form

 

x' = A(t) x + b(t) (1.2)

where A(t) = (a_ij(t)), $1 \leq i$, $j \leq n$, is an $n \times n$ matrix whose entries are functions of t and $b(t) = (b_{1}(t), \ldots , b_{n}(t))^{T}$. This, of course, is very disappointing. However, it is not necessary, in most applications to find the solutions of (1.1) explicitly. For example, let x₁(t) and x₂(t) denote the populations, at time t, of two species competing amongst themselves for the limited food and living space in their microcosm. Suppose, moreover, that the rates of growth of x₁(t) and x₂(t) are governed by system (1.1) with n=2. In this case, we are not really interested in the values of x₁(t) and x₂(t) at every time t. Rather, we are interested in the qualitative properties of x₁(t) and x₂(t). Specifically, we wish to answer the following questions: Do there exist values $\eta_{1}$ and $\eta_{2}$ for the populations of each species at which both species co-exist together in a steady state? In other words are there numbers $\eta_{1}, \, \eta_{2}$ such that $x_{1}(t) = \eta_{1}$, $x_{2}(t) = \eta_{2}$ is a solution of (1.1)? Such values $\eta_{1}, \, \eta_{2}$, if they exist, are called equilibrium values of (1.1). Suppose that the two species are co-existing in equilibrium and we suddenly add a few members of species 1 to the microcosm. Will x₁(t) and x₂(t) remain close to their equilibrium values for all future times? Or perhaps the extra few members give species 1 an added advantage and cause the extinction of species 2. Suppose that x₁(t) and x₂(t) have arbitrary values at t=0. What happens as t approaches infinity? Will one species ultimately emerge victorious or will the struggle for existence end in a draw? Other examples of a similar nature can be found in physical sciences, engineering, etc. More generally, we are interested in determining the following properties of solutions of (1.1):

1.

Do there exist equilibrium values for which $x(t) \equiv \ol{x}$ is a solution of (1.1)?

2.

Let $\phi (t)$ be a solution of (1.1). Suppose that $\psi (t)$ is a second solution with $\psi (t_{0})$ very close to $\phi (t_{0})$; i.e. $\psi_{j} (t_{0})$ is very close to $\phi_{j} (t_{0})$, $j = 1,2, \ldots , n$. Will $\psi (t)$ remain close to $\phi (t)$ as t approaches infinity? This question is often referred to as the problem of stability. It is the most fundamental problem in the qualitative theory of differential equations and has occupied the attention of many mathematicians for the past hundred years.

3.

What happens to solution x(t) of (1.1) as t approaches infinity? Do all solutions approach equilibrium values? If they don't approach equilibrium values, do they at least approach a periodic solution?