Vector Fields Possessing an Integral

Abstract

For a general vector field ˙ x = f (x), x∈ R n , a scalar valued function I(x) is said to be an integral (sometimes the term first integral is used) if it is constant on trajectories, i.e., ˙ I(x) = ∇I(x) · ˙ x = ∇I(x) · f (x) = 0, where " · " denotes the usual Euclidean inner product. From this relation we see that the level sets of I(x) (which are generally (n − 1)-dimensional) are invariant sets. For two-dimensional vector fields the level sets actually give the trajectories of the system. We examine this case in more detail. 5.1 Vector Fields on Two-Manifolds Having an Integral In applications, three types of two-dimensional phase spaces occur frequently ; they are (1) the plane, R 2 = R 1 × R 1 , (2) the cylinder, R 1 × S 1 , and (3) the two-torus, T 2 = S 1 × S 1. The vector field can be written as ˙ x = f (x, y), ˙ y = g(x, y), (5.1.1) where f and g are C r (r ≥ 1), and as (x, y) ∈ R 1 × R 1 for a vector field on the plane, as (x, y) ∈ R 1 × S 1 for a vector field on the cylinder, and as (x, y) ∈ S 1 × S 1 for a vector field on the torus, where S 1 denotes the circle (which is sometimes referred to as a 1-torus, T 1). We now want to give some examples of how these different phase spaces arise and at the same time go into more detail concering the idea of an integrable vector field. We begin with the unforced Duffing oscillator. Example 5.1.1 (The Unforced Duffing Oscillator). We have been slowly discovering the global structure of the phase space of the unforced Duffing oscillator given by¨x by¨ by¨x − x + δ ˙ x + x 3 = 0, (5.1.2)