By Ulf Grenander
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Example text
The evolution of the system in time is described by the motion of this point along an orbit of the DE in IRn , with t as time. In this interpretation, the DE implies that the vector field f is the velocity of the moving point in state space (this should not be confused with the physical velocity of a physical particle). One cannot hope to find exact solutions of a non-linear DE (1) for n ;::: 2 (except in very special cases). One thus has to use either qualitative methods, perturbative methods, or numerical methods, in order to deduce the behavior of the physical system.
Following Lienard, consider the following conditions: Ll: G is an odd function, L2: lim"..... oo G(x) = +00, L3: There exists an 0:, x> (3. and there exists (3 > 0 such that G(x) > 0, G1(x) > 0 for with 0 < 0: s:; (3, such that G( x) < 0 for 0 < X < 0:. The corollary to the Poincare-Bendixson theorem can then be used to prove the existence of a periodic orbit of the DE (89). Theorem (Lienard). If G satisfies conditions L1-L3, tben tbe DE (88) admits a periodic orbit. o Proof. [ef. Hale, pages 57-59, [4]] Comment: If 0: = (3, in conditions L2 and L3, it can further be shown that there is a unique periodic orbit, which attracts nearby orbits.
Proof. [ef. 3, page 296, [4]] 0 Comments: 1. A function V : jRn -+ jR which satisfies the above theorem for xES C lR" is called a Liapunov function on S. 2. One can often use the level sets of the function V to define the trapping set S in the theorem. 3. In applying the theorem, we note that we simply have to find whole orbits that are contained in the set {x E SIV(x) = O} to obtain the w-limit set w(a). Example. Consider the DE x~ = -O:X2 - xi, 0: > O. We note that when 0: = 0, the DE. x~ + lxi.