Abstract. We discuss the phenomenon of cycling for noise induced escape to a unstable periodic orbit. The presence of cycling is shown to follow from qualitative properties of two quasipotential functions. A method of numerically evaluating these quasipotential functions is described, and applied to the Van der Pol oscillator as an example. Figures resulting from these calculations reveal that nonconvergent cycling of exit measures does occur for the Van der Pol example.
x1: IntroductionThe Van der Pol equation We are going to consider < 0 (which is equivalent t o r e v ersing time for > 0). This makes (1.2) a system with a stable critical point at the origin, surrounded by an unstable limit cycle. The e ects of adding an asymptotically small random perturbation to such a system have been discussed in recent w ork, such as 2] and other references cited there. Speci cally we compare the solution x( ) o f ( 1 . 2 ) , with initial condition x(0) = x 0 , to the solution x (t) of the following Itô equation:The parameter > 0 is viewed as the strength of the random perturbation. Our interest is in asymptotic behavior as # 0. It can be shown in several ways that x (t) ! x(t) uniformly (in probability) over t in any xed time interval 0 T ]. The story is di erent however when we l o o k at the whole time axis, t 2 0 1).Let D be the region enclosed by the limit cycle of (1.2), with @ Dbeing the limit cycle itself. For any x 0 2 D the solution x(t) of (1.2) remains forever in D and converges to 0 as t ! 1 . In contrast to this, x (t) w i l l e v entually (with probability 1 ) w ander out to @ D , r e a c hing it rst at some random time < 1.A particularly interesting phenomena occurs in the study of the distribution of this point x ( ) o f r s t \noise induced" escape to the limit cycle @ D :(1.4) (A) = P x ( ) 2 A] A @ D :1991 Mathematics Subject Classi cation. 60H30, 60H10, 60J70, 60J60.