The Van der Pol Equation An important kind of second-order non-linear autonomous equation has the form x �� + u( x ) x � + v( x ) = 0 (Liénard equation) . (1) One might think of this as a model for a spring-mass system where the damping force u( x ) depends on position (for example, the mass might be moving through a viscous medium of varying density), and the spring con stant v( x ) depends on how much the spring is stretched—this last is true of all springs, to some extent. We also allow for the possibility that u( x ) < 0 (i.e., that there is "negative damping"). The system equivalent to (1) is x� = y y� = −v( x ) − u( x ) y (2) Under certain conditions, the system (2) has a unique stable limit cycle, or what is the same thing, the equation (1) has a unique periodic solution; and all nearby solutions tend towards this periodic solution as t → ∞. The conditions which guarantee this were given by Liénard, and generalized in the following theorem. Levinson-Smith Theorem Suppose the following conditions are satisﬁed. (a) u( x ) is even and continuous, (b) v( x ) is odd, v( x ) > 0 i f x > 0, and v( x ) is continuous for all x, �x (c) V ( x ) → ∞ as x → ∞, where V ( x ) = 0 v(t) dt , (d) for some k > 0, we have U ( x ) < 0, U ( x ) > 0 and increasing, U ( x ) → ∞, ⎫ for 0 < x < k, ⎬ for x > k, ⎭ as x → ∞, where U ( x ) = � x Then, the system (2) has i) a unique critical point at the origin; ii) a unique non-zero closed trajectory C, which is a stable limit cycle around the origin; iii) all other non-zero trajectories spiralling towards C as t → ∞ . 0 u(t) dt. 18.03SC We omit the proof, as too difﬁcult. A classic application is to the equa tion x �� − a(1 − x2 ) x � + x = 0 (van der Pol equation) (3) which describes the current x (t) in a certain type of vacuum tube. (The constant a is a positive parameter depending on the tube constants.) The equation has a unique non-zero periodic solution. Intuitively, think of it as modeling a non-linear spring-mass system. When | x | is large, the restoring and damping forces are large, so that | x | should decrease with time. But when | x | gets small, the damping becomes negative, which should make | x | tend to increase with time. Thus it is plausible that the solutions should oscillate; that it has exactly one periodic solution is a more subtle fact. There is a lot of interest in limit cycles, because of their appearance in systems which model processes exhibiting periodicity. But not a great deal is known about them – this is still an area of active research.