Deterministic and stochastic Duffing-van der Pol oscillators are non-explosive

AbstractThis paper is concerned with the (non-)explosion behavior of solutions of non-linear random and stochastic differential equations.We primarily investigate the Duffing-van der Pol oscillator $$\ddot x = (\alpha + \sigma _1 \xi _1 )x + \beta \dot x - x^3 - x^2 \dot x + \sigma _2 \xi _2 ,$$ where α, β are bifurcation parameters, ξ1,ξ2 are either real or white noise processes, and σ1, σ2 are intensity parameters.The notion of (strict) completeness (the rigorous mathematical formulation of “non-explosiveness”) is introduced, and its scope is explained in detail. On the basis of the Duffing-van der Pol equation techniques for proving or disproving (strict) completeness are presented. It will turn out that the forward solution of (1) is strictly complete, but the backward solution is not complete in both the real and white noise case. This is in particular true for the deterministic Duffing-van der Pol oscillator.In addition, some general results on the completeness of stochastic differential equations are given.