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This paper addresses dynamical behaviors of switching van der Pol circuits by investigating stability issues of autonomous nonlinear dynamical systems. Firstly, a kind of nonlinear dynamical systems, i.e., a piecewise/ switching autonomous nonlinear system, is used to formulate a class of van der Pol circuits that are composed of at least two alternative electrical sub-circuits. Secondly, based on the concept of system stability in the sense of Lyapunov and continuously positive definite functions, this paper proposes two new necessary and sufficient conditions of stability/ asymptotic stability for autonomous nonlinear systems. Thirdly, based on the new stability results obtained in this paper, we present several criteria for globally asymptotical stability and local instability of the unique zero equilibrium of such kind of electrical circuits. Meanwhile, the existence and uniqueness of a stable limit cycle are also given for the switching van der Pol circuit. These novel results obtained in this paper show that switching actions involved in van der Pol circuits may vanish the classical nonlinear dynamical phenomena of relaxation oscillations. Finally, numerical simulations of several nonlinear systems and a switching van der Pol circuit illustrate the effectiveness and practicality of our new results.