Limit Cycle Equation Van Der Pol and Duffing Equation

Abstract

Van der Pol Equation and Duffing Equation are the second order differential equation which are read as x '' µ ( x 2  1 ) x ' + x = 0 and x ' '  kx' - x + x³ = F cos( w t ) . The aim of this paper is to study about the limit cycle of both equations. The discussion will be focused on the existence and the stability of the limit cycle and also the difference between the limit cycle of van der Pol equation and the Duffing equation. The results are limit cycle exist in van der Pol equation and Duffing equation. The form of limit cycle of the van der Pol equatuon is depended to the µ values and the the form of limit cycle of the Duffing equation is depended to the k, F, and w values. The difference of both equations is that van der Pol equation is globally stable while the Duffing equation is locally stable. Duffing equation may be having more than one limit cycle while van der Pol equation can only has single limit cycle.