| The integrability of dynamical systems has been an active subject and it at-tracts lots of mathematician and physicists. For higher dimensional differential systems their dynamics are difficult to study. If the systems have first integrals, then it can be reduced to lower dimensional differential systems on the energy sur-faces. So it is possible to study the global dynamics of the systems. In practice, some realistic systems, especially the mechanical systems, usually do have first integrals. How to search them is one of the interesting problems.In this paper we mainly consider the integrability of a class of circuit system-s. Llibre and Valls [9] studied the Muthuswamy-Chua systems [21], which is an autonomous system driven by a circuit consisting of a linear circuit inductor, a capacitor and a nonlinear memory linear resistor. This system can present chaos attractors for suitable choice of parameters. For studying its dynamics further with other parameters, Llibre and Valls investigated the existence of rational first inte-grals, polynomial first integral, Darboux polynomials and Darboux first integrals. But Llibre and Valls’s result is not complete. By using the characteristic curve for solving linear differential equations and quasihomogeneous polynomials we obtain Darboux polynomials without nonzero cofactors.Muthuswamy and Chua [21] suggested a more general systems, the second part of this paper choose a class of differential systems based on Muthuswamy-Chua’s suggestions. The systems are the following: We study the existence of rational first integrals, polynomial first integrals, Dar-boux polynomials and Darboux first integrals of these last systems. The main results are the following: .If β-γ=0,c≠0,a+b-1/2≠0,the system has a Darboux polynomials of the form f=J0(x)(z-1)r,the correspnding cofactors are k=-ry where r are arbitrary positive integers;.If β＝0，γ≠0,the system has only polynomial first integrals of the form F=F(x). and so on. |
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