Qualitative Analysis Of An Three Dimensional Autonomous System

In this paper,we investigate Darboux integrability and dynamics of a three dimensional nonlinear system with three parameters where(s,b,r)?(1,1,1).If(s,b,r)=(1,1,1),then the system is a chaotic system which was proposed by J.C.Sprott through numerical simulation in 1994 18].According to different values of parameters,we divide the parameter s into two classes s? 0 and s=0.When s is not 0,i.e.s?0,by the method of characteristic curves and quasi-homogeneous theory,we obtain that the system doesn t have nontrivial Darboux polynomials.Moreover,the system has trivial Darboux polynomials,i.e.polynomial first integrals,if and only if one of the following statements holds.(1)s?0,b?0,r=0.Then the polynomial first integral is H(x,y,z)=x2+z2;(2)s?0,b=0,r=0.Then H1(x,y,z)=x2+z2 and H2(X,y,z)=y2+2sz are two polynomial first integrals.When s equals to 0,i.e.s= 0,the system has the other polynomial first integrals if one of the following statements holds.(1)s=0,b=0,r=0.Then H1(x,y,z)=x2+z2 and H2(x,y,z)=y are two polynomial first integrals.(2)s=0,b=0,r?0.Then H(x,y,z)=y is polynomial first integral.(3)s=0,b?0,r=0.Then H1(x,y,z)=x2+z2 and H2(x,y,z,t)=ln y+bt are two polynomial first integrals.(4)s=0,b?0,r?0.Then H(x,y,z,t)=Iny+bt is polynomial first integral.Under the above conditions,we systematically study the dynamics and topo-logical phase portraits of the system,which is two dimensional systems on the equipotential surfaces of the first integrals.