Generalized Lorenz-type systems (Q1916805)

The author studies generalization of the Lorenz system of the form \(x'=a(y-x)\), \(y'=f(x)- ny-xz\), \(z'=xy-bz\). He first shows that for the classical choice \(f(x)=rx\) and some special choices of parameters and initial values the exact solutions of the Lorenz system may be obtained in terms of Jacobian elliptic functions. Then he shows that for a few choices of \(f\), in particular \(f(x)= rx+x^3/(2a)\) it is possible to obtain exact formulas for all solutions of the considered system in terms of Bessel functions or modified Bessel functions. Moreover, for some parameter values these solutions exhibit chaotic behavior.