Integrability and zero-Hopf bifurcation in the Sprott A system
Consider the polynomial system \[ \frac{dx}{dt}= y, \quad \frac{dy}{dt}=-x-yz, \quad \frac{dz}{dt}=y^2-a ,\tag{1} \] where \( a \) is a real parameter. It is known that system (1) has no equilibrium for \( a \neq 0 \), but it exhibits chaotic behavior for \( a=1 \). The authors contribute to the understanding of the behavior of system (1) by proving (i). (1) has no Darboux first integral for \( a \neq 0 \). (ii). In case \( a=0 \), the unique Darboux first integrals are functions of \( x^2+y^2+z^2 \). (iii). Hopf bifurcation occurs from the non-isolated equilibrium \( (0,0,0) \) for increasing \( a \) and crossing \( a=1 \).
scientific article; zbMATH DE number 7229227
scientific article