Continuum of limit cycles in a class of three-dimensional quadratic polynomial systems

It is well-known that planar autonomous polynomial dynamical systems have only a finite number of limit cycles. We present a class of parameter depending three-dimensional autonomous systems whose right-hand side consists of quadratic polynomials and establish the bifurcation of a continuum of limit cycles from a period annulus and from an equilibrium point. A limit cycle <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi mathvariant="normal"> Γ </a:mi> </a:math> is understood as a closed orbit without equilibrium point which is the limit set of a trajectory different from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi mathvariant="normal"> Γ </c:mi> </c:math> . The systems under consideration are distinguished by the existence of two straight lines of equilibria and the existence of a family of invariant manifolds foliating <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msup> <e:mrow class="MJX-TeXAtom-ORD"> <e:mi mathvariant="double-struck">R</e:mi> </e:mrow> <e:mn>3</e:mn> </e:msup> <e:mo class="MJX-variant"> ∖ </e:mo> <e:mo fence="false" stretchy="false">{</e:mo> <e:mn>0</e:mn> <e:mo fence="false" stretchy="false">}</e:mo> </e:math> .