Dynamics of the double scroll circuit

Consideration is given to a system of differential equations which has two homoclinic solutions to a same saddle-focus point. The authors prove the existence of a subsystem of solutions which remain in a neighborhood of those homoclinic orbits. In addition to the upper and lower subsystem already described by L. Slinikov´s (1965) theorem, this theorem contains solutions that switch in every turn from the neighborhood of one homoclinic to the neighborhood of the other