On detecting periodic solutions and chaos in the time periodically forced ODEs

The geometric method for detecting periodic solutions and chaotic dynamics in ordinary differential equations was studied. The approach was based on the existence of some sets (called periodic isolating segments) in the extended phase space, satisfying some topological conditions. A basic property of the segment is that in any point on the boundary of the segment, the vector field is directed either outward or inward with respect to the segment. By chaos, it means the existence of a compact invariant set such that the Poincaré map is semiconjugate to the shift on two symbols and the counter-image of any periodic point in the shift contains a periodic point of the Poincaré map.