Lecture
This lecture covers special types of systems, focusing on gradient systems and limit cycles. It explains the properties of gradient systems and the conditions for equilibrium points. The instructor discusses the stability of equilibrium points and the concept of limit cycles in detail, emphasizing the behavior of trajectories. The lecture also delves into the Poincaré-Bendixson theorem and the implications of closed orbits in phase space. Furthermore, it explores the phenomenon of homoclinic orbits and the criteria for the existence of closed orbits. The session concludes with a study of chaotic behavior in systems, particularly the Transcritical bifurcation and Pitchfork bifurcation.