One-Directional Grid Scroll Hidden Attractor on the Plane

The Poincaré-Bendixson theorem states that a continuous planar dynamical system, which contains only finitely many fixed points, can only generate an orbit that converge to a fixed point or a periodic orbit. Therefore, there are constraints in the plane to generate chaotic behavior. In this work, we introduce a planar system to display a one-directional (1D) grid scroll hidden attractor on the plane. This continuous-time planar system uses the hysteresis phenomenon to overcome the in-plane constraint and thus get stretch-and-fold behavior to get chaotic behavior. The mechanism of generation of chaotic behavior in the plane is based on unstable focus equilibrium points and hysteresis.