Piecewise smooth (non-smooth) systems refer to mathematical models or dynamical systems that exhibit different behaviours or dynamics in different regions or intervals of their state space. These systems are characterized by having smooth dynamics within each region but experiencing sudden jumps at the boundaries between these regions, known as switching surfaces or discontinuity surfaces. The system either switches between different regions or slides along the switching surface. Sliding occurs when the system's trajectory moves along the boundary between regions for some time before entering a new region. These systems are found in many different areas, like the refraction of light, stick-slip vibrations induced by friction, impacts of rigid bodies, and switches in electrical circuits, as well as applications in economics, medicine, biology, and ecology. 

• Discontinuous of Filippov-type: In certain systems, the transition between different dynamical regions occurs continuously, implying that the state variables evolve smoothly across the switching surface. This could involve a gradual change in parameters or dynamics as the system moves from one region to another. Examples are systems with dry friction and visco-elastic supports.

Such systems can be analyzed using Filippov's Convex Method, which constructs a convex combination of the vector fields on either side of the manifold to define the system's behaviour on the manifold.

• Impulsive: This phenomenon arises when the system's behaviour changes suddenly upon crossing a switching boundary. The system's dynamics can instantaneously alter at these boundaries, resulting in a discontinuity in the state variables.

Examples of discontinuous switching include impact systems, where the velocity or other state variables experience an instantaneous jump when a collision occurs, or relay control systems, which switch between different control modes based on threshold conditions.

• Hybrid System: These systems are composed of continuous dynamics governed by differential equations with discrete dynamics governed by rules or events. Hybrid systems are used to model various real-world phenomena, such as in robotics, control systems, biological systems, and communication networks.

Stick-slip oscillations are observed in various systems, characterized by intermittent motion involving alternating periods of sticking and slipping. These oscillations typically occur when there is relative motion between two surfaces in contact. The term "stick-slip" describes the pattern of motion: the object sticks due to static friction until the applied force overcomes it, causing a sudden slip. Once slipping starts, the friction decreases, allowing the object to accelerate until it is slowed down again by static friction, restarting the cycle. Stick-slip oscillations are also called "self-excited vibrations" because the motion sustains itself due to the interaction of friction forces within the system without needing continuous external input.

Stick-slip oscillations are common in various real-world scenarios, ranging from everyday occurrences such as chalk squeaking on a blackboard, the noise of a creaking door, and the sound produced by violin string vibration to critical applications in mechanical systems like disc brake squeal, drill-string motions, and geological processes like earthquakes. They are also the subject of study in materials science and tribology (the study of friction, lubrication, and wear). Understanding stick-slip behaviour is crucial in designing systems to minimize friction-related issues and optimize performance.