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A singular point, also known as an equilibrium point or critical point, is a point in the phase plane where the system's state does not change; the derivatives of the state variables are zero. In control systems, these points represent system's steady-state behaviors.

A vector field is a representation in the phase plane of the instantaneous rate of change of the state variables. In control system context, it visualizes the direction and speed of the state trajectory at any given point in the phase plane.

A Lyapunov function is a scalar function used to prove the stability of an equilibrium point in the phase plane. In control systems, it indicates the presence of a function whose value decreases along trajectories for a stable system and helps in analyzing stability.

A saddle point is a type of singular point in the phase plane where trajectories approach the point along one axis (stable manifold), but move away along another axis (unstable manifold). In control systems, it symbolizes an unstable behavior where any disturbance can cause the system to leave this state.

A homoclinic orbit is a trajectory that starts and ends at the same saddle point. In control systems, such an orbit indicates complex dynamic behavior, where the system's phase path leaves a saddle point and returns to it over time.

A node is an equilibrium point in the phase plane where trajectories are either all drawn into it (stable node) or repelled away from it (unstable node). In control systems, it represents a state where the system exhibits monotonic behavior towards or away from an equilibrium state.

A focus is an equilibrium point where nearby trajectories spiral towards or away from it. In a control system context, it represents a state that exhibits oscillatory behavior before stabilizing at (stable focus) or moving away from (unstable focus) the equilibrium point.

A separatrix is a trajectory in the phase plane that marks the boundary between different types of motion. In a control system, it separates different behaviors of the system, such as converging to a stable point or diverging to infinity.

A stable equilibrium point is a singular point where trajectories approach as time goes to infinity. In control systems, this indicates a system that tends to return to this point after a disturbance.

A center point is a type of equilibrium point in the phase plane analysis where neighboring trajectories circulate around it but neither converge nor diverge. In control systems, this suggests a system that oscillates indefinitely with a constant amplitude around an equilibrium state.

Bifurcation in phase plane analysis refers to a qualitative change in the structure of the system's trajectories as a parameter is varied. In the context of control systems, it describes how the system's behavior can drastically change due to small changes in system parameters.

A trajectory in phase plane analysis is a path that a state vector takes over time starting from an initial condition. In control systems, it represents the time evolution of the system's states.

An unstable equilibrium point is a singular point where nearby trajectories diverge away as time progresses. In control systems, this means that the system will move away from this point after a slight disturbance.

A limit cycle is an isolated closed trajectory in the phase plane, towards which neighboring trajectories converge (stable limit cycle) or away from which they diverge (unstable limit cycle). In control system terms, this behavior represents periodic responses of self-sustained oscillations.