phaseplot

A phase plot, also known as a phase portrait or state-space plot, is a graphical tool used in mathematics and physics to visualize the behavior of dynamical systems. It represents the evolution of a system's state over time. A dynamical system is described by a set of differential equations that dictate how its state variables change. The state of a system at any given moment is defined by a point in its state space, where each dimension of the state space corresponds to a state variable.

A phase plot plots these state variables against each other. For a system with two state variables, x and y, a phase plot would have x on one axis and y on the other. The trajectory of the system is represented by a curve on this plot, showing how the values of x and y change together over time. These trajectories are often called phase trajectories or orbits.

Phase plots are invaluable for understanding the qualitative behavior of a system without explicitly solving the differential equations. They can reveal important features such as equilibrium points (where the system's state does not change), limit cycles (closed trajectories that represent periodic oscillations), and the stability of these features. Stable equilibrium points attract nearby trajectories, while unstable ones repel them. The overall pattern of trajectories in a phase plot provides a comprehensive overview of all possible behaviors of the system under different initial conditions. They are widely used in fields like engineering, economics, biology, and fluid dynamics.