The Phase Plane 2025

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The document discusses the concept of phase planes and phase portraits in the context of linear systems of differential equations. It explains how to visualize solutions through parametric curves, classify equilibrium solutions based on their stability and type, and provides detailed classifications for various cases such as distinct real eigenvalues, repeated eigenvalues, and complex conjugate eigenvalues. The document also touches on nonhomogeneous linear systems and nonlinear systems, emphasizing the importance of Jacobian matrices for determining critical points and their stability.

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What is the phase plane?

In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc.

What is a phase portrait?

A phase portrait graph of a dynamical system depicts the systems trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a phase space. The axes are of state variables.

What is the difference between phase portrait and trajectory?

The path travelled by the point in a solution is called a trajectory of the system. A picture of the trajectories is called a phase portrait of the system.

How to determine direction of phase plane?

4:18 14:13 So its going this. Way. All right now for uh for the point 01. All right well x is zero. So theMoreSo its going this. Way. All right now for uh for the point 01. All right well x is zero. So the change in x is zero y is one so change in y is -2. Change y over the change in x is -2 over 0.

What is the difference between phase space and phase plane?

In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane.

What is the difference between phase space and phase portrait?

Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value.

What is the difference between phase plane and phase portrait?

The graphic of a trajectory drawn as a parametric curve in the xy-plane is called a phase portrait and the xy-plane in which it is drawn is called the phase plane.

The Phase Plane 2025

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