“Phase portrait”的意思、由来-开放百科全书

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Phase portrait

释义

Examples
Phase Portraits to Visualize Behavior of Systems of Ordinary Differential Equations
See also
References
External links

A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve, or point.

Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the state space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An attractor is a stable point which is also called 'sink'. The repellor is considered as an unstable point, which is also known as 'source'.

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

Examples

Simple pendulum, see picture (right).
Simple harmonic oscillator where the phase portrait is made up of ellipses centred at the origin, which is a fixed point.
Van der Pol oscillator see picture (bottom right).
Parameter plane (c-plane) and Mandelbrot set

Phase Portraits to Visualize Behavior of Systems of Ordinary Differential Equations

A phase portrait represents the directional behavior of a system of ODEs. The phase portrait can indicate the stability of the system. ^[1]

Stability^[1]
Unstable	Most of the system's solutions tend towards ∞ over time
Asymptotically stable	All of the system's solutions tend to 0 over time
Neutrally stable	None of the system's solutions tend towards ∞ over time, but most solutions do not tend towards 0 either

The phase portrait behavior of a system of ODEs can be determined by the eigenvalues or the trace and determinant (trace = λ₁ + λ₂, determinant = λ₁ x λ₂) of the system.^[1]

Phase Portrait Behavior^[1]
Eigenvalue, Trace, Determinant	Phase Portrait Shape
λ₁ & λ₂ are real and of opposite sign; Determinant < 0	Saddle (unstable)
λ₁ & λ₂ are real and of the same sign, and λ₁ ≠ λ₂; 0 < determinant < (trace² / 4)	Node (stable if trace < 0, unstable if trace > 0)
λ₁ & λ₂ have both a real and imaginary component; 0 < (trace² / 4) < determinant	Spiral (stable if trace < 0, unstable if trace > 0)

References

1. ^¹²³Haynes Miller, and Arthur Mattuck. 18.03 Differential Equations. Spring 2010. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. (Supplementary Notes 26 by Haynes Miller: https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/readings/supp_notes/MIT18_03S10_chapter_26.pdf)

{{cite book

}} Chapter 1.

{{cite book|author=Steven Strogatz|title=Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering|year=2001|isbn=9780738204536}}

External links

Linear Phase Portraits, an MIT Mathlet.

1 : Dynamical systems

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Examples

Phase Portraits to Visualize Behavior of Systems of Ordinary Differential Equations

See also

References

External links