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In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space.
The signs of the eigenvalues indicate the phase plane's behaviour: If the signs are opposite, the intersection of the eigenvectors is a saddle point . If the signs are both positive, the eigenvectors represent stable situations that the system diverges away from, and the intersection is an unstable node .
Phase portrait of a two-well Duffing oscillator (a differential equation, rather than a map) showing chaotic behavior. The Duffing map (also called as 'Holmes map') is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior.
Some typical examples of the time series and phase portraits of the Duffing equation, showing the appearance of subharmonics through period-doubling bifurcation – as well chaotic behavior – are shown in the figures below.
Phase portrait showing saddle-node bifurcation. Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
The phase portrait of the pendulum equation x ″ + sin x = 0.The highlighted curve shows the heteroclinic orbit from (x, x′) = (–π, 0) to (x, x′) = (π, 0).This orbit corresponds with the (rigid) pendulum starting upright, making one revolution through its lowest position, and ending upright again.
The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor (possibly negative). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. Its eigenvectors are those ...