Phase plane
Encyclopedia
A phase plane is a visual display of certain characteristics of certain kinds of differential equation
s; it is a 2-dimensional version of the general n-dimensional phase space
.
Phase planes are useful in visualizing the behavior of physical systems; in particular, of oscillatory systems such as predator-prey models (see Lotka–Volterra equations). These models can "spiral in" towards zero, "spiral out" towards infinity, or reach neutrally stable situations called centres where the path traced out can be either circular, elliptical, or ovoid, or some variant thereof. This is useful in determining if the dynamics are stable or not.
Other examples of oscillatory systems are certain chemical reactions with multiple steps, some of which involve equilibria rather than reactions that go to completion. In such cases one can model the rise and fall of reactant and product concentration (or mass, or amount of substance) with the correct differential equations and a good understanding of chemical kinetics.
Certain systems of differential equation
s can be written in the form:
where c may be any combination of constants in order to create linear combinations with x on the right side; here x is in bold to indicate it is actually a vector, not a scalar.
Such systems may be solved algebraically (as seen here). More commonly they are solved with the coefficients of the right hand side written in matrix form using eigenvalues and eigenvectors. The eigenvalues represent the powers of the exponential components and the eigenvectors are coefficients. If the solutions are written in algebraic form, they express the fundamental multiplicative factor of the exponential term. Due to the nonuniqueness of eigenvectors, every solution arrived at in this way has undetermined constants c1, c2, and so on, up to the number of eigenvectors.
For the special case of a two-by-two matrix representing a system of differential equations, the solutions are:
Here,
1 and 
2 are the eigenvalues, and the two matrices containing (k1, k2), (k3, k4) are the basic eigenvectors. The constants c1 and c2 account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system.
The phase plane is then first set-up by drawing straight lines representing the two eigenvectors (which represent stable situations where the system either converges towards those lines or diverges away from them). Then the phase plane is plotted by using full lines instead of direction field dashes. The signs of the eigenvalues will tell how the system's phase plane behaves:
The above can be visualized by recalling the behavior of exponential terms in differential equation solutions.
This page covers only the case for real, separate eigenvalues. Real, repeated eigenvalues require solving the coefficient matrix with an unknown vector and the first eigenvector to generate the second solution of a two-by-two system. However, if the matrix is symmetric, it is possible to use the orthogonal eigenvector to generate the second solution.
Complex eigenvalues and eigenvectors generate solutions in the form of sine
s and cosines
as well as exponentials. One of the simplicities in this situation is that only one of the eigenvalues and one of the eigenvectors is needed to generate the full solution set for the system.
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s; it is a 2-dimensional version of the general n-dimensional phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
.
Phase planes are useful in visualizing the behavior of physical systems; in particular, of oscillatory systems such as predator-prey models (see Lotka–Volterra equations). These models can "spiral in" towards zero, "spiral out" towards infinity, or reach neutrally stable situations called centres where the path traced out can be either circular, elliptical, or ovoid, or some variant thereof. This is useful in determining if the dynamics are stable or not.
Other examples of oscillatory systems are certain chemical reactions with multiple steps, some of which involve equilibria rather than reactions that go to completion. In such cases one can model the rise and fall of reactant and product concentration (or mass, or amount of substance) with the correct differential equations and a good understanding of chemical kinetics.
Certain systems of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s can be written in the form:
where c may be any combination of constants in order to create linear combinations with x on the right side; here x is in bold to indicate it is actually a vector, not a scalar.
Such systems may be solved algebraically (as seen here). More commonly they are solved with the coefficients of the right hand side written in matrix form using eigenvalues and eigenvectors. The eigenvalues represent the powers of the exponential components and the eigenvectors are coefficients. If the solutions are written in algebraic form, they express the fundamental multiplicative factor of the exponential term. Due to the nonuniqueness of eigenvectors, every solution arrived at in this way has undetermined constants c1, c2, and so on, up to the number of eigenvectors.
For the special case of a two-by-two matrix representing a system of differential equations, the solutions are:
Here,
1 and 2 are the eigenvalues, and the two matrices containing (k1, k2), (k3, k4) are the basic eigenvectors. The constants c1 and c2 account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system.The phase plane is then first set-up by drawing straight lines representing the two eigenvectors (which represent stable situations where the system either converges towards those lines or diverges away from them). Then the phase plane is plotted by using full lines instead of direction field dashes. The signs of the eigenvalues will tell how the system's phase plane behaves:
- If the signs are opposite, the intersection of the eigenvectors is a saddle pointSaddle pointIn mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
. - If the signs are both positive, the eigenvectors represent stable situations that the system diverges away from, and the intersection is an unstable nodeNode (autonomous system)The behaviour of a linear autonomous system around a critical point is a node if the following conditions are satisfied:Each path converges to the critical point as t \rightarrow \infty . Furthermore, each path approaches the point asymptotically through a line....
. - If the signs are both negative, the eigenvectors represent stable situations that the system converges towards, and the intersection is a stable node.
The above can be visualized by recalling the behavior of exponential terms in differential equation solutions.
This page covers only the case for real, separate eigenvalues. Real, repeated eigenvalues require solving the coefficient matrix with an unknown vector and the first eigenvector to generate the second solution of a two-by-two system. However, if the matrix is symmetric, it is possible to use the orthogonal eigenvector to generate the second solution.
Complex eigenvalues and eigenvectors generate solutions in the form of sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
s and cosines
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
as well as exponentials. One of the simplicities in this situation is that only one of the eigenvalues and one of the eigenvectors is needed to generate the full solution set for the system.
See also
- Phase linePhase line (mathematics)In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, dy/dt = ƒ...
, 1-dimensional case - Phase spacePhase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
, n-dimensional case - Phase portraitPhase portraitA phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is representated by a different curve, or point....
- Phase plane methodPhase plane methodIn the context of nonlinear system analysis, the phase plane method refers to graphically determining the existence of limit cycles. The phase plane, applicable for second order systems only, is a plot with axes being the values of the two state variables, x_2 vs. x_1. Vectors representing the...