By replacing each derivative of order higher than the 1st with a new variable, we can rewrite any set of nonlinear differential equations of arbitrary order by a larger set of 1st order nonlinear differential equations. Therefore, the study of stability can be restricted to the latter kind of equations without loss of generality. Treating the derivative as variables and all else as constants, we can solve for the derivatives by standard algebraic means. A unique solution is expected if the original differential equations is to admit a unique solution. Thus, we can further restrict our attention to equations of the form
where and .
The existence of a unique solution near a point is ensured if the Lipschitz conditions are satisfied, i.e., if the partial derivatives of f exist and are continuous, and if there exists such that
for all (2)
where is the norm of the vector x. We shall assume (2) is always satisfied in the following discussion.
The steady state solutions with
are often called fixed, singular, or critical points. If we plot the solutions of (1) in the n-dimensional phase plane using t as parameter, they will trace out trajectories except for the 's, which remain stationary. The trajectories are given by the equations
for all j. (4)
The stability of a steady state is studied by the introduction of a small perturbation. To begin, we write the solutions of (1) in the neighborhood of as
Taking the time derivatives of (5) gives, with the help of (3),
or, in component form
In linear stability theory, the terms are dropped so that (6) or (7) can be written in matrix form as
where is a matrix with elements
Now, the Lipschitz conditions ensure the existence of n independent eigenvectors to A so that it can be diagonalized. The eigenvalues of A are solutions to the secular equation
Let the right eigenvector belonging to the eigenvalue be , i.e.,
Setting in (8) gives
1. All real and negative Þ stable.
2. All real with some positive and some negative Þ is a saddle point.
3. All imaginary Þ Nonlinear analysis required to determine exact behavior. If motion is purely oscillatory, is called a center.
4. All complex with negative real part Þ stable and is called a focus.
5. All complex with positive real part Þ unstable.