C.1. Linear Stability Theory

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

        dy dt =f( y )                                         (1)

where y=( y 1 ,, y n )  and f=( f 1 ,, f n )

The existence of a unique solution near a point y 0 =( y 1 0 ,, y n 0 )  is ensured if the Lipschitz conditions are satisfied, i.e., if the partial derivatives of f exist and are continuous, and if there exists M,δ>0  such that

        | f( y )f( y 0 ) |M| y y 0 |         for all          | y y 0 |<δ                (2)

where | x |  is the norm of the vector x.  We shall assume (2) is always satisfied in the following discussion.

 

The steady state solutions y ¯  with

y ¯ ˙ =0=f( y ¯ )                                                    (3)

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 y ¯  's, which remain stationary.  The trajectories are given by the equations

        d y j f j =dt=const               for all j.                   (4)

 

The stability of a steady state y ¯  is studied by the introduction of a small perturbation.  To begin, we write the solutions of (1) in the neighborhood of y ¯  as

        y= y ¯ +x                                   (5)

Taking the time derivatives of (5) gives, with the help of (3),

        y ˙ = x ˙ =f( y )   = j=1 n f y j | y ¯ x j +O( x 2 )   = ( x y )f | y ¯ +O( x 2 )              (6)

or, in component form

        x ˙ i = j=1 n f i y j | y ¯ x j +O( x 2 )                    (7)

In linear stability theory, the O( x 2 )  terms are dropped so that (6) or (7) can be written in matrix form as

        x ˙ =A  x                             (8)

where A=( a ij )  is a matrix with elements

a ij = f i y j | y ¯                 (9)

Now, the Lipschitz conditions ensure the existence of n independent eigenvectors to A so that it can be diagonalized.  The eigenvalues λ k  of A are solutions to the secular equation

        det|   AλI   |=0                       (10)

Let the right eigenvector belonging to the eigenvalue λ k  be z k , i.e.,

        A   z k = λ k    z k                              (11)

Setting x= z k  in (8) gives

        z ˙ k =A   z k = λ k    z k              for   k=1,,n                (12)

Þ            z k ( t )= z k ( 0 )exp( λ k   t )                                    (13)

Analysis

1.         All λ k  real and negative  Þ stable.

2.         All λ k  real with some positive and some negative  Þ  y ¯  is a saddle point.

3.         All λ k  imaginary  Þ    Nonlinear analysis required to determine exact behavior.  If motion is purely oscillatory, y ¯  is called a center.

4.         All λ k  complex with negative real part  Þ  stable and y ¯  is called a focus.

5.         All λ k  complex with positive real part  Þ  unstable.